Rational groups

A group G is rational if all of its characters are rational valued. (Equivalently every g and g^k are conjugate for k coprime to |G|.) It is an unsolved problem to determine what cyclic groups of order p can be composition factors of rational groups.

Rational groups are sometimes also known as Q-groups, though some authors reserve that term for a stronger property that every representation of G can be realized over ℚ. (According to this terminology, Q₈ is rational but not a Q-group.)

Groups of order 1

		d	ρ	Label	ID
C₁	Trivial group	1	1+	C1	1,1

Groups of order 2

		d	ρ	Label	ID
C₂	Cyclic group	2	1+	C2	2,1

Groups of order 4

		d	ρ	Label	ID
C₂²	Klein 4-group V₄ = elementary abelian group of type [2,2]; = rectangle symmetries	4		C2^2	4,2

Groups of order 6

		d	ρ	Label	ID
S₃	Symmetric group on 3 letters; = D₃ = GL₂(𝔽₂) = triangle symmetries = 1^st non-abelian group	3	2+	S3	6,1

Groups of order 8

		d	ρ	Label	ID
D₄	Dihedral group; = He₂ = AΣL₁(𝔽₄) = 2₊¹⁺² = square symmetries	4	2+	D4	8,3
Q₈	Quaternion group; = C₄ .C₂ = Dic₂ = 2_-¹⁺²	8	2-	Q8	8,4
C₂³	Elementary abelian group of type [2,2,2]	8		C2^3	8,5

Groups of order 12

		d	ρ	Label	ID
D₆	Dihedral group; = C₂×S₃ = hexagon symmetries	6	2+	D6	12,4

Groups of order 16

		d	Label	ID
C₂⁴	Elementary abelian group of type [2,2,2,2]	16	C2^4	16,14
C₂×D₄	Direct product of C₂ and D₄	8	C2xD4	16,11
C₂×Q₈	Direct product of C₂ and Q₈	16	C2xQ8	16,12

Groups of order 18

		d	ρ	Label	ID
C₃⋊S₃	The semidirect product of C₃ and S₃ acting via S₃/C₃=C₂	9		C3:S3	18,4

Groups of order 24

		d	ρ	Label	ID
S₄	Symmetric group on 4 letters; = PGL₂(𝔽₃) = Aut(Q₈) = Hol(C₂²) = tetrahedron symmetries = cube/octahedron rotations	4	3+	S4	24,12
C₂²×S₃	Direct product of C₂² and S₃	12		C2^2xS3	24,14

Groups of order 32

		d	ρ	Label	ID
2₊¹⁺⁴	Extraspecial group; = D₄ ○D₄	8	4+	ES+(2,2)	32,49
2_-¹⁺⁴	Gamma matrices = Extraspecial group; = D₄ ○Q₈	16	4-	ES-(2,2)	32,50
C₂²≀C₂	Wreath product of C₂² by C₂	8		C2^2wrC2	32,27
C₈⋊C₂²	The semidirect product of C₈ and C₂² acting faithfully; = Aut(D₈) = Hol(C₈)	8	4+	C8:C2^2	32,43
C₄⋊₁D₄	The semidirect product of C₄ and D₄ acting via D₄/C₄=C₂	16		C4:1D4	32,34
C₄⋊Q₈	The semidirect product of C₄ and Q₈ acting via Q₈/C₄=C₂	32		C4:Q8	32,35
C₈.C₂²	The non-split extension by C₈ of C₂² acting faithfully	16	4-	C8.C2^2	32,44
C₂⁵	Elementary abelian group of type [2,2,2,2,2]	32		C2^5	32,51
C₂²×D₄	Direct product of C₂² and D₄	16		C2^2xD4	32,46
C₂²×Q₈	Direct product of C₂² and Q₈	32		C2^2xQ8	32,47

Groups of order 36

		d	ρ	Label	ID
S₃²	Direct product of S₃ and S₃; = Spin+₄(𝔽₂) = Hol(S₃)	6	4+	S3^2	36,10
C₂×C₃⋊S₃	Direct product of C₂ and C₃⋊S₃	18		C2xC3:S3	36,13

Groups of order 48

		d	ρ	Label	ID
C₂×S₄	Direct product of C₂ and S₄; = O₃(𝔽₃) = cube/octahedron symmetries	6	3+	C2xS4	48,48
S₃×D₄	Direct product of S₃ and D₄; = Aut(D₁₂) = Hol(C₁₂)	12	4+	S3xD4	48,38
S₃×Q₈	Direct product of S₃ and Q₈	24	4-	S3xQ8	48,40
S₃×C₂³	Direct product of C₂³ and S₃	24		S3xC2^3	48,51

Groups of order 54

		d	ρ	Label	ID
C₃³⋊C₂	3^rd semidirect product of C₃³ and C₂ acting faithfully	27		C3^3:C2	54,14

