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G = C24⋊He3  order 432 = 24·33

The semidirect product of C24 and He3 acting via He3/C3=C32

metabelian, soluble, monomial

Aliases: C24⋊He3, (C3×A4)⋊A4, C3.3A42, C22⋊1(C32⋊A4), (C23×C6).3C32, (A4×C2×C6)⋊1C3, (C2×C6).3(C3×A4), (C3×C22⋊A4)⋊1C3, SmallGroup(432,526)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23×C6 — C24⋊He3
Chief series
C1 — C22 — C24 — C23×C6 — A4×C2×C6 — C24⋊He3
Lower central
C24 — C23×C6 — C24⋊He3
Upper central
C1 — C3

Generators and relations for C24⋊He3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e3=f3=g3=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=b, af=fa, ag=ga, bc=cb, bd=db, bf=fb, bg=gb, gdg-1=cd=dc, ce=ec, cf=fc, gcg-1=d, de=ed, df=fd, ef=fe, geg-1=ef-1, fg=gf >

Subgroups: 655 in 90 conjugacy classes, 15 normal (5 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C32, A4, C2×C6, C2×C6, C24, C3×C6, C2×A4, C22×C6, He3, C3×A4, C3×A4, C62, C22×A4, C22⋊A4, C23×C6, C6×A4, C32⋊A4, A4×C2×C6, C3×C22⋊A4, C24⋊He3
Quotients: C1, C3, C32, A4, He3, C3×A4, C32⋊A4, A42, C24⋊He3

Smallest permutation representation of C24⋊He3
►On 36 points
Generators in S36
(1 17)(2 24)(3 19)(4 25)(5 14)(6 7)(8 13)(9 26)(10 28)(11 32)(12 36)(15 27)(16 22)(18 21)(20 23)(29 35)(30 33)(31 34)

(1 23)(2 21)(3 16)(4 13)(5 9)(6 27)(7 15)(8 25)(10 31)(11 35)(12 30)(14 26)(17 20)(18 24)(19 22)(28 34)(29 32)(33 36)

(1 23)(2 24)(3 22)(4 25)(5 26)(6 27)(7 15)(8 13)(9 14)(10 34)(11 35)(12 36)(16 19)(17 20)(18 21)(28 31)(29 32)(30 33)

(1 20)(2 21)(3 19)(4 13)(5 14)(6 15)(7 27)(8 25)(9 26)(10 28)(11 29)(12 30)(16 22)(17 23)(18 24)(31 34)(32 35)(33 36)

(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 33)(34 35 36)

(1 15 32)(2 13 33)(3 14 31)(4 36 21)(5 34 19)(6 35 20)(7 29 23)(8 30 24)(9 28 22)(10 16 26)(11 17 27)(12 18 25)

(1 3 33)(2 15 14)(4 29 10)(5 24 27)(6 9 18)(7 26 21)(8 11 34)(12 20 22)(13 32 31)(16 36 23)(17 19 30)(25 35 28)
 
G:=sub<Sym(36)| (1,17)(2,24)(3,19)(4,25)(5,14)(6,7)(8,13)(9,26)(10,28)(11,32)(12,36)(15,27)(16,22)(18,21)(20,23)(29,35)(30,33)(31,34), (1,23)(2,21)(3,16)(4,13)(5,9)(6,27)(7,15)(8,25)(10,31)(11,35)(12,30)(14,26)(17,20)(18,24)(19,22)(28,34)(29,32)(33,36), (1,23)(2,24)(3,22)(4,25)(5,26)(6,27)(7,15)(8,13)(9,14)(10,34)(11,35)(12,36)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,20)(2,21)(3,19)(4,13)(5,14)(6,15)(7,27)(8,25)(9,26)(10,28)(11,29)(12,30)(16,22)(17,23)(18,24)(31,34)(32,35)(33,36), (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)(31,32,33)(34,35,36), (1,15,32)(2,13,33)(3,14,31)(4,36,21)(5,34,19)(6,35,20)(7,29,23)(8,30,24)(9,28,22)(10,16,26)(11,17,27)(12,18,25), (1,3,33)(2,15,14)(4,29,10)(5,24,27)(6,9,18)(7,26,21)(8,11,34)(12,20,22)(13,32,31)(16,36,23)(17,19,30)(25,35,28)>;
 
