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## G = He3⋊3M4(2)  order 432 = 24·33

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊3M4(2)
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊3C8 — He3⋊3M4(2)
 Lower central He3 — C2×He3 — He3⋊3M4(2)
 Upper central C1 — C4

Generators and relations for He33M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece=c-1, ede=d5 >

Subgroups: 403 in 86 conjugacy classes, 23 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, C3×C12, S3×C6, C8⋊S3, C4.Dic3, He3⋊C2, C2×He3, C3×C3⋊C8, C324C8, S3×C12, He33C4, C4×He3, C2×He3⋊C2, D6.Dic3, He33C8, C4×He3⋊C2, He33M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), C4×S3, S32, C8⋊S3, C6.D6, C32⋊D6, C12.31D6, He3⋊(C2×C4), He33M4(2)

Smallest permutation representation of He33M4(2)
On 72 points
Generators in S72
(1 22 61)(2 62 23)(3 24 63)(4 64 17)(5 18 57)(6 58 19)(7 20 59)(8 60 21)(9 70 52)(10 53 71)(11 72 54)(12 55 65)(13 66 56)(14 49 67)(15 68 50)(16 51 69)(25 39 48)(26 41 40)(27 33 42)(28 43 34)(29 35 44)(30 45 36)(31 37 46)(32 47 38)
(1 15 45)(2 46 16)(3 9 47)(4 48 10)(5 11 41)(6 42 12)(7 13 43)(8 44 14)(17 39 71)(18 72 40)(19 33 65)(20 66 34)(21 35 67)(22 68 36)(23 37 69)(24 70 38)(25 53 64)(26 57 54)(27 55 58)(28 59 56)(29 49 60)(30 61 50)(31 51 62)(32 63 52)
(1 50 68)(2 51 69)(3 52 70)(4 53 71)(5 54 72)(6 55 65)(7 56 66)(8 49 67)(9 32 38)(10 25 39)(11 26 40)(12 27 33)(13 28 34)(14 29 35)(15 30 36)(16 31 37)(17 48 64)(18 41 57)(19 42 58)(20 43 59)(21 44 60)(22 45 61)(23 46 62)(24 47 63)
(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 60)(18 57)(19 62)(20 59)(21 64)(22 61)(23 58)(24 63)(25 35)(26 40)(27 37)(28 34)(29 39)(30 36)(31 33)(32 38)(42 46)(44 48)(49 71)(50 68)(51 65)(52 70)(53 67)(54 72)(55 69)(56 66)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 4 6 6 6 type + + + + + + + + - image C1 C2 C2 C4 C4 S3 D6 M4(2) C4×S3 C8⋊S3 S32 C6.D6 C12.31D6 C32⋊D6 He3⋊(C2×C4) He3⋊3M4(2) kernel He3⋊3M4(2) He3⋊3C8 C4×He3⋊C2 He3⋊3C4 C2×He3⋊C2 C32⋊4C8 C3×C12 He3 C3×C6 C32 C12 C6 C3 C4 C2 C1 # reps 1 2 1 2 2 2 2 2 4 8 1 1 2 2 2 4

Matrix representation of He33M4(2) in GL6(𝔽73)

 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 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
,
 0 0 64 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 8 0 0
,
 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 46 0 0 0 0 0 0 0 46 0 0 0 0 46 0 0 0 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 72 0

G:=sub<GL(6,GF(73))| [0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[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],[0,8,0,0,0,0,0,0,1,0,0,0,64,0,0,0,0,0,0,0,0,0,0,8,0,0,0,64,0,0,0,0,0,0,1,0],[0,0,0,46,0,0,0,0,0,0,0,46,0,0,0,0,46,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

He33M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3M_4(2)
% in TeX

G:=Group("He3:3M4(2)");
// GroupNames label

G:=SmallGroup(432,82);
// by ID

G=gap.SmallGroup(432,82);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,58,571,4037,537,14118,7069]);
// 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=e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

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