Groups of order 64

		d	ρ	Label	ID
C₂≀C₂²	Wreath product of C₂ by C₂²; = Hol(C₂×C₄)	8	4+	C2wrC2^2	64,138
D₄⋊₄D₄	3^rd semidirect product of D₄ and D₄ acting via D₄/C₂²=C₂; = Hol(D₄)	8	4+	D4:4D4	64,134
C₂³⋊₃D₄	2^nd semidirect product of C₂³ and D₄ acting via D₄/C₂=C₂²	16		C2^3:3D4	64,215
C₂³⋊₂Q₈	2^nd semidirect product of C₂³ and Q₈ acting via Q₈/C₂=C₂²	16		C2^3:2Q8	64,224
C₂⁴⋊C₂²	4^th semidirect product of C₂⁴ and C₂² acting faithfully	16		C2^4:C2^2	64,242
C₈⋊₃D₄	3^rd semidirect product of C₈ and D₄ acting via D₄/C₂=C₂²	32		C8:3D4	64,177
C₈⋊Q₈	The semidirect product of C₈ and Q₈ acting via Q₈/C₂=C₂²	64		C8:Q8	64,182
D₄.₁₀D₄	5^th non-split extension by D₄ of D₄ acting via D₄/C₂²=C₂	16	4-	D4.10D4	64,137
C₂².₂₉C₂⁴	15^th central stem extension by C₂² of C₂⁴	16		C2^2.29C2^4	64,216
C₂².₅₄C₂⁴	40^th central stem extension by C₂² of C₂⁴	16		C2^2.54C2^4	64,241
C₈.₂D₄	2^nd non-split extension by C₈ of D₄ acting via D₄/C₂=C₂²	32		C8.2D4	64,178
C₂³.₃₈C₂³	11^st non-split extension by C₂³ of C₂³ acting via C₂³/C₂²=C₂	32		C2^3.38C2^3	64,217
C₂².₃₁C₂⁴	17^th central stem extension by C₂² of C₂⁴	32		C2^2.31C2^4	64,218
C₂³.₄₁C₂³	14^th non-split extension by C₂³ of C₂³ acting via C₂³/C₂²=C₂	32		C2^3.41C2^3	64,225
C₂².₅₆C₂⁴	42^nd central stem extension by C₂² of C₂⁴	32		C2^2.56C2^4	64,243
C₂².₅₇C₂⁴	43^rd central stem extension by C₂² of C₂⁴	32		C2^2.57C2^4	64,244
C₂².₅₈C₂⁴	44^th central stem extension by C₂² of C₂⁴	64		C2^2.58C2^4	64,245
C₂⁶	Elementary abelian group of type [2,2,2,2,2,2]	64		C2^6	64,267
D₄²	Direct product of D₄ and D₄	16		D4^2	64,226
C₂×2₊¹⁺⁴	Direct product of C₂ and 2₊¹⁺⁴	16		C2xES+(2,2)	64,264
D₄×Q₈	Direct product of D₄ and Q₈	32		D4xQ8	64,230
D₄×C₂³	Direct product of C₂³ and D₄	32		D4xC2^3	64,261
C₂×2_-¹⁺⁴	Direct product of C₂ and 2_-¹⁺⁴	32		C2xES-(2,2)	64,265
Q₈²	Direct product of Q₈ and Q₈	64		Q8^2	64,239
Q₈×C₂³	Direct product of C₂³ and Q₈	64		Q8xC2^3	64,262
C₂×C₈⋊C₂²	Direct product of C₂ and C₈⋊C₂²	16		C2xC8:C2^2	64,254
C₂×C₂²≀C₂	Direct product of C₂ and C₂²≀C₂	16		C2xC2^2wrC2	64,202
C₂×C₄⋊₁D₄	Direct product of C₂ and C₄⋊₁D₄	32		C2xC4:1D4	64,211
C₂×C₈.C₂²	Direct product of C₂ and C₈.C₂²	32		C2xC8.C2^2	64,255
C₂×C₄⋊Q₈	Direct product of C₂ and C₄⋊Q₈	64		C2xC4:Q8	64,212

Groups of order 72

		d	ρ	Label	ID
PSU₃(𝔽₂)	Projective special unitary group on 𝔽₂³; = C₃² ⋊Q₈ = M₉	9	8+	PSU(3,2)	72,41
S₃≀C₂	Wreath product of S₃ by C₂; = SO+₄(𝔽₂)	6	4+	S3wrC2	72,40
C₃⋊S₄	The semidirect product of C₃ and S₄ acting via S₄/A₄=C₂	12	6+	C3:S4	72,43
C₂×S₃²	Direct product of C₂, S₃ and S₃	12	4+	C2xS3^2	72,46
C₂²×C₃⋊S₃	Direct product of C₂² and C₃⋊S₃	36		C2^2xC3:S3	72,49