G:=Group( (1,17)(2,24)(3,19)(4,25)(5,14)(6,7)(8,13)(9,26)(10,28)(11,32)(12,36)(15,27)(16,22)(18,21)(20,23)(29,35)(30,33)(31,34), (1,23)(2,21)(3,16)(4,13)(5,9)(6,27)(7,15)(8,25)(10,31)(11,35)(12,30)(14,26)(17,20)(18,24)(19,22)(28,34)(29,32)(33,36), (1,23)(2,24)(3,22)(4,25)(5,26)(6,27)(7,15)(8,13)(9,14)(10,34)(11,35)(12,36)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,20)(2,21)(3,19)(4,13)(5,14)(6,15)(7,27)(8,25)(9,26)(10,28)(11,29)(12,30)(16,22)(17,23)(18,24)(31,34)(32,35)(33,36), (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)(31,32,33)(34,35,36), (1,15,32)(2,13,33)(3,14,31)(4,36,21)(5,34,19)(6,35,20)(7,29,23)(8,30,24)(9,28,22)(10,16,26)(11,17,27)(12,18,25), (1,3,33)(2,15,14)(4,29,10)(5,24,27)(6,9,18)(7,26,21)(8,11,34)(12,20,22)(13,32,31)(16,36,23)(17,19,30)(25,35,28) );
 
G=PermutationGroup([[(1,17),(2,24),(3,19),(4,25),(5,14),(6,7),(8,13),(9,26),(10,28),(11,32),(12,36),(15,27),(16,22),(18,21),(20,23),(29,35),(30,33),(31,34)], [(1,23),(2,21),(3,16),(4,13),(5,9),(6,27),(7,15),(8,25),(10,31),(11,35),(12,30),(14,26),(17,20),(18,24),(19,22),(28,34),(29,32),(33,36)], [(1,23),(2,24),(3,22),(4,25),(5,26),(6,27),(7,15),(8,13),(9,14),(10,34),(11,35),(12,36),(16,19),(17,20),(18,21),(28,31),(29,32),(30,33)], [(1,20),(2,21),(3,19),(4,13),(5,14),(6,15),(7,27),(8,25),(9,26),(10,28),(11,29),(12,30),(16,22),(17,23),(18,24),(31,34),(32,35),(33,36)], [(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),(31,32,33),(34,35,36)], [(1,15,32),(2,13,33),(3,14,31),(4,36,21),(5,34,19),(6,35,20),(7,29,23),(8,30,24),(9,28,22),(10,16,26),(11,17,27),(12,18,25)], [(1,3,33),(2,15,14),(4,29,10),(5,24,27),(6,9,18),(7,26,21),(8,11,34),(12,20,22),(13,32,31),(16,36,23),(17,19,30),(25,35,28)]])
 

32 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F3G3H3I3J6A6B6C6D6E6F6G···6R
order122233333333336666666···6
size133911121212124848484833339912···12

32 irreducible representations

dim111333399
type+++
imageC1C3C3A4He3C3×A4C32⋊A4A42C24⋊He3
kernelC24⋊He3A4×C2×C6C3×C22⋊A4C3×A4C24C2×C6C22C3C1
# reps1442241212

Matrix representation of C24⋊He3 ►in GL9(𝔽7)

100000000
010000000
001000000
000010000
000100000
000666000
000000100
000000010
000000001
,
100000000
010000000
001000000
000001000
000666000
000100000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000001000
000000010
000000100
000000666
,
100000000
010000000
001000000
000100000
000010000
000001000
000000001
000000666
000000100
,
010000000
663000000
001000000
000200000
000002000
000555000
000000100
000000010
000000001
,
400000000
040000000
004000000
000100000
000010000
000001000
000000100
000000010
000000001
,
335000000
100000000
644000000
000400000
000040000
000004000
000000100
000000001
000000666

G:=sub<GL(9,GF(7))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,6,0],[0,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,2,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[3,1,6,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,5,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,6] >;
 

C24⋊He3 in GAP, Magma, Sage, TeX 

C_2^4\rtimes {\rm He}_3
 
% in TeX
 
G:=Group("C2^4:He3");
 
// GroupNames label
 
G:=SmallGroup(432,526);
 
// by ID
 
G=gap.SmallGroup(432,526);
 
# by ID
 
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,169,766,326,13613,5298]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^3=f^3=g^3=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=b,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,g*d*g^-1=c*d=d*c,c*e=e*c,c*f=f*c,g*c*g^-1=d,d*e=e*d,d*f=f*d,e*f=f*e,g*e*g^-1=e*f^-1,f*g=g*f>;
 
// generators/relations
 

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