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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, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C22≀C2, C4⋊D4, C22.D4, C23×C4, C22×D4, C22×D4, C25, C25, C22×C22⋊C4, C2×C22≀C2, C2×C4⋊D4, C2×C22.D4, C233D4, D4×C23, C2×C233D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C25, C233D4, D4×C23, C2×2+ 1+4, 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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