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G = C2×C23.9D4  order 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⋊C4⋊33(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

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, C2, C4, C22, C22, C22, C2×C4, C23, C23, C23, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C25, C23.9D4, C22×C22⋊C4, C22×C22⋊C4, C2×C23.9D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C23⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23.9D4, C2×C2.C42, C2×C23⋊C4, 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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