Groups of order 96

		d	ρ	Label	ID
C₂²⋊S₄	The semidirect product of C₂² and S₄ acting via S₄/C₂²=S₃	8	6+	C2^2:S4	96,227
C₂²×S₄	Direct product of C₂² and S₄	12		C2^2xS4	96,226
S₃×C₂⁴	Direct product of C₂⁴ and S₃	48		S3xC2^4	96,230
C₂×S₃×D₄	Direct product of C₂, S₃ and D₄	24		C2xS3xD4	96,209
C₂×S₃×Q₈	Direct product of C₂, S₃ and Q₈	48		C2xS3xQ8	96,212

Groups of order 108

		d	ρ	Label	ID
C₃²⋊D₆	The semidirect product of C₃² and D₆ acting faithfully	9	6+	C3^2:D6	108,17
S₃×C₃⋊S₃	Direct product of S₃ and C₃⋊S₃	18		S3xC3:S3	108,39
C₂×C₃³⋊C₂	Direct product of C₂ and C₃³⋊C₂	54		C2xC3^3:C2	108,44

Groups of order 120

		d	ρ	Label	ID
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	5	4+	S5	120,34

Groups of order 128

		d	ρ	Label	ID
2₊¹⁺⁶	Extraspecial group; = D₄ ○2₊¹⁺⁴	16	8+	ES+(2,3)	128,2326
2_-¹⁺⁶	Extraspecial group; = D₄ ○2_-¹⁺⁴	32	8-	ES-(2,3)	128,2327
D₄≀C₂	Wreath product of D₄ by C₂	8	4+	D4wrC2	128,928
Q₈≀C₂	Wreath product of Q₈ by C₂	16	4-	Q8wrC2	128,937
C₂³≀C₂	Wreath product of C₂³ by C₂	16		C2^3wrC2	128,1578
D₈⋊₁₁D₄	5^th semidirect product of D₈ and D₄ acting via D₄/C₂²=C₂	16	8+	D8:11D4	128,2020
C₄²⋊₅D₄	5^th semidirect product of C₄² and D₄ acting faithfully	16	8+	C4^2:5D4	128,931
C₄²⋊₆D₄	6^th semidirect product of C₄² and D₄ acting faithfully	16	8+	C4^2:6D4	128,932
D₈⋊C₂³	6^th semidirect product of D₈ and C₂³ acting via C₂³/C₂²=C₂	16	8+	D8:C2^3	128,2317
C₂⁴⋊C₂³	2^nd semidirect product of C₂⁴ and C₂³ acting faithfully	16	8+	C2^4:C2^3	128,1758
C₄²⋊C₂³	4^th semidirect product of C₄² and C₂³ acting faithfully	16		C4^2:C2^3	128,2264
M₄(2)⋊C₂³	3^rd semidirect product of M₄(2) and C₂³ acting via C₂³/C₂=C₂²	16	8+	M4(2):C2^3	128,1751
C₂⁴⋊₇D₄	2^nd semidirect product of C₂⁴ and D₄ acting via D₄/C₂=C₂²	32		C2^4:7D4	128,1135
C₂⁴⋊₉D₄	4^th semidirect product of C₂⁴ and D₄ acting via D₄/C₂=C₂²	32		C2^4:9D4	128,1345
C₂⁴⋊₆Q₈	5^th semidirect product of C₂⁴ and Q₈ acting via Q₈/C₂=C₂²	32		C2^4:6Q8	128,1572
C₂⁴⋊₁₁D₄	6^th semidirect product of C₂⁴ and D₄ acting via D₄/C₂=C₂²	32		C2^4:11D4	128,1544
M₄(2)⋊₇D₄	1^st semidirect product of M₄(2) and D₄ acting via D₄/C₄=C₂	32		M4(2):7D4	128,1883
C₂⁵⋊C₂²	2^nd semidirect product of C₂⁵ and C₂² acting faithfully	32		C2^5:C2^2	128,1411
C₄³⋊₁₅C₂	15^th semidirect product of C₄³ and C₂ acting faithfully	64		C4^3:15C2	128,1599
C₄²⋊₃₁D₄	25^th semidirect product of C₄² and D₄ acting via D₄/C₂=C₂²	64		C4^2:31D4	128,1389
C₄²⋊₃₃D₄	27^th semidirect product of C₄² and D₄ acting via D₄/C₂=C₂²	64		C4^2:33D4	128,1550
C₄²⋊₃₄D₄	28^th semidirect product of C₄² and D₄ acting via D₄/C₂=C₂²	64		C4^2:34D4	128,1551
C₄²⋊₃₅D₄	29^th semidirect product of C₄² and D₄ acting via D₄/C₂=C₂²	64		C4^2:35D4	128,1555
M₄(2)⋊₈D₄	2^nd semidirect product of M₄(2) and D₄ acting via D₄/C₄=C₂	64		M4(2):8D4	128,1884
M₄(2)⋊₅Q₈	3^rd semidirect product of M₄(2) and Q₈ acting via Q₈/C₄=C₂	64		M4(2):5Q8	128,1897
C₄²⋊₇Q₈	7^th semidirect product of C₄² and Q₈ acting via Q₈/C₂=C₂²	128		C4^2:7Q8	128,1283
C₄²⋊₁₀Q₈	10^th semidirect product of C₄² and Q₈ acting via Q₈/C₂=C₂²	128		C4^2:10Q8	128,1392
C₄²⋊₁₂Q₈	12^nd semidirect product of C₄² and Q₈ acting via Q₈/C₂=C₂²	128		C4^2:12Q8	128,1575
C₄²⋊₁₃Q₈	13^rd semidirect product of C₄² and Q₈ acting via Q₈/C₂=C₂²	128		C4^2:13Q8	128,1576
C₄²⋊₁₉Q₈	6^th semidirect product of C₄² and Q₈ acting via Q₈/C₄=C₂	128		C4^2:19Q8	128,1600
C₄².₁₅D₄	15^th non-split extension by C₄² of D₄ acting faithfully	16	8+	C4^2.15D4	128,934
C₂⁴.₁₇₇D₄	32^nd non-split extension by C₂⁴ of D₄ acting via D₄/C₂²=C₂	16		C2^4.177D4	128,1735
C₂³.₉C₂⁴	9^th non-split extension by C₂³ of C₂⁴ acting via C₂⁴/C₂²=C₂²	16	8+	C2^3.9C2^4	128,1759
C₄².₁₂C₂³	12^nd non-split extension by C₄² of C₂³ acting faithfully	16	8+	C4^2.12C2^3	128,1753
