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## G = C24⋊He3order 432 = 24·33

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

Aliases: C24⋊He3, (C3×A4)⋊A4, C3.3A42, C221(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 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 6A 6B 6C 6D 6E 6F 6G ··· 6R order 1 2 2 2 3 3 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 ··· 6 size 1 3 3 9 1 1 12 12 12 12 48 48 48 48 3 3 3 3 9 9 12 ··· 12

32 irreducible representations

 dim 1 1 1 3 3 3 3 9 9 type + + + image C1 C3 C3 A4 He3 C3×A4 C32⋊A4 A42 C24⋊He3 kernel C24⋊He3 A4×C2×C6 C3×C22⋊A4 C3×A4 C24 C2×C6 C22 C3 C1 # reps 1 4 4 2 2 4 12 1 2

Matrix representation of C24⋊He3 in GL9(𝔽7)

 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 6 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 0 1 0 0 0 0 0 0 6 6 6 0 0 0 0 0 0 1 0 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 6 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 0 1 0 0 0 0 0 0 6 6 6 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 0 0 0 6 6 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 5 5 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 3 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 4 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 6 6 6

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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