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G = C23⋊C42  order 128 = 27

2nd semidirect product of C23 and C42 acting via C42/C22=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23⋊3C42, C25.13C22, C24.523C23, C23.155C24, C22.262+ 1+4, C24.64(C2×C4), (C2×C42)⋊1C22, C2.8(C22×C42), (C23×C4).30C22, C22.27(C23×C4), C22.14(C2×C42), C23.204(C22×C4), C2.C42⋊70C22, (C22×C4).1231C23, C2.1(C22.11C24), (C4×C22⋊C4)⋊2C2, (C2×C22⋊C4)⋊16C4, C22⋊C4⋊45(C2×C4), (C22×C4)⋊11(C2×C4), C22⋊C4○2(C22⋊C4), (C2×C4).288(C22×C4), (C22×C22⋊C4).8C2, (C2×C22⋊C4).551C22, C2.C42○(C2.C42), C22⋊C4○(C2×C22⋊C4), SmallGroup(128,1005)

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

Derived series
C1 — C2 — C23⋊C42
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23⋊C42
Lower central
C1 — C2 — C23⋊C42
Upper central
C1 — C23 — C23⋊C42
Jennings
C1 — C23 — C23⋊C42

Generators and relations for C23⋊C42
 G = < a,b,c,d,e | a2=b2=c2=d4=e4=1, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 828 in 456 conjugacy classes, 260 normal (5 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C25, C4×C22⋊C4, C22×C22⋊C4, C23⋊C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, 2+ 1+4, C22×C42, C22.11C24, C23⋊C42

Smallest permutation representation of C23⋊C42
►On 32 points
Generators in S32
(1 3)(2 4)(5 23)(6 24)(7 21)(8 22)(9 11)(10 12)(13 15)(14 16)(17 30)(18 31)(19 32)(20 29)(25 27)(26 28)

(2 14)(4 16)(6 22)(8 24)(10 28)(12 26)(17 32)(19 30)

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

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

68 conjugacy classes

class 1 2A···2G2H···2S4A···4AV
order12···22···24···4
size11···12···22···2

68 irreducible representations

dim11114
type++++
imageC1C2C2C42+ 1+4
kernelC23⋊C42C4×C22⋊C4C22×C22⋊C4C2×C22⋊C4C22
# reps1123484

Matrix representation of C23⋊C42 ►in GL6(𝔽5)

400000
010000
004000
000100
000040
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
020000
000040
000001
001000
000400
,
200000
030000
000100
004000
000004
000010

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
 

C23⋊C42 in GAP, Magma, Sage, TeX 

C_2^3\rtimes C_4^2
 
% in TeX
 
G:=Group("C2^3:C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,1005);
 
// by ID
 
G=gap.SmallGroup(128,1005);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,219,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=e^4=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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