C₂².₇₃C₂⁵	54^th central stem extension by C₂² of C₂⁵	16		C2^2.73C2^5	128,2216
D₈.₁₃D₄	5^th non-split extension by D₈ of D₄ acting via D₄/C₂²=C₂	32	8-	D8.13D4	128,2021
C₄.C₂⁵	13^rd non-split extension by C₄ of C₂⁵ acting via C₂⁵/C₂⁴=C₂	32	8-	C4.C2^5	128,2318
C₄².₁₄D₄	14^th non-split extension by C₄² of D₄ acting faithfully	32	8-	C4^2.14D4	128,933
C₄².₁₆D₄	16^th non-split extension by C₄² of D₄ acting faithfully	32	8-	C4^2.16D4	128,935
C₂⁴.₁₅Q₈	14^th non-split extension by C₂⁴ of Q₈ acting via Q₈/C₂=C₂²	32		C2^4.15Q8	128,1574
C₂⁴.₁₇₈D₄	33^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂²=C₂	32		C2^4.178D4	128,1736
C₂⁴.₁₂₅D₄	80^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²	32		C2^4.125D4	128,1924
C₂⁴.₁₂₆D₄	81^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²	32		C2^4.126D4	128,1925
C₂⁴.₁₂₈D₄	83^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²	32		C2^4.128D4	128,1927
C₄².₂₇₅D₄	257^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	32		C4^2.275D4	128,1949
C₄².C₂³	1^st non-split extension by C₄² of C₂³ acting faithfully	32		C4^2.C2^3	128,387
C₄².₁₃C₂³	13^rd non-split extension by C₄² of C₂³ acting faithfully	32	8-	C4^2.13C2^3	128,1754
C₂³.₁₀C₂⁴	10^th non-split extension by C₂³ of C₂⁴ acting via C₂⁴/C₂²=C₂²	32	8-	C2^3.10C2^4	128,1760
C₂².₇₅C₂⁵	56^th central stem extension by C₂² of C₂⁵	32		C2^2.75C2^5	128,2218
C₂².₈₇C₂⁵	68^th central stem extension by C₂² of C₂⁵	32		C2^2.87C2^5	128,2230
C₂³.₃₃₃C₂⁴	50^th central stem extension by C₂³ of C₂⁴	32		C2^3.333C2^4	128,1165
C₂³.₅₆₉C₂⁴	286^th central stem extension by C₂³ of C₂⁴	32		C2^3.569C2^4	128,1401
C₂².₁₂₅C₂⁵	106^th central stem extension by C₂² of C₂⁵	32		C2^2.125C2^5	128,2268
C₂².₁₂₆C₂⁵	107^th central stem extension by C₂² of C₂⁵	32		C2^2.126C2^5	128,2269
C₂².₁₂₇C₂⁵	108^th central stem extension by C₂² of C₂⁵	32		C2^2.127C2^5	128,2270
C₂².₁₃₂C₂⁵	113^rd central stem extension by C₂² of C₂⁵	32		C2^2.132C2^5	128,2275
C₂².₁₃₈C₂⁵	119^th central stem extension by C₂² of C₂⁵	32		C2^2.138C2^5	128,2281
M₄(2).C₂³	4^th non-split extension by M₄(2) of C₂³ acting via C₂³/C₂=C₂²	32	8-	M4(2).C2^3	128,1752
C₄².₁₉₆D₄	178^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.196D4	128,1390
C₄².₁₉₉D₄	181^st non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.199D4	128,1552
C₄².₂₀₀D₄	182^nd non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.200D4	128,1553
C₄².₂₇₆D₄	258^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.276D4	128,1950
C₄².₂₉₀D₄	272^nd non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.290D4	128,1970
C₄².₂₉₁D₄	273^rd non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.291D4	128,1971
C₄².₃₀₁D₄	283^rd non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.301D4	128,1985
C₄².₃₀₂D₄	284^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.302D4	128,1986
C₄².₃₀₃D₄	285^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	64		C4^2.303D4	128,1987
C₄².₆C₂³	6^th non-split extension by C₄² of C₂³ acting faithfully	64		C4^2.6C2^3	128,392
C₂².₈₈C₂⁵	69^th central stem extension by C₂² of C₂⁵	64		C2^2.88C2^5	128,2231
C₂².₉₂C₂⁵	73^rd central stem extension by C₂² of C₂⁵	64		C2^2.92C2^5	128,2235
C₂³.₃₃₄C₂⁴	51^st central stem extension by C₂³ of C₂⁴	64		C2^3.334C2^4	128,1166
C₂³.₅₁₄C₂⁴	231^st central stem extension by C₂³ of C₂⁴	64		C2^3.514C2^4	128,1346
C₂⁴.₃₆₀C₂³	200^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²	64		C2^4.360C2^3	128,1347
C₂⁴.₃₆₁C₂³	201^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²	64		C2^4.361C2^3	128,1348
C₂³.₅₅₆C₂⁴	273^rd central stem extension by C₂³ of C₂⁴	64		C2^3.556C2^4	128,1388
C₂³.₅₅₉C₂⁴	276^th central stem extension by C₂³ of C₂⁴	64		C2^3.559C2^4	128,1391
