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## G = He3⋊1M4(2)  order 432 = 24·33

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for He31M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=a-1b-1c-1, eae=a-1b, bc=cb, bd=db, be=eb, dcd-1=a-1bc, ece=c-1, ede=d5 >

Subgroups: 329 in 62 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C24, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, C3×M4(2), He3⋊C2, C2×He3, S3×C12, He33C4, C4×He3, C2×He3⋊C2, He32C8, C4×He3⋊C2, He31M4(2)
Quotients: C1, C2, C4, C22, C2×C4, M4(2), C32⋊C4, C2×C32⋊C4, He3⋊C4, C32⋊M4(2), C2×He3⋊C4, He31M4(2)

Smallest permutation representation of He31M4(2)
On 72 points
Generators in S72
(1 27 68)(2 37 62)(3 63 24)(4 53 30)(5 31 72)(6 33 58)(7 59 20)(8 49 26)(9 52 29)(10 71 39)(11 40 57)(12 19 55)(13 56 25)(14 67 35)(15 36 61)(16 23 51)(17 48 64)(18 54 41)(21 44 60)(22 50 45)(28 69 46)(32 65 42)(34 43 66)(38 47 70)
(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 30 39)(18 31 40)(19 32 33)(20 25 34)(21 26 35)(22 27 36)(23 28 37)(24 29 38)(49 67 60)(50 68 61)(51 69 62)(52 70 63)(53 71 64)(54 72 57)(55 65 58)(56 66 59)
(1 27 61)(3 63 29)(5 31 57)(7 59 25)(9 52 38)(11 40 54)(13 56 34)(15 36 50)(17 39 30)(18 72 41)(19 32 33)(20 43 66)(21 35 26)(22 68 45)(23 28 37)(24 47 70)(49 67 60)(51 62 69)(53 71 64)(55 58 65)
(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 67)(18 72)(19 69)(20 66)(21 71)(22 68)(23 65)(24 70)(25 59)(26 64)(27 61)(28 58)(29 63)(30 60)(31 57)(32 62)(33 51)(34 56)(35 53)(36 50)(37 55)(38 52)(39 49)(40 54)(42 46)(44 48)

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

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

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

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 1 1 12 12 2 9 9 1 1 12 12 18 18 18 18 18 18 2 2 9 9 9 9 12 12 12 12 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 C32⋊M4(2) He3⋊1M4(2) kernel He3⋊1M4(2) He3⋊2C8 C4×He3⋊C2 C4×He3 C2×He3⋊C2 He3 C4 C2 C12 C6 C3 C1 # reps 1 2 1 2 2 2 8 8 2 2 4 4

Matrix representation of He31M4(2) in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 1 63 0 0 0 0 72 1 0 0 0 72 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 8 66 0 0 0 9 65 64 0 0 8 65 0
,
 0 1 0 0 0 46 0 0 0 0 0 0 8 9 65 0 0 9 0 64 0 0 0 9 64
,
 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 1 0 64 0 0 65 8 0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,63,72,72,0,0,0,1,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,8,9,8,0,0,66,65,65,0,0,0,64,0],[0,46,0,0,0,1,0,0,0,0,0,0,8,9,0,0,0,9,0,9,0,0,65,64,64],[1,0,0,0,0,0,72,0,0,0,0,0,1,1,65,0,0,0,0,8,0,0,0,64,0] >;

He31M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_1M_4(2)
% in TeX

G:=Group("He3:1M4(2)");
// GroupNames label

G:=SmallGroup(432,274);
// by ID

G=gap.SmallGroup(432,274);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,64,58,3924,298,5381,2539,537]);
// 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^-1*b^-1*c^-1,e*a*e=a^-1*b,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1*b*c,e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

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