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## G = C2×C23.9D4order 128 = 27

### Direct product of C2 and C23.9D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23.9D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C2×C23.9D4
 Lower central C1 — C2 — C22 — C2×C23.9D4
 Upper central C1 — C23 — C25 — C2×C23.9D4
 Jennings C1 — C2 — C24 — C2×C23.9D4

Generators and relations for C2×C23.9D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bde-1 >

Subgroups: 772 in 340 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×12], C22 [×3], C22 [×16], C22 [×52], C2×C4 [×48], C23 [×5], C23 [×30], C23 [×36], C22⋊C4 [×8], C22⋊C4 [×20], C22×C4 [×4], C22×C4 [×28], C24 [×3], C24 [×12], C24 [×4], C2×C22⋊C4 [×16], C2×C22⋊C4 [×10], C23×C4 [×2], C23×C4 [×2], C25, C23.9D4 [×4], C22×C22⋊C4, C22×C22⋊C4 [×2], C2×C23.9D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C23⋊C4 [×4], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C23.9D4 [×4], C2×C2.C42, C2×C23⋊C4 [×2], C2×C23.9D4

Smallest permutation representation of C2×C23.9D4
On 32 points
Generators in S32
(1 21)(2 22)(3 23)(4 24)(5 11)(6 12)(7 9)(8 10)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)
(1 11)(2 27)(3 9)(4 25)(5 21)(6 14)(7 23)(8 16)(10 30)(12 32)(13 20)(15 18)(17 22)(19 24)(26 31)(28 29)
(1 31)(2 32)(3 29)(4 30)(5 20)(6 17)(7 18)(8 19)(9 28)(10 25)(11 26)(12 27)(13 21)(14 22)(15 23)(16 24)
(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 20 31 5)(2 16)(3 18 29 7)(4 14)(6 10)(8 12)(9 23 28 15)(11 21 26 13)(17 25)(19 27)(22 30)(24 32)

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

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

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

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 1 2 2 4 type + + + + - + image C1 C2 C2 C4 C4 D4 Q8 C23⋊C4 kernel C2×C23.9D4 C23.9D4 C22×C22⋊C4 C2×C22⋊C4 C23×C4 C24 C24 C22 # reps 1 4 3 20 4 6 2 4

Matrix representation of C2×C23.9D4 in GL8(𝔽5)

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

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

C2×C23.9D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._9D_4
% in TeX

G:=Group("C2xC2^3.9D4");
// GroupNames label

G:=SmallGroup(128,471);
// by ID

G=gap.SmallGroup(128,471);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,1411]);
// Polycyclic

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

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