C₂⁴.₃₈₄C₂³	224^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²	64		C2^4.384C2^3	128,1407
C₂⁴.₃₈₅C₂³	225^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²	64		C2^4.385C2^3	128,1409
C₂³.₅₈₃C₂⁴	300^th central stem extension by C₂³ of C₂⁴	64		C2^3.583C2^4	128,1415
C₂⁴.₄₅₉C₂³	299^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²	64		C2^4.459C2^3	128,1545
C₂³.₇₁₄C₂⁴	431^st central stem extension by C₂³ of C₂⁴	64		C2^3.714C2^4	128,1546
C₂³.₇₁₅C₂⁴	432^nd central stem extension by C₂³ of C₂⁴	64		C2^3.715C2^4	128,1547
C₂³.₇₁₆C₂⁴	433^rd central stem extension by C₂³ of C₂⁴	64		C2^3.716C2^4	128,1548
C₂⁴.₄₆₂C₂³	302^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²	64		C2^4.462C2^3	128,1549
C₂³.₇₄₁C₂⁴	458^th central stem extension by C₂³ of C₂⁴	64		C2^3.741C2^4	128,1573
C₂².₁₃₃C₂⁵	114^th central stem extension by C₂² of C₂⁵	64		C2^2.133C2^5	128,2276
C₂².₁₃₇C₂⁵	118^th central stem extension by C₂² of C₂⁵	64		C2^2.137C2^5	128,2280
C₂².₁₃₉C₂⁵	120^th central stem extension by C₂² of C₂⁵	64		C2^2.139C2^5	128,2282
C₂².₁₄₁C₂⁵	122^nd central stem extension by C₂² of C₂⁵	64		C2^2.141C2^5	128,2284
C₂².₁₄₅C₂⁵	126^th central stem extension by C₂² of C₂⁵	64		C2^2.145C2^5	128,2288
C₄².₄₀Q₈	40^th non-split extension by C₄² of Q₈ acting via Q₈/C₂=C₂²	128		C4^2.40Q8	128,1577
C₄².₂₀₁D₄	183^rd non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²	128		C4^2.201D4	128,1554
C₂³.₆₃₄C₂⁴	351^st central stem extension by C₂³ of C₂⁴	128		C2^3.634C2^4	128,1466
C₂³.₇₁₁C₂⁴	428^th central stem extension by C₂³ of C₂⁴	128		C2^3.711C2^4	128,1543
C₂⁷	Elementary abelian group of type [2,2,2,2,2,2,2]	128		C2^7	128,2328
C₂²×2₊¹⁺⁴	Direct product of C₂² and 2₊¹⁺⁴	32		C2^2xES+(2,2)	128,2323
D₄×C₂⁴	Direct product of C₂⁴ and D₄	64		D4xC2^4	128,2320
C₂²×2_-¹⁺⁴	Direct product of C₂² and 2_-¹⁺⁴	64		C2^2xES-(2,2)	128,2324
Q₈×C₂⁴	Direct product of C₂⁴ and Q₈	128		Q8xC2^4	128,2321
C₂×C₂≀C₂²	Direct product of C₂ and C₂≀C₂²	16		C2xC2wrC2^2	128,1755
C₂×D₄⋊₄D₄	Direct product of C₂ and D₄⋊₄D₄	16		C2xD4:4D4	128,1746
C₂×D₄²	Direct product of C₂, D₄ and D₄	32		C2xD4^2	128,2194
C₂×C₂³⋊₃D₄	Direct product of C₂ and C₂³⋊₃D₄	32		C2xC2^3:3D4	128,2177
C₂²×C₈⋊C₂²	Direct product of C₂² and C₈⋊C₂²	32		C2^2xC8:C2^2	128,2310
C₂×C₂³⋊₂Q₈	Direct product of C₂ and C₂³⋊₂Q₈	32		C2xC2^3:2Q8	128,2188
C₂×D₄.₁₀D₄	Direct product of C₂ and D₄.₁₀D₄	32		C2xD4.10D4	128,1749
C₂²×C₂²≀C₂	Direct product of C₂² and C₂²≀C₂	32		C2^2xC2^2wrC2	128,2163
C₂×C₂⁴⋊C₂²	Direct product of C₂ and C₂⁴⋊C₂²	32		C2xC2^4:C2^2	128,2258
C₂×C₂².₂₉C₂⁴	Direct product of C₂ and C₂².₂₉C₂⁴	32		C2xC2^2.29C2^4	128,2178
C₂×C₂².₅₄C₂⁴	Direct product of C₂ and C₂².₅₄C₂⁴	32		C2xC2^2.54C2^4	128,2257
C₂×D₄×Q₈	Direct product of C₂, D₄ and Q₈	64		C2xD4xQ8	128,2198
C₂×C₈⋊₃D₄	Direct product of C₂ and C₈⋊₃D₄	64		C2xC8:3D4	128,1880
C₂×C₈.₂D₄	Direct product of C₂ and C₈.₂D₄	64		C2xC8.2D4	128,1881
C₂²×C₄⋊₁D₄	Direct product of C₂² and C₄⋊₁D₄	64		C2^2xC4:1D4	128,2172
C₂²×C₈.C₂²	Direct product of C₂² and C₈.C₂²	64		C2^2xC8.C2^2	128,2311
C₂×C₂³.₃₈C₂³	Direct product of C₂ and C₂³.₃₈C₂³	64		C2xC2^3.38C2^3	128,2179
C₂×C₂².₃₁C₂⁴	Direct product of C₂ and C₂².₃₁C₂⁴	64		C2xC2^2.31C2^4	128,2180
C₂×C₂³.₄₁C₂³	Direct product of C₂ and C₂³.₄₁C₂³	64		C2xC2^3.41C2^3	128,2189
C₂×C₂².₅₆C₂⁴	Direct product of C₂ and C₂².₅₆C₂⁴	64		C2xC2^2.56C2^4	128,2259
C₂×C₂².₅₇C₂⁴	Direct product of C₂ and C₂².₅₇C₂⁴	64		C2xC2^2.57C2^4	128,2260
C₂×Q₈²	Direct product of C₂, Q₈ and Q₈	128		C2xQ8^2	128,2209
C₂×C₈⋊Q₈	Direct product of C₂ and C₈⋊Q₈	128		C2xC8:Q8	128,1893
C₂²×C₄⋊Q₈	Direct product of C₂² and C₄⋊Q₈	128		C2^2xC4:Q8	128,2173
C₂×C₂².₅₈C₂⁴	Direct product of C₂ and C₂².₅₈C₂⁴	128		C2xC2^2.58C2^4	128,2262

