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## G = C2×C23⋊3D4order 128 = 27

### Direct product of C2 and C23⋊3D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23⋊3D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×C23⋊3D4
 Lower central C1 — C22 — C2×C23⋊3D4
 Upper central C1 — C23 — C2×C23⋊3D4
 Jennings C1 — C22 — C2×C23⋊3D4

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 ··· 2G 2H ··· 2S 2T ··· 2AA 4A ··· 4P order 1 2 ··· 2 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 2+ 1+4 kernel C2×C23⋊3D4 C22×C22⋊C4 C2×C22≀C2 C2×C4⋊D4 C2×C22.D4 C23⋊3D4 D4×C23 C24 C22 # reps 1 1 4 4 4 16 2 8 4

Matrix representation of C2×C233D4 in GL8(ℤ)

 -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 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 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 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 -2 -1 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 0 2 0 0 0 0 0 2 1 0 -2 0 0 0 0 -1 0 -1 0 0 0 0 0 2 1 2 -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 0 2 0 0 0 0 0 -2 -1 0 2 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 2 1

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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