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G = C22×C23⋊C4  order 128 = 27

Direct product of C22 and C23⋊C4

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

Aliases: C22×C23⋊C4, C25⋊3C4, C24.167D4, C23.1C24, C25.66C22, C24.468C23, C24⋊6(C2×C4), (C23×C4)⋊10C4, (C22×D4)⋊24C4, C23⋊3(C22×C4), C22⋊C4⋊13C23, (D4×C23).13C2, C23.635(C2×D4), (C2×D4).349C23, C22.10(C23×C4), C22.23(C22×D4), C23.128(C22⋊C4), (C22×D4).546C22, (C2×D4)⋊43(C2×C4), (C22×C4)⋊7(C2×C4), (C2×C4)⋊3(C22×C4), (C22×C22⋊C4)⋊13C2, (C2×C22⋊C4)⋊76C22, C22.77(C2×C22⋊C4), C2.24(C22×C22⋊C4), SmallGroup(128,1613)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

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

Generators and relations for C22×C23⋊C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 1228 in 556 conjugacy classes, 180 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C22×D4, C22×D4, C25, C2×C23⋊C4, C22×C22⋊C4, D4×C23, C22×C23⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C23⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C2×C23⋊C4, C22×C22⋊C4, C22×C23⋊C4

Smallest permutation representation of C22×C23⋊C4
►On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 12)(6 9)(7 10)(8 11)(13 29)(14 30)(15 31)(16 32)(17 24)(18 21)(19 22)(20 23)

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

(1 14)(2 31)(3 25)(5 12)(7 23)(8 17)(10 20)(11 24)(15 28)(16 32)(18 21)(27 30)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2S2T2U2V2W4A···4T
order12···22···222224···4
size11···12···244444···4

44 irreducible representations

dim111111124
type++++++
imageC1C2C2C2C4C4C4D4C23⋊C4
kernelC22×C23⋊C4C2×C23⋊C4C22×C22⋊C4D4×C23C23×C4C22×D4C25C24C22
# reps11221212284

Matrix representation of C22×C23⋊C4 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
41000000
00400000
00010000
00001000
00000400
00000010
00002014
,
40000000
04000000
00400000
00040000
00001000
00000100
00000040
00002304
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
34000000
02000000
00030000
00200000
00000010
00002313
00000100
00000002

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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[3,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,2] >;
 

C22×C23⋊C4 in GAP, Magma, Sage, TeX 

C_2^2\times C_2^3\rtimes C_4
 
% in TeX
 
G:=Group("C2^2xC2^3:C4");
 
// GroupNames label
 
G:=SmallGroup(128,1613);
 
// by ID
 
G=gap.SmallGroup(128,1613);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
 
// generators/relations
 

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