Groups of order 144

		d	ρ	Label	ID
Dic₃⋊D₆	2^nd semidirect product of Dic₃ and D₆ acting via D₆/S₃=C₂; = Hol(Dic₃)	12	4+	Dic3:D6	144,154
S₃×S₄	Direct product of S₃ and S₄; = Hol(C₂×C₆)	12	6+	S3xS4	144,183
C₂×PSU₃(𝔽₂)	Direct product of C₂ and PSU₃(𝔽₂)	18	8+	C2xPSU(3,2)	144,187
C₂×S₃≀C₂	Direct product of C₂ and S₃≀C₂	12	4+	C2xS3wrC2	144,186
C₂×C₃⋊S₄	Direct product of C₂ and C₃⋊S₄	18	6+	C2xC3:S4	144,189
C₂²×S₃²	Direct product of C₂², S₃ and S₃	24		C2^2xS3^2	144,192
D₄×C₃⋊S₃	Direct product of D₄ and C₃⋊S₃	36		D4xC3:S3	144,172
Q₈×C₃⋊S₃	Direct product of Q₈ and C₃⋊S₃	72		Q8xC3:S3	144,174
C₂³×C₃⋊S₃	Direct product of C₂³ and C₃⋊S₃	72		C2^3xC3:S3	144,196

