C2^4.21D4 - GroupNames

G = C₂⁴.₂₁D₄ order 128 = 2⁷

21^st non-split extension by C₂⁴ of D₄ acting faithfully

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

Aliases: C₂⁴.₂₁D₄, C₄.₅₀(C₄×D₄), (C₂×D₄).₅Q₈, C₄⋊C₄.₃₀₀D₄, C₄.D₄⋊₂C₄, C₄²⋊₆C₄⋊₁C₂, C₂³.₄(C₄⋊C₄), M₄(2)⋊₁(C₂×C₄), C₂.₂(D₄⋊₄D₄), C₂³.₅₅₂(C₂×D₄), C₄.₂₃(C₂²⋊Q₈), M₄(2)⋊C₄⋊₁C₂, C₂.₂(D₄.₉D₄), C₂².₈₀C₂²≀C₂, (C₂²×C₄).₆₇₂C₂³, C₄²⋊C₂.₈C₂², C₂².₁₁C₂⁴.₃C₂, (C₂×C₄²).₂₆₃C₂², (C₂²×D₄).₁₃C₂², (C₂×M₄(2)).₅C₂², C₂⁴.₃C₂².₄C₂, C₂.₁₂(C₂³.₈Q₈), C₂².₄₅(C₂².D₄), (C₂×C₄).₇(C₂×Q₈), (C₂×D₄).₆₆(C₂×C₄), (C₂×C₄).₉₈₂(C₂×D₄), C₂².₂₂(C₂×C₄⋊C₄), (C₂×C₄).₆(C₂²×C₄), (C₂×C₄⋊C₄).₅₄C₂², (C₂×C₄.D₄).₁C₂, (C₂×C₄).₃₁₅(C₄○D₄), SmallGroup(128,588)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₂₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂²×D₄ — C₂².₁₁C₂⁴ — C₂⁴.₂₁D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₂₁D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₂₁D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₂₁D₄

Generators and relations for C₂⁴.₂₁D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=c, eae^-1=faf^-1=ab=ba, ac=ca, ad=da, bc=cb, ebe^-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de³ >

Subgroups: 372 in 162 conjugacy classes, 56 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₄.D₄, C₄.Q₈, C₂.D₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂×M₄(2), C₂²×D₄, C₄²⋊₆C₄, C₂⁴.₃C₂², C₂×C₄.D₄, M₄(2)⋊C₄, C₂².₁₁C₂⁴, C₂⁴.₂₁D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂³.₈Q₈, D₄⋊₄D₄, D₄.₉D₄, C₂⁴.₂₁D₄

Smallest permutation representation of C₂⁴.₂₁D₄
►On 32 points

Generators in S₃₂

(2 25)(3 7)(4 31)(6 29)(8 27)(9 24)(10 14)(11 22)(13 20)(15 18)(17 21)(26 30)

(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)

(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)

(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)

(1 22 32 11)(2 21 25 10)(3 20 26 9)(4 19 27 16)(5 18 28 15)(6 17 29 14)(7 24 30 13)(8 23 31 12)

G:=sub<Sym(32)| (2,25)(3,7)(4,31)(6,29)(8,27)(9,24)(10,14)(11,22)(13,20)(15,18)(17,21)(26,30), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23), (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22,32,11)(2,21,25,10)(3,20,26,9)(4,19,27,16)(5,18,28,15)(6,17,29,14)(7,24,30,13)(8,23,31,12)>;

G:=Group( (2,25)(3,7)(4,31)(6,29)(8,27)(9,24)(10,14)(11,22)(13,20)(15,18)(17,21)(26,30), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23), (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22,32,11)(2,21,25,10)(3,20,26,9)(4,19,27,16)(5,18,28,15)(6,17,29,14)(7,24,30,13)(8,23,31,12) );

G=PermutationGroup([[(2,25),(3,7),(4,31),(6,29),(8,27),(9,24),(10,14),(11,22),(13,20),(15,18),(17,21),(26,30)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,20),(10,17),(11,22),(12,19),(13,24),(14,21),(15,18),(16,23)], [(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,22,32,11),(2,21,25,10),(3,20,26,9),(4,19,27,16),(5,18,28,15),(6,17,29,14),(7,24,30,13),(8,23,31,12)]])

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+		+	-	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	Q₈	D₄	C₄○D₄	D₄⋊₄D₄	D₄.₉D₄
kernel	C₂⁴.₂₁D₄	C₄²⋊₆C₄	C₂⁴.₃C₂²	C₂×C₄.D₄	M₄(2)⋊C₄	C₂².₁₁C₂⁴	C₄.D₄	C₄⋊C₄	C₂×D₄	C₂⁴	C₂×C₄	C₂	C₂
# reps	1	2	1	1	2	1	8	4	2	2	4	2	2

Matrix representation of C₂⁴.₂₁D₄ ►in GL₆(𝔽₁₇)

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

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

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

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

7	3	0	0	0	0
6	10	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	16
0	0	0	1	0	0
0	0	1	0	0	0

0	14	0	0	0	0
6	0	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

C₂⁴.₂₁D₄ in GAP, Magma, Sage, TeX

C_2^4._{21}D_4

% in TeX

G:=Group("C2^4.21D4");

// GroupNames label

G:=SmallGroup(128,588);

// by ID

G=gap.SmallGroup(128,588);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,1411]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,e*a*e^-1=f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^3>;

// generators/relations

G = C24.21D4 order 128 = 27

21st non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₂₁D₄ order 128 = 2⁷

21^st non-split extension by C₂⁴ of D₄ acting faithfully