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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 [×6], C2 [×12], C4 [×24], C22, C22 [×18], C22 [×52], C2×C4 [×24], C2×C4 [×48], Q8 [×16], C23, C23 [×34], C23 [×36], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×36], C22×C4 [×12], C2×Q8 [×16], C2×Q8 [×8], C24 [×15], C24 [×4], C2×C22⋊C4 [×36], C2×C4⋊C4 [×12], C22⋊Q8 [×96], C23×C4 [×6], C22×Q8 [×4], C25, C22×C22⋊C4 [×3], C2×C22⋊Q8 [×12], C232Q8 [×16], C2×C232Q8
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C24 [×31], C22×Q8 [×14], 2+ 1+4 [×4], C25, C232Q8 [×4], Q8×C23, C2×2+ 1+4 [×2], 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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