Groups of order 162

		d	ρ	Label	ID
C₃⁴⋊C₂	4^th semidirect product of C₃⁴ and C₂ acting faithfully	81		C3^4:C2	162,54

Groups of order 192

		d	ρ	Label	ID
Q₈⋊₂S₄	2^nd semidirect product of Q₈ and S₄ acting via S₄/C₂²=S₃; = Hol(Q₈)	8	4+	Q8:2S4	192,1494
C₂³⋊S₄	2^nd semidirect product of C₂³ and S₄ acting faithfully; = Aut(C₂²×C₄)	8	4+	C2^3:S4	192,1493
C₂⁴⋊D₆	1^st semidirect product of C₂⁴ and D₆ acting faithfully; = Aut(C₂×Q₈)	8	6+	C2^4:D6	192,955
C₄²⋊D₆	The semidirect product of C₄² and D₆ acting faithfully	12	6+	C4^2:D6	192,956
D₄×S₄	Direct product of D₄ and S₄	12	6+	D4xS4	192,1472
Q₈×S₄	Direct product of Q₈ and S₄	24	6-	Q8xS4	192,1477
C₂³×S₄	Direct product of C₂³ and S₄	24		C2^3xS4	192,1537
S₃×2₊¹⁺⁴	Direct product of S₃ and 2₊¹⁺⁴	24	8+	S3xES+(2,2)	192,1524
S₃×2_-¹⁺⁴	Direct product of S₃ and 2_-¹⁺⁴	48	8-	S3xES-(2,2)	192,1526
S₃×C₂⁵	Direct product of C₂⁵ and S₃	96		S3xC2^5	192,1542
C₂×C₂²⋊S₄	Direct product of C₂ and C₂²⋊S₄	12	6+	C2xC2^2:S4	192,1538
S₃×C₈⋊C₂²	Direct product of S₃ and C₈⋊C₂²; = Aut(D₂₄) = Hol(C₂₄)	24	8+	S3xC8:C2^2	192,1331
S₃×C₂²≀C₂	Direct product of S₃ and C₂²≀C₂	24		S3xC2^2wrC2	192,1147
C₂²×S₃×D₄	Direct product of C₂², S₃ and D₄	48		C2^2xS3xD4	192,1514
S₃×C₄⋊₁D₄	Direct product of S₃ and C₄⋊₁D₄	48		S3xC4:1D4	192,1273
S₃×C₈.C₂²	Direct product of S₃ and C₈.C₂²	48	8-	S3xC8.C2^2	192,1335
S₃×C₄⋊Q₈	Direct product of S₃ and C₄⋊Q₈	96		S3xC4:Q8	192,1282
C₂²×S₃×Q₈	Direct product of C₂², S₃ and Q₈	96		C2^2xS3xQ8	192,1517

Groups of order 200

		d	ρ	Label	ID
C₅²⋊Q₈	The semidirect product of C₅² and Q₈ acting faithfully	10	8+	C5^2:Q8	200,44

Groups of order 216

		d	ρ	Label	ID
C₃²⋊₄S₄	2^nd semidirect product of C₃² and S₄ acting via S₄/A₄=C₂	36		C3^2:4S4	216,165
S₃³	Direct product of S₃, S₃ and S₃; = Hol(C₃×S₃)	12	8+	S3^3	216,162
C₂×C₃²⋊D₆	Direct product of C₂ and C₃²⋊D₆	18	6+	C2xC3^2:D6	216,102
C₂²×C₃³⋊C₂	Direct product of C₂² and C₃³⋊C₂	108		C2^2xC3^3:C2	216,176
C₂×S₃×C₃⋊S₃	Direct product of C₂, S₃ and C₃⋊S₃	36		C2xS3xC3:S3	216,171

