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## G = He3⋊8M4(2)  order 432 = 24·33

### 2nd semidirect product of He3 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊8M4(2)
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4C8 — He3⋊8M4(2)
 Lower central He3 — C2×He3 — He3⋊8M4(2)
 Upper central C1 — C12 — C2×C12

Generators and relations for He38M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede=d5 >

Subgroups: 281 in 110 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, He3, C3×C12, C62, C4.Dic3, C3×M4(2), C2×He3, C2×He3, C3×C3⋊C8, C6×C12, C4×He3, C22×He3, C3×C4.Dic3, He34C8, C2×C4×He3, He38M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), C3⋊S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4.Dic3, He3⋊C2, C2×C3⋊Dic3, He33C4, C2×He3⋊C2, C12.58D6, C2×He33C4, He38M4(2)

Smallest permutation representation of He38M4(2)
On 72 points
Generators in S72
(1 59 47)(2 48 60)(3 61 41)(4 42 62)(5 63 43)(6 44 64)(7 57 45)(8 46 58)(9 51 26)(10 27 52)(11 53 28)(12 29 54)(13 55 30)(14 31 56)(15 49 32)(16 25 50)(17 38 70)(18 71 39)(19 40 72)(20 65 33)(21 34 66)(22 67 35)(23 36 68)(24 69 37)
(1 25 18)(2 26 19)(3 27 20)(4 28 21)(5 29 22)(6 30 23)(7 31 24)(8 32 17)(9 40 48)(10 33 41)(11 34 42)(12 35 43)(13 36 44)(14 37 45)(15 38 46)(16 39 47)(49 70 58)(50 71 59)(51 72 60)(52 65 61)(53 66 62)(54 67 63)(55 68 64)(56 69 57)
(1 59 16)(2 9 60)(3 61 10)(4 11 62)(5 63 12)(6 13 64)(7 57 14)(8 15 58)(17 46 70)(18 71 47)(19 48 72)(20 65 41)(21 42 66)(22 67 43)(23 44 68)(24 69 45)(25 50 39)(26 40 51)(27 52 33)(28 34 53)(29 54 35)(30 36 55)(31 56 37)(32 38 49)
(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)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 type + + + + - + - image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 He3⋊C2 He3⋊3C4 C2×He3⋊C2 He3⋊3C4 He3⋊8M4(2) kernel He3⋊8M4(2) He3⋊4C8 C2×C4×He3 C4×He3 C22×He3 C6×C12 C3×C12 C3×C12 C62 He3 C32 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 4 4 4 4 2 16 4 4 4 4 4

Matrix representation of He38M4(2) in GL5(𝔽73)

 8 0 0 0 0 49 64 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 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 8 0 0 1 0 0
,
 3 66 0 0 0 60 70 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0
,
 1 0 0 0 0 53 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72

G:=sub<GL(5,GF(73))| [8,49,0,0,0,0,64,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,64,0,0,0,0,0,8,0],[3,60,0,0,0,66,70,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0],[1,53,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72] >;

He38M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_8M_4(2)
% in TeX

G:=Group("He3:8M4(2)");
// GroupNames label

G:=SmallGroup(432,185);
// by ID

G=gap.SmallGroup(432,185);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,58,1124,4037,537]);
// 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^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^5>;
// generators/relations

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