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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

Aliases: C2×C23.9D4, C24.8Q8, C24.153D4, C25.6C22, C23.30C42, C24.161C23, (C23×C4)⋊2C4, C24.46(C2×C4), C23.50(C2×Q8), C23.63(C4⋊C4), C23.538(C2×D4), C22.4(C2×C42), C23.62(C22×C4), C23.70(C22⋊C4), C22.47(C23⋊C4), C22.31(C2.C42), (C22×C4)⋊5(C2×C4), C2.4(C2×C23⋊C4), C22⋊C433(C2×C4), (C2×C22⋊C4)⋊13C4, C22.12(C2×C4⋊C4), (C22×C22⋊C4).6C2, C22.26(C2×C22⋊C4), C2.16(C2×C2.C42), (C2×C22⋊C4).406C22, SmallGroup(128,471)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C23.9D4
C1C2C22C23C24C25C22×C22⋊C4 — C2×C23.9D4
C1C2C22 — C2×C23.9D4
C1C23C25 — C2×C23.9D4
C1C2C24 — 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···2G2H···2S4A···4X
order12···22···24···4
size11···12···24···4

44 irreducible representations

dim11111224
type++++-+
imageC1C2C2C4C4D4Q8C23⋊C4
kernelC2×C23.9D4C23.9D4C22×C22⋊C4C2×C22⋊C4C23×C4C24C24C22
# reps143204624

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

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00000100
00001000
00001143
00000001
,
40000000
04000000
00400000
00040000
00000100
00001000
00004412
00001104
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
04000000
40000000
00040000
00100000
00000010
00001143
00001000
00000004
,
10000000
04000000
00300000
00020000
00000400
00001000
00000010
00001044

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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