C4^2.22C2^3 - GroupNames

G = C₄².₂₂C₂³ order 128 = 2⁷

22^nd non-split extension by C₄² of C₂³ acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄².₂₂C₂³, C₄○D₄⋊₁Q₈, Q₈.₆(C₂×Q₈), D₄.₆(C₂×Q₈), C₄⋊C₄.₃₄₃D₄, D₄.Q₈⋊₁₅C₂, Q₈.Q₈⋊₁₅C₂, D₄⋊Q₈⋊₂₁C₂, C₄.Q₁₆⋊₂₁C₂, C₂.₁₄(D₄○D₈), C₄⋊C₈.₄₆C₂², C₄⋊C₄.₄₆C₂³, (C₂×C₈).₃₀C₂³, C₂.₁₄(Q₈○D₈), C₄.₃₄(C₂²×Q₈), (C₂×C₄).₂₈₁C₂⁴, C₂²⋊C₄.₁₄₄D₄, (C₄×D₄).₇₀C₂², C₂³.₄₅₀(C₂×D₄), C₄⋊Q₈.₁₀₃C₂², C₄.₁₉(C₂²⋊Q₈), (C₄×Q₈).₆₇C₂², C₄.Q₈.₁₁C₂², (C₂×D₄).₃₉₈C₂³, (C₂×Q₈).₃₆₉C₂³, M₄(2)⋊C₄⋊₂₀C₂, C₂.D₈.₁₆₈C₂², D₄⋊C₄.₂₇C₂², (C₂²×C₈).₁₄₄C₂², Q₈⋊C₄.₂₈C₂², C₂³.₂₄D₄.₄C₂, C₂³.₃₆D₄.₃C₂, C₂².₅₄₁(C₂²×D₄), C₂².₂₀(C₂²⋊Q₈), C₄².C₂.₁₀C₂², C₄².₆C₂²⋊₁₀C₂, C₂³.₄₁C₂³⋊₄C₂, (C₂²×C₄).₁₀₀₀C₂³, (C₂×M₄(2)).₇₀C₂², C₄²⋊C₂.₁₂₀C₂², C₂³.₃₃C₂³.₇C₂, (C₂×C₂.D₈)⋊₂₂C₂, C₄.₉₁(C₂×C₄○D₄), (C₂×C₄).₄₈₃(C₂×D₄), (C₂×C₄).₁₀₅(C₂×Q₈), C₂.₆₂(C₂×C₂²⋊Q₈), (C₂×C₄).₄₈₃(C₄○D₄), (C₂×C₄⋊C₄).₆₀₇C₂², (C₂×C₄○D₄).₁₃₄C₂², SmallGroup(128,1815)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — C₄².₂₂C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂³.₃₃C₂³ — C₄².₂₂C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₂₂C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₂₂C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₂C₂³

Generators and relations for C₄².₂₂C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=1, c²=e²=a², ab=ba, cac^-1=a^-1, ad=da, eae^-1=ab², cbc^-1=dbd=b^-1, be=eb, dcd=bc, ece^-1=a²c, de=ed >

Subgroups: 332 in 188 conjugacy classes, 100 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, D₄⋊C₄, D₄⋊C₄, Q₈⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×C₄○D₄, C₂³.₂₄D₄, C₂³.₃₆D₄, C₄².₆C₂², C₂×C₂.D₈, M₄(2)⋊C₄, D₄⋊Q₈, C₄.Q₁₆, D₄.Q₈, Q₈.Q₈, C₂³.₃₃C₂³, C₂³.₄₁C₂³, C₄².₂₂C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₂²⋊Q₈, D₄○D₈, Q₈○D₈, C₄².₂₂C₂³

Smallest permutation representation of C₄².₂₂C₂³
►On 64 points

Generators in S₆₄

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 42 46 17)(2 43 47 18)(3 44 48 19)(4 41 45 20)(5 62 33 39)(6 63 34 40)(7 64 35 37)(8 61 36 38)(9 15 23 49)(10 16 24 50)(11 13 21 51)(12 14 22 52)(25 56 60 31)(26 53 57 32)(27 54 58 29)(28 55 59 30)

(1 26 3 28)(2 25 4 27)(5 52 7 50)(6 51 8 49)(9 40 11 38)(10 39 12 37)(13 36 15 34)(14 35 16 33)(17 53 19 55)(18 56 20 54)(21 61 23 63)(22 64 24 62)(29 43 31 41)(30 42 32 44)(45 58 47 60)(46 57 48 59)

(5 62)(6 63)(7 64)(8 61)(13 51)(14 52)(15 49)(16 50)(17 42)(18 43)(19 44)(20 41)(25 31)(26 32)(27 29)(28 30)(33 39)(34 40)(35 37)(36 38)(53 57)(54 58)(55 59)(56 60)

(1 22 3 24)(2 9 4 11)(5 30 7 32)(6 56 8 54)(10 46 12 48)(13 43 15 41)(14 19 16 17)(18 49 20 51)(21 47 23 45)(25 38 27 40)(26 62 28 64)(29 34 31 36)(33 55 35 53)(37 57 39 59)(42 52 44 50)(58 63 60 61)

