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G = C23⋊C42order 128 = 27

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

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

Aliases: C233C42, 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.C4270C22, (C22×C4).1231C23, C2.1(C22.11C24), (C4×C22⋊C4)⋊2C2, (C2×C22⋊C4)⋊16C4, C22⋊C445(C2×C4), (C22×C4)⋊11(C2×C4), C22⋊C42(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

C1C2 — C23⋊C42
C1C2C22C23C24C25C22×C22⋊C4 — C23⋊C42
C1C2 — C23⋊C42
C1C23 — C23⋊C42
C1C23 — 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 [×6], C2 [×12], C4 [×24], C22 [×19], C22 [×52], C2×C4 [×24], C2×C4 [×48], C23, C23 [×34], C23 [×36], C42 [×8], C22⋊C4 [×48], C22×C4 [×36], C22×C4 [×12], C24 [×15], C24 [×4], C2.C42 [×8], C2×C42 [×8], C2×C22⋊C4 [×36], C23×C4 [×6], C25, C4×C22⋊C4 [×12], C22×C22⋊C4 [×3], C23⋊C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], 2+ 1+4 [×4], C22×C42, C22.11C24 [×6], C23⋊C42

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

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

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

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

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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