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## G = He3⋊4M4(2)  order 432 = 24·33

### The semidirect product of He3 and M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊4M4(2)
 Chief series C1 — C3 — He3 — C2×He3 — He3⋊3C4 — He3⋊2C8 — He3⋊4M4(2)
 Lower central He3 — C2×He3 — He3⋊4M4(2)
 Upper central C1 — C6 — C2×C6

Generators and relations for He34M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ac-1, ce=ec, ede=d5 >

Subgroups: 281 in 62 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, M4(2), C3×C6, C24, C2×Dic3, C2×C12, He3, C3×Dic3, C62, C3×M4(2), C2×He3, C2×He3, C6×Dic3, He33C4, C22×He3, He32C8, C2×He33C4, He34M4(2)
Quotients: C1, C2, C4, C22, C2×C4, M4(2), C32⋊C4, C2×C32⋊C4, He3⋊C4, C62.C4, C2×He3⋊C4, He34M4(2)

Smallest permutation representation of He34M4(2)
On 72 points
Generators in S72
(1 61 22)(2 31 62)(3 37 69)(4 53 38)(5 57 18)(6 27 58)(7 33 65)(8 49 34)(9 32 63)(10 70 25)(11 54 39)(12 19 55)(13 28 59)(14 66 29)(15 50 35)(16 23 51)(17 48 64)(20 56 43)(21 44 60)(24 52 47)(26 41 71)(30 45 67)(36 68 46)(40 72 42)
(1 15 45)(2 16 46)(3 9 47)(4 10 48)(5 11 41)(6 12 42)(7 13 43)(8 14 44)(17 38 25)(18 39 26)(19 40 27)(20 33 28)(21 34 29)(22 35 30)(23 36 31)(24 37 32)(49 66 60)(50 67 61)(51 68 62)(52 69 63)(53 70 64)(54 71 57)(55 72 58)(56 65 59)
(1 61 35)(3 37 63)(5 57 39)(7 33 59)(9 32 52)(11 54 26)(13 28 56)(15 50 30)(17 25 38)(18 41 71)(19 40 27)(20 65 43)(21 29 34)(22 45 67)(23 36 31)(24 69 47)(49 66 60)(51 62 68)(53 70 64)(55 58 72)
(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 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)

G:=sub<Sym(72)| (1,61,22)(2,31,62)(3,37,69)(4,53,38)(5,57,18)(6,27,58)(7,33,65)(8,49,34)(9,32,63)(10,70,25)(11,54,39)(12,19,55)(13,28,59)(14,66,29)(15,50,35)(16,23,51)(17,48,64)(20,56,43)(21,44,60)(24,52,47)(26,41,71)(30,45,67)(36,68,46)(40,72,42), (1,15,45)(2,16,46)(3,9,47)(4,10,48)(5,11,41)(6,12,42)(7,13,43)(8,14,44)(17,38,25)(18,39,26)(19,40,27)(20,33,28)(21,34,29)(22,35,30)(23,36,31)(24,37,32)(49,66,60)(50,67,61)(51,68,62)(52,69,63)(53,70,64)(54,71,57)(55,72,58)(56,65,59), (1,61,35)(3,37,63)(5,57,39)(7,33,59)(9,32,52)(11,54,26)(13,28,56)(15,50,30)(17,25,38)(18,41,71)(19,40,27)(20,65,43)(21,29,34)(22,45,67)(23,36,31)(24,69,47)(49,66,60)(51,62,68)(53,70,64)(55,58,72), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)>;

G:=Group( (1,61,22)(2,31,62)(3,37,69)(4,53,38)(5,57,18)(6,27,58)(7,33,65)(8,49,34)(9,32,63)(10,70,25)(11,54,39)(12,19,55)(13,28,59)(14,66,29)(15,50,35)(16,23,51)(17,48,64)(20,56,43)(21,44,60)(24,52,47)(26,41,71)(30,45,67)(36,68,46)(40,72,42), (1,15,45)(2,16,46)(3,9,47)(4,10,48)(5,11,41)(6,12,42)(7,13,43)(8,14,44)(17,38,25)(18,39,26)(19,40,27)(20,33,28)(21,34,29)(22,35,30)(23,36,31)(24,37,32)(49,66,60)(50,67,61)(51,68,62)(52,69,63)(53,70,64)(54,71,57)(55,72,58)(56,65,59), (1,61,35)(3,37,63)(5,57,39)(7,33,59)(9,32,52)(11,54,26)(13,28,56)(15,50,30)(17,25,38)(18,41,71)(19,40,27)(20,65,43)(21,29,34)(22,45,67)(23,36,31)(24,69,47)(49,66,60)(51,62,68)(53,70,64)(55,58,72), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72) );

G=PermutationGroup([[(1,61,22),(2,31,62),(3,37,69),(4,53,38),(5,57,18),(6,27,58),(7,33,65),(8,49,34),(9,32,63),(10,70,25),(11,54,39),(12,19,55),(13,28,59),(14,66,29),(15,50,35),(16,23,51),(17,48,64),(20,56,43),(21,44,60),(24,52,47),(26,41,71),(30,45,67),(36,68,46),(40,72,42)], [(1,15,45),(2,16,46),(3,9,47),(4,10,48),(5,11,41),(6,12,42),(7,13,43),(8,14,44),(17,38,25),(18,39,26),(19,40,27),(20,33,28),(21,34,29),(22,35,30),(23,36,31),(24,37,32),(49,66,60),(50,67,61),(51,68,62),(52,69,63),(53,70,64),(54,71,57),(55,72,58),(56,65,59)], [(1,61,35),(3,37,63),(5,57,39),(7,33,59),(9,32,52),(11,54,26),(13,28,56),(15,50,30),(17,25,38),(18,41,71),(19,40,27),(20,65,43),(21,29,34),(22,45,67),(23,36,31),(24,69,47),(49,66,60),(51,62,68),(53,70,64),(55,58,72)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64),(66,70),(68,72)]])

38 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E ··· 6J 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 3 3 3 3 4 4 4 6 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 2 1 1 12 12 9 9 18 1 1 2 2 12 ··· 12 18 18 18 18 9 9 9 9 18 18 18 ··· 18

38 irreducible representations

 dim 1 1 1 1 1 2 3 3 4 4 4 6 type + + + + + - image C1 C2 C2 C4 C4 M4(2) He3⋊C4 C2×He3⋊C4 C32⋊C4 C2×C32⋊C4 C62.C4 He3⋊4M4(2) kernel He3⋊4M4(2) He3⋊2C8 C2×He3⋊3C4 He3⋊3C4 C22×He3 He3 C22 C2 C2×C6 C6 C3 C1 # reps 1 2 1 2 2 2 8 8 2 2 4 4

Matrix representation of He34M4(2) in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 64 0 0 1 0 0
,
 0 42 0 0 0 15 0 0 0 0 0 0 7 56 7 0 0 10 10 7 0 0 7 10 10
,
 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,64,0],[0,15,0,0,0,42,0,0,0,0,0,0,7,10,7,0,0,56,10,10,0,0,7,7,10],[1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

He34M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4M_4(2)
% in TeX

G:=Group("He3:4M4(2)");
// GroupNames label

G:=SmallGroup(432,278);
// by ID

G=gap.SmallGroup(432,278);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,58,3924,298,5381,2539,537]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a*c^-1,c*e=e*c,e*d*e=d^5>;
// generators/relations

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