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G = C2×C233D4order 128 = 27

Direct product of C2 and C233D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C2×C233D4, C2412D4, C254C22, C247C23, C22.34C25, C23.115C24, C22.992+ 1+4, C4⋊C44C23, C236(C2×D4), (C2×D4)⋊5C23, (D4×C23)⋊12C2, C22⋊C44C23, (C2×C4).37C24, C2.13(D4×C23), C4⋊D460C22, (C22×C4)⋊13C23, (C23×C4)⋊29C22, C22≀C225C22, (C22×D4)⋊29C22, C22.47(C22×D4), C2.4(C2×2+ 1+4), C22.D430C22, (C2×C4⋊D4)⋊53C2, (C2×C4⋊C4)⋊63C22, (C2×C22≀C2)⋊20C2, (C2×C22⋊C4)⋊39C22, (C22×C22⋊C4)⋊30C2, (C2×C22.D4)⋊48C2, SmallGroup(128,2177)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C233D4
C1C2C22C23C24C25D4×C23 — C2×C233D4
C1C22 — C2×C233D4
C1C23 — C2×C233D4
C1C22 — C2×C233D4

Generators and relations for C2×C233D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >

Subgroups: 1932 in 992 conjugacy classes, 436 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×20], C4 [×16], C22, C22 [×18], C22 [×124], C2×C4 [×16], C2×C4 [×40], D4 [×80], C23, C23 [×42], C23 [×124], C22⋊C4 [×48], C4⋊C4 [×16], C22×C4 [×24], C22×C4 [×12], C2×D4 [×48], C2×D4 [×104], C24, C24 [×26], C24 [×20], C2×C22⋊C4 [×20], C2×C4⋊C4 [×4], C22≀C2 [×32], C4⋊D4 [×32], C22.D4 [×32], C23×C4 [×4], C22×D4 [×28], C22×D4 [×16], C25, C25 [×2], C22×C22⋊C4, C2×C22≀C2 [×4], C2×C4⋊D4 [×4], C2×C22.D4 [×4], C233D4 [×16], D4×C23 [×2], C2×C233D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ 1+4 [×4], C25, C233D4 [×4], D4×C23, C2×2+ 1+4 [×2], C2×C233D4

Smallest permutation representation of C2×C233D4
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 10)(6 11)(7 12)(8 9)(13 24)(14 21)(15 22)(16 23)(17 32)(18 29)(19 30)(20 31)
(1 31)(2 32)(3 29)(4 30)(5 24)(6 21)(7 22)(8 23)(9 16)(10 13)(11 14)(12 15)(17 26)(18 27)(19 28)(20 25)
(1 3)(2 12)(4 10)(5 28)(6 8)(7 26)(9 11)(13 30)(14 16)(15 32)(17 22)(18 20)(19 24)(21 23)(25 27)(29 31)
(1 9)(2 10)(3 11)(4 12)(5 26)(6 27)(7 28)(8 25)(13 32)(14 29)(15 30)(16 31)(17 24)(18 21)(19 22)(20 23)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 29)(14 32)(15 31)(16 30)(17 21)(18 24)(19 23)(20 22)(25 28)(26 27)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2S2T···2AA4A···4P
order12···22···22···24···4
size11···12···24···44···4

44 irreducible representations

dim111111124
type+++++++++
imageC1C2C2C2C2C2C2D42+ 1+4
kernelC2×C233D4C22×C22⋊C4C2×C22≀C2C2×C4⋊D4C2×C22.D4C233D4D4×C23C24C22
# reps1144416284

Matrix representation of C2×C233D4 in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00-100000
000-10000
00000-100
0000-1000
0000-1001
00000-110
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
00001010
0000-2-101
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
0-1000000
10000000
000-10000
00100000
00001020
0000210-2
0000-10-10
0000212-1
,
01000000
10000000
00010000
00100000
00001020
0000-2-102
000000-10
00000021

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,1,-2,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,2,-1,2,0,0,0,0,0,1,0,1,0,0,0,0,2,0,-1,2,0,0,0,0,0,-2,0,-1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-2,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,2,0,-1,2,0,0,0,0,0,2,0,1] >;

C2×C233D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_3D_4
% in TeX

G:=Group("C2xC2^3:3D4");
// GroupNames label

G:=SmallGroup(128,2177);
// by ID

G=gap.SmallGroup(128,2177);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123]);
// Polycyclic

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

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