G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,26,3,28)(2,25,4,27)(5,52,7,50)(6,51,8,49)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,53,19,55)(18,56,20,54)(21,61,23,63)(22,64,24,62)(29,43,31,41)(30,42,32,44)(45,58,47,60)(46,57,48,59), (5,62)(6,63)(7,64)(8,61)(13,51)(14,52)(15,49)(16,50)(17,42)(18,43)(19,44)(20,41)(25,31)(26,32)(27,29)(28,30)(33,39)(34,40)(35,37)(36,38)(53,57)(54,58)(55,59)(56,60), (1,22,3,24)(2,9,4,11)(5,30,7,32)(6,56,8,54)(10,46,12,48)(13,43,15,41)(14,19,16,17)(18,49,20,51)(21,47,23,45)(25,38,27,40)(26,62,28,64)(29,34,31,36)(33,55,35,53)(37,57,39,59)(42,52,44,50)(58,63,60,61)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,26,3,28)(2,25,4,27)(5,52,7,50)(6,51,8,49)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,53,19,55)(18,56,20,54)(21,61,23,63)(22,64,24,62)(29,43,31,41)(30,42,32,44)(45,58,47,60)(46,57,48,59), (5,62)(6,63)(7,64)(8,61)(13,51)(14,52)(15,49)(16,50)(17,42)(18,43)(19,44)(20,41)(25,31)(26,32)(27,29)(28,30)(33,39)(34,40)(35,37)(36,38)(53,57)(54,58)(55,59)(56,60), (1,22,3,24)(2,9,4,11)(5,30,7,32)(6,56,8,54)(10,46,12,48)(13,43,15,41)(14,19,16,17)(18,49,20,51)(21,47,23,45)(25,38,27,40)(26,62,28,64)(29,34,31,36)(33,55,35,53)(37,57,39,59)(42,52,44,50)(58,63,60,61) );

G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,42,46,17),(2,43,47,18),(3,44,48,19),(4,41,45,20),(5,62,33,39),(6,63,34,40),(7,64,35,37),(8,61,36,38),(9,15,23,49),(10,16,24,50),(11,13,21,51),(12,14,22,52),(25,56,60,31),(26,53,57,32),(27,54,58,29),(28,55,59,30)], [(1,26,3,28),(2,25,4,27),(5,52,7,50),(6,51,8,49),(9,40,11,38),(10,39,12,37),(13,36,15,34),(14,35,16,33),(17,53,19,55),(18,56,20,54),(21,61,23,63),(22,64,24,62),(29,43,31,41),(30,42,32,44),(45,58,47,60),(46,57,48,59)], [(5,62),(6,63),(7,64),(8,61),(13,51),(14,52),(15,49),(16,50),(17,42),(18,43),(19,44),(20,41),(25,31),(26,32),(27,29),(28,30),(33,39),(34,40),(35,37),(36,38),(53,57),(54,58),(55,59),(56,60)], [(1,22,3,24),(2,9,4,11),(5,30,7,32),(6,56,8,54),(10,46,12,48),(13,43,15,41),(14,19,16,17),(18,49,20,51),(21,47,23,45),(25,38,27,40),(26,62,28,64),(29,34,31,36),(33,55,35,53),(37,57,39,59),(42,52,44,50),(58,63,60,61)]])

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	4R	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4	8	8	8	8	8	8
size	1	1	1	1	2	2	4	4	2	2	2	2	4	···	4	8	8	8	8	4	4	4	4	8	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	Q₈	C₄○D₄	D₄○D₈	Q₈○D₈
kernel	C₄².₂₂C₂³	C₂³.₂₄D₄	C₂³.₃₆D₄	C₄².₆C₂²	C₂×C₂.D₈	M₄(2)⋊C₄	D₄⋊Q₈	C₄.Q₁₆	D₄.Q₈	Q₈.Q₈	C₂³.₃₃C₂³	C₂³.₄₁C₂³	C₂²⋊C₄	C₄⋊C₄	C₄○D₄	C₂×C₄	C₂	C₂
# reps	1	1	1	1	1	1	2	2	2	2	1	1	2	2	4	4	2	2

Matrix representation of C₄².₂₂C₂³ ►in GL₆(𝔽₁₇)

0	1	0	0	0	0
16	0	0	0	0	0
0	0	0	0	4	0
0	0	0	0	0	4
0	0	4	0	0	0
0	0	0	4	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	15	0	0
0	0	1	1	0	0
0	0	0	0	16	15
0	0	0	0	1	1

10	16	0	0	0	0
16	7	0	0	0	0
0	0	0	0	11	11
0	0	0	0	3	6
0	0	6	6	0	0
0	0	14	11	0	0

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	16	16	0	0
0	0	0	0	1	0
0	0	0	0	16	16

0	16	0	0	0	0
1	0	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	4	0
0	0	0	0	0	4

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,4,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,0,6,14,0,0,0,0,6,11,0,0,11,3,0,0,0,0,11,6,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C₄².₂₂C₂³ in GAP, Magma, Sage, TeX

C_4^2._{22}C_2^3

% in TeX

G:=Group("C4^2.22C2^3");

// GroupNames label

G:=SmallGroup(128,1815);

// by ID

G=gap.SmallGroup(128,1815);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,1018,248,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=e^2=a^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;

// generators/relations

G = C42.22C23 order 128 = 27

22nd non-split extension by C42 of C23 acting faithfully

G = C₄².₂₂C₂³ order 128 = 2⁷

22^nd non-split extension by C₄² of C₂³ acting faithfully