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## G = He3⋊5M4(2)  order 432 = 24·33

### 1st semidirect product of He3 and M4(2) acting via M4(2)/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — He3⋊5M4(2)
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — C4×C32⋊C6 — He3⋊5M4(2)
 Lower central C32 — C3×C6 — He3⋊5M4(2)
 Upper central C1 — C4 — C8

Generators and relations for He35M4(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=ebe=b-1, cd=dc, ce=ec, ede=d5 >

Subgroups: 321 in 78 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C3×M4(2), C32⋊C6, C2×He3, C3×C3⋊C8, C324C8, C3×C24, C3×C24, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C3×C8⋊S3, C24⋊S3, He33C8, C8×He3, C4×C32⋊C6, He35M4(2)
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), C3×S3, C4×S3, C2×C12, S3×C6, C8⋊S3, C3×M4(2), C32⋊C6, S3×C12, C2×C32⋊C6, C3×C8⋊S3, C4×C32⋊C6, He35M4(2)

Smallest permutation representation of He35M4(2)
On 72 points
Generators in S72
(9 58 17)(10 18 59)(11 60 19)(12 20 61)(13 62 21)(14 22 63)(15 64 23)(16 24 57)(33 42 55)(34 56 43)(35 44 49)(36 50 45)(37 46 51)(38 52 47)(39 48 53)(40 54 41)
(1 25 67)(2 68 26)(3 27 69)(4 70 28)(5 29 71)(6 72 30)(7 31 65)(8 66 32)(9 58 17)(10 18 59)(11 60 19)(12 20 61)(13 62 21)(14 22 63)(15 64 23)(16 24 57)(33 55 42)(34 43 56)(35 49 44)(36 45 50)(37 51 46)(38 47 52)(39 53 48)(40 41 54)
(1 37 17)(2 38 18)(3 39 19)(4 40 20)(5 33 21)(6 34 22)(7 35 23)(8 36 24)(9 25 51)(10 26 52)(11 27 53)(12 28 54)(13 29 55)(14 30 56)(15 31 49)(16 32 50)(41 61 70)(42 62 71)(43 63 72)(44 64 65)(45 57 66)(46 58 67)(47 59 68)(48 60 69)
(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 58)(10 63)(11 60)(12 57)(13 62)(14 59)(15 64)(16 61)(18 22)(20 24)(25 67)(26 72)(27 69)(28 66)(29 71)(30 68)(31 65)(32 70)(34 38)(36 40)(41 50)(42 55)(43 52)(44 49)(45 54)(46 51)(47 56)(48 53)

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

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

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

62 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12L 12M 12N 24A 24B 24C 24D 24E ··· 24T 24U 24V 24W 24X order 1 2 2 3 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 18 2 3 3 6 6 6 1 1 18 2 3 3 6 6 6 18 18 2 2 18 18 2 2 3 3 3 3 6 ··· 6 18 18 2 2 2 2 6 ··· 6 18 18 18 18

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D6 M4(2) C3×S3 C4×S3 S3×C6 C8⋊S3 C3×M4(2) S3×C12 C3×C8⋊S3 C32⋊C6 C2×C32⋊C6 C4×C32⋊C6 He3⋊5M4(2) kernel He3⋊5M4(2) He3⋊3C8 C8×He3 C4×C32⋊C6 C24⋊S3 C32⋊C12 C2×C32⋊C6 C32⋊4C8 C3×C24 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3×C24 C3×C12 He3 C24 C3×C6 C12 C32 C32 C6 C3 C8 C4 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 4 4 4 8 1 1 2 4

Matrix representation of He35M4(2) in GL8(𝔽73)

 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 70 67 0 0 0 0 0 0 70 3 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0
,
 72 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(73))| [72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[70,70,0,0,0,0,0,0,67,3,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0],[72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

He35M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5M_4(2)
% in TeX

G:=Group("He3:5M4(2)");
// GroupNames label

G:=SmallGroup(432,116);
// by ID

G=gap.SmallGroup(432,116);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,4037,2035,14118]);
// 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=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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