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## G = He3⋊6M4(2)  order 432 = 24·33

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊6M4(2)
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — C4×He3⋊C2 — He3⋊6M4(2)
 Lower central He3 — C2×He3 — He3⋊6M4(2)
 Upper central C1 — C12 — C24

Generators and relations for He36M4(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, bd=db, be=eb, dcd-1=ece=c-1, ede=d5 >

Subgroups: 377 in 110 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), He3⋊C2, C2×He3, C3×C3⋊C8, C3×C24, S3×C12, He33C4, C4×He3, C2×He3⋊C2, C3×C8⋊S3, He34C8, C8×He3, C4×He3⋊C2, He36M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), C3⋊S3, C4×S3, C2×C3⋊S3, C8⋊S3, He3⋊C2, C4×C3⋊S3, C2×He3⋊C2, C24⋊S3, C4×He3⋊C2, He36M4(2)

Smallest permutation representation of He36M4(2)
On 72 points
Generators in S72
(1 47 69)(2 70 48)(3 41 71)(4 72 42)(5 43 65)(6 66 44)(7 45 67)(8 68 46)(9 26 22)(10 23 27)(11 28 24)(12 17 29)(13 30 18)(14 19 31)(15 32 20)(16 21 25)(33 58 55)(34 56 59)(35 60 49)(36 50 61)(37 62 51)(38 52 63)(39 64 53)(40 54 57)
(1 61 18)(2 62 19)(3 63 20)(4 64 21)(5 57 22)(6 58 23)(7 59 24)(8 60 17)(9 43 40)(10 44 33)(11 45 34)(12 46 35)(13 47 36)(14 48 37)(15 41 38)(16 42 39)(25 72 53)(26 65 54)(27 66 55)(28 67 56)(29 68 49)(30 69 50)(31 70 51)(32 71 52)
(1 13 30)(2 31 14)(3 15 32)(4 25 16)(5 9 26)(6 27 10)(7 11 28)(8 29 12)(17 49 35)(18 36 50)(19 51 37)(20 38 52)(21 53 39)(22 40 54)(23 55 33)(24 34 56)(41 71 63)(42 64 72)(43 65 57)(44 58 66)(45 67 59)(46 60 68)(47 69 61)(48 62 70)
(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 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)(17 21)(19 23)(33 51)(34 56)(35 53)(36 50)(37 55)(38 52)(39 49)(40 54)(41 71)(42 68)(43 65)(44 70)(45 67)(46 72)(47 69)(48 66)(58 62)(60 64)

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

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

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

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 ··· 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 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 18 1 1 6 6 6 6 1 1 18 1 1 6 6 6 6 18 18 2 2 18 18 1 1 1 1 6 ··· 6 18 18 2 2 2 2 6 ··· 6 18 18 18 18

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 6 type + + + + + + image C1 C2 C2 C2 C4 C4 S3 D6 M4(2) C4×S3 C8⋊S3 He3⋊C2 C2×He3⋊C2 C4×He3⋊C2 He3⋊6M4(2) kernel He3⋊6M4(2) He3⋊4C8 C8×He3 C4×He3⋊C2 He3⋊3C4 C2×He3⋊C2 C3×C24 C3×C12 He3 C3×C6 C32 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 2 8 16 4 4 8 4

Matrix representation of He36M4(2) in GL5(𝔽73)

 0 72 0 0 0 1 72 0 0 0 0 0 0 0 1 0 0 72 0 0 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 0 0 8 0 0 72 0 0 0 0 0 9 0
,
 65 65 0 0 0 57 8 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 72
,
 1 72 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(5,GF(73))| [0,1,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,9,0,0,8,0,0],[65,57,0,0,0,65,8,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,72] >;

He36M4(2) in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_6M_4(2)
% in TeX

G:=Group("He3:6M4(2)");
// GroupNames label

G:=SmallGroup(432,174);
// by ID

G=gap.SmallGroup(432,174);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,1124,4037,537]);
// 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=e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

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