Groups of order 240

		d	ρ	Label	ID
C₂×S₅	Direct product of C₂ and S₅; = O₃(𝔽₅)	10	4+	C2xS5	240,189

Groups of order 288

		d	ρ	Label	ID
PSO₄⁺ (𝔽₃)	Projective special orthogonal group of + type on 𝔽₃⁴; = A₄ ⋊S₄ = Hol(A₄)	12	9+	PSO+(4,3)	288,1026
D₆≀C₂	Wreath product of D₆ by C₂	12	4+	D6wrC2	288,889
C₄⋊S₃≀C₂	The semidirect product of C₄ and S₃≀C₂ acting via S₃≀C₂/C₃²⋊C₄=C₂	24	8+	C4:S3wrC2	288,879
C₃²⋊C₄⋊Q₈	1^st semidirect product of C₃²⋊C₄ and Q₈ acting via Q₈/C₄=C₂	48	8-	C3^2:C4:Q8	288,870
C₂²×PSU₃(𝔽₂)	Direct product of C₂² and PSU₃(𝔽₂)	36		C2^2xPSU(3,2)	288,1032
C₂×S₃×S₄	Direct product of C₂, S₃ and S₄; = Aut(S₃×SL₂(𝔽₃))	18	6+	C2xS3xS4	288,1028
S₃²×D₄	Direct product of S₃, S₃ and D₄	24	8+	S3^2xD4	288,958
C₂²×S₃≀C₂	Direct product of C₂² and S₃≀C₂	24		C2^2xS3wrC2	288,1031
C₂×Dic₃⋊D₆	Direct product of C₂ and Dic₃⋊D₆	24		C2xDic3:D6	288,977
C₂²×C₃⋊S₄	Direct product of C₂² and C₃⋊S₄	36		C2^2xC3:S4	288,1034
S₃²×Q₈	Direct product of S₃, S₃ and Q₈	48	8-	S3^2xQ8	288,965
S₃²×C₂³	Direct product of C₂³, S₃ and S₃	48		S3^2xC2^3	288,1040
C₂⁴×C₃⋊S₃	Direct product of C₂⁴ and C₃⋊S₃	144		C2^4xC3:S3	288,1044
C₂×D₄×C₃⋊S₃	Direct product of C₂, D₄ and C₃⋊S₃	72		C2xD4xC3:S3	288,1007
C₂×Q₈×C₃⋊S₃	Direct product of C₂, Q₈ and C₃⋊S₃	144		C2xQ8xC3:S3	288,1010

Groups of order 324

		d	ρ	Label	ID
He₃⋊D₆	The semidirect product of He₃ and D₆ acting faithfully	9	6+	He3:D6	324,39
He₃⋊₅D₆	1^st semidirect product of He₃ and D₆ acting via D₆/C₃=C₂²	18	12+	He3:5D6	324,121
S₃×C₃³⋊C₂	Direct product of S₃ and C₃³⋊C₂	54		S3xC3^3:C2	324,168
C₂×C₃⁴⋊C₂	Direct product of C₂ and C₃⁴⋊C₂	162		C2xC3^4:C2	324,175
C₃⋊S₃²	Direct product of C₃⋊S₃ and C₃⋊S₃	18		C3:S3^2	324,169

Groups of order 400

		d	ρ	Label	ID
C₂×C₅²⋊Q₈	Direct product of C₂ and C₅²⋊Q₈	20	8+	C2xC5^2:Q8	400,212

Groups of order 432

		d	ρ	Label	ID
C₆²⋊₅D₆	5^th semidirect product of C₆² and D₆ acting faithfully	18	6+	C6^2:5D6	432,523
C₆²⋊₂₃D₆	4^th semidirect product of C₆² and D₆ acting via D₆/C₃=C₂²	36		C6^2:23D6	432,686
S₃×PSU₃(𝔽₂)	Direct product of S₃ and PSU₃(𝔽₂)	24	16+	S3xPSU(3,2)	432,742
S₃×S₃≀C₂	Direct product of S₃ and S₃≀C₂	12	8+	S3xS3wrC2	432,741
S₃×C₃⋊S₄	Direct product of S₃ and C₃⋊S₄	24	12+	S3xC3:S4	432,747
C₃⋊S₃×S₄	Direct product of C₃⋊S₃ and S₄	36		C3:S3xS4	432,746
C₂²×C₃²⋊D₆	Direct product of C₂² and C₃²⋊D₆	36		C2^2xC3^2:D6	432,545
C₂×C₃²⋊₄S₄	Direct product of C₂ and C₃²⋊₄S₄	54		C2xC3^2:4S4	432,762
D₄×C₃³⋊C₂	Direct product of D₄ and C₃³⋊C₂	108		D4xC3^3:C2	432,724
Q₈×C₃³⋊C₂	Direct product of Q₈ and C₃³⋊C₂	216		Q8xC3^3:C2	432,726
C₂³×C₃³⋊C₂	Direct product of C₂³ and C₃³⋊C₂	216		C2^3xC3^3:C2	432,774
C₂×S₃³	Direct product of C₂, S₃, S₃ and S₃	24	8+	C2xS3^3	432,759
C₂²×S₃×C₃⋊S₃	Direct product of C₂², S₃ and C₃⋊S₃	72		C2^2xS3xC3:S3	432,768

Groups of order 480

		d	ρ	Label	ID
C₂²×S₅	Direct product of C₂² and S₅	20		C2^2xS5	480,1186

Groups of order 486

		d	ρ	Label	ID
C₃⁵⋊C₂	5^th semidirect product of C₃⁵ and C₂ acting faithfully	243		C3^5:C2	486,260

⋊

ℤ

𝔽

○

≀

ℚ

◁