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## G = He3⋊7D8order 432 = 24·33

### 2nd semidirect product of He3 and D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3 — He3⋊7D8
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊5D4 — He3⋊7D8
 Lower central He3 — C2×He3 — C4×He3 — He3⋊7D8
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for He37D8
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=d-1 >

Subgroups: 577 in 121 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, D4, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, He3, C3×C12, S3×C6, C62, D4⋊S3, C3×D8, He3⋊C2, C2×He3, C2×He3, C3×C3⋊C8, C3×D12, D4×C32, C4×He3, C2×He3⋊C2, C22×He3, C3×D4⋊S3, He34C8, He35D4, D4×He3, He37D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, C3⋊D4, C2×C3⋊S3, D4⋊S3, He3⋊C2, C327D4, C2×He3⋊C2, C327D8, He37D4, He37D8

Smallest permutation representation of He37D8
On 72 points
Generators in S72
(1 43 59)(2 60 44)(3 45 61)(4 62 46)(5 47 63)(6 64 48)(7 41 57)(8 58 42)(9 26 37)(10 38 27)(11 28 39)(12 40 29)(13 30 33)(14 34 31)(15 32 35)(16 36 25)(17 54 67)(18 68 55)(19 56 69)(20 70 49)(21 50 71)(22 72 51)(23 52 65)(24 66 53)
(1 31 20)(2 32 21)(3 25 22)(4 26 23)(5 27 24)(6 28 17)(7 29 18)(8 30 19)(9 65 46)(10 66 47)(11 67 48)(12 68 41)(13 69 42)(14 70 43)(15 71 44)(16 72 45)(33 56 58)(34 49 59)(35 50 60)(36 51 61)(37 52 62)(38 53 63)(39 54 64)(40 55 57)
(1 70 49)(2 50 71)(3 72 51)(4 52 65)(5 66 53)(6 54 67)(7 68 55)(8 56 69)(9 23 37)(10 38 24)(11 17 39)(12 40 18)(13 19 33)(14 34 20)(15 21 35)(16 36 22)(25 45 61)(26 62 46)(27 47 63)(28 64 48)(29 41 57)(30 58 42)(31 43 59)(32 60 44)
(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 8)(3 7)(4 6)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(41 61)(42 60)(43 59)(44 58)(45 57)(46 64)(47 63)(48 62)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4 6A 6B 6C 6D 6E 6F 6G 6H 6I ··· 6P 6Q 6R 8A 8B 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 3 3 3 3 3 3 4 6 6 6 6 6 6 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 4 36 1 1 6 6 6 6 2 1 1 4 4 6 6 6 6 12 ··· 12 36 36 18 18 2 2 12 12 12 12 18 18 18 18

41 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 4 6 6 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D8 C3⋊D4 He3⋊C2 C2×He3⋊C2 D4⋊S3 He3⋊7D4 He3⋊7D8 kernel He3⋊7D8 He3⋊4C8 He3⋊5D4 D4×He3 D4×C32 C2×He3 C3×C12 He3 C3×C6 D4 C4 C32 C2 C1 # reps 1 1 1 1 4 1 4 2 8 4 4 4 2 4

Matrix representation of He37D8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 64 0 0
,
 0 14 0 0 0 26 41 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 29 72 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,64,0,0,8,0,0,0,0,0,1,0],[0,26,0,0,0,14,41,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,29,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

He37D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_7D_8
% in TeX

G:=Group("He3:7D8");
// GroupNames label

G:=SmallGroup(432,192);
// by ID

G=gap.SmallGroup(432,192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,1124,4037,537]);
// 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^-1>;
// generators/relations

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