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## G = He3⋊7M4(2)  order 432 = 24·33

### 1st 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×C6 — He3⋊7M4(2)
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3C8 — He3⋊7M4(2)
 Lower central C32 — C3×C6 — He3⋊7M4(2)
 Upper central C1 — C4 — C2×C4

Generators and relations for He37M4(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, dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 257 in 86 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C3 [×3], C4 [×2], C22, C6, C6 [×8], C8 [×2], C2×C4, C32 [×2], C32, C12 [×2], C12 [×6], C2×C6, C2×C6 [×3], M4(2), C3×C6 [×2], C3×C6 [×5], C3⋊C8 [×4], C24 [×2], C2×C12, C2×C12 [×3], He3, C3×C12 [×4], C3×C12 [×2], C62 [×2], C62, C4.Dic3 [×2], C3×M4(2), C2×He3, C2×He3, C3×C3⋊C8 [×2], C324C8 [×2], C6×C12 [×2], C6×C12, C4×He3 [×2], C22×He3, C3×C4.Dic3, C12.58D6, He33C8 [×2], C2×C4×He3, He37M4(2)
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, M4(2), C3×S3, C2×Dic3, C2×C12, C3×Dic3 [×2], S3×C6, C4.Dic3, C3×M4(2), C32⋊C6, C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C3×C4.Dic3, C2×C32⋊C12, He37M4(2)

Smallest permutation representation of He37M4(2)
On 72 points
Generators in S72
(1 68 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 22 27)(10 28 23)(11 24 29)(12 30 17)(13 18 31)(14 32 19)(15 20 25)(16 26 21)(33 59 56)(34 49 60)(35 61 50)(36 51 62)(37 63 52)(38 53 64)(39 57 54)(40 55 58)
(1 61 18)(2 19 62)(3 63 20)(4 21 64)(5 57 22)(6 23 58)(7 59 24)(8 17 60)(9 43 39)(10 40 44)(11 45 33)(12 34 46)(13 47 35)(14 36 48)(15 41 37)(16 38 42)(25 70 52)(26 53 71)(27 72 54)(28 55 65)(29 66 56)(30 49 67)(31 68 50)(32 51 69)
(1 47 31)(2 48 32)(3 41 25)(4 42 26)(5 43 27)(6 44 28)(7 45 29)(8 46 30)(9 54 22)(10 55 23)(11 56 24)(12 49 17)(13 50 18)(14 51 19)(15 52 20)(16 53 21)(33 66 59)(34 67 60)(35 68 61)(36 69 62)(37 70 63)(38 71 64)(39 72 57)(40 65 58)
(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)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 type + + + + - + - + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) C3×C4.Dic3 C32⋊C6 C32⋊C12 C2×C32⋊C6 C32⋊C12 He3⋊7M4(2) kernel He3⋊7M4(2) He3⋊3C8 C2×C4×He3 C12.58D6 C4×He3 C22×He3 C32⋊4C8 C6×C12 C3×C12 C62 C6×C12 C3×C12 C3×C12 C62 He3 C2×C12 C12 C12 C2×C6 C32 C32 C3 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 1 1 1 2 2 2 2 2 4 4 8 1 1 1 1 4

Matrix representation of He37M4(2) in GL6(𝔽73)

 1 7 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 7 0 0 0 0 0 72 0 0 0 0 1 72
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64
,
 8 0 56 0 0 0 72 0 65 0 0 0 8 64 65 0 0 0 0 0 0 8 0 56 0 0 0 72 0 65 0 0 0 8 64 65
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,7,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,7,72,72],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[8,72,8,0,0,0,0,0,64,0,0,0,56,65,65,0,0,0,0,0,0,8,72,8,0,0,0,0,0,64,0,0,0,56,65,65],[0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

He37M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_7M_4(2)
% in TeX

G:=Group("He3:7M4(2)");
// GroupNames label

G:=SmallGroup(432,137);
// by ID

G=gap.SmallGroup(432,137);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,4037,2035,14118]);
// 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,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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