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## G = C2×C23⋊2Q8order 128 = 27

### Direct product of C2 and C23⋊2Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23⋊2Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C2×C23⋊2Q8
 Lower central C1 — C22 — C2×C23⋊2Q8
 Upper central C1 — C23 — C2×C23⋊2Q8
 Jennings C1 — C22 — C2×C23⋊2Q8

Generators and relations for C2×C232Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1068 in 648 conjugacy classes, 436 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C25, C22×C22⋊C4, C2×C22⋊Q8, C232Q8, C2×C232Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, C25, C232Q8, Q8×C23, C2×2+ 1+4, C2×C232Q8

Smallest permutation representation of C2×C232Q8
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 23)(6 24)(7 21)(8 22)(13 17)(14 18)(15 19)(16 20)(25 32)(26 29)(27 30)(28 31)
(1 3)(2 21)(4 23)(5 12)(6 8)(7 10)(9 11)(13 28)(14 16)(15 26)(17 31)(18 20)(19 29)(22 24)(25 27)(30 32)
(13 26)(14 27)(15 28)(16 25)(17 29)(18 30)(19 31)(20 32)
(1 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 26)(14 27)(15 28)(16 25)(17 29)(18 30)(19 31)(20 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 19 3 17)(2 18 4 20)(5 27 7 25)(6 26 8 28)(9 15 11 13)(10 14 12 16)(21 32 23 30)(22 31 24 29)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4X order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 2 4 type + + + + - + image C1 C2 C2 C2 Q8 2+ 1+4 kernel C2×C23⋊2Q8 C22×C22⋊C4 C2×C22⋊Q8 C23⋊2Q8 C24 C22 # reps 1 3 12 16 8 4

Matrix representation of C2×C232Q8 in GL8(𝔽5)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 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 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 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 4 0 0 0 0 0 0 0 0 4
,
 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 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0
,
 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0] >;

C2×C232Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2Q_8
% in TeX

G:=Group("C2xC2^3:2Q8");
// GroupNames label

G:=SmallGroup(128,2188);
// by ID

G=gap.SmallGroup(128,2188);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,387,352,1123]);
// Polycyclic

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

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