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## G = He3⋊6D8order 432 = 24·33

### 1st semidirect product of He3 and D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — He3⋊6D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊4D4 — He3⋊6D8
 Lower central C32 — C3×C6 — C3×C12 — He3⋊6D8
 Upper central C1 — C2 — C4 — D4

Generators and relations for He36D8
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=d-1 >

Subgroups: 505 in 93 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, D4, C32, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D4⋊S3, C3×D8, C32⋊C6, C2×He3, C2×He3, C3×C3⋊C8, C324C8, C3×D12, C12⋊S3, D4×C32, D4×C32, C4×He3, C2×C32⋊C6, C22×He3, C3×D4⋊S3, C327D8, He33C8, He34D4, D4×He3, He36D8
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, S3×C6, D4⋊S3, C3×D8, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×D4⋊S3, He36D4, He36D8

Smallest permutation representation of He36D8
On 72 points
Generators in S72
(1 68 56)(2 49 69)(3 70 50)(4 51 71)(5 72 52)(6 53 65)(7 66 54)(8 55 67)(9 57 37)(10 38 58)(11 59 39)(12 40 60)(13 61 33)(14 34 62)(15 63 35)(16 36 64)(17 48 29)(18 30 41)(19 42 31)(20 32 43)(21 44 25)(22 26 45)(23 46 27)(24 28 47)
(1 15 41)(2 42 16)(3 9 43)(4 44 10)(5 11 45)(6 46 12)(7 13 47)(8 48 14)(17 62 67)(18 68 63)(19 64 69)(20 70 57)(21 58 71)(22 72 59)(23 60 65)(24 66 61)(25 38 51)(26 52 39)(27 40 53)(28 54 33)(29 34 55)(30 56 35)(31 36 49)(32 50 37)
(1 68 35)(2 69 36)(3 70 37)(4 71 38)(5 72 39)(6 65 40)(7 66 33)(8 67 34)(9 57 32)(10 58 25)(11 59 26)(12 60 27)(13 61 28)(14 62 29)(15 63 30)(16 64 31)(17 55 48)(18 56 41)(19 49 42)(20 50 43)(21 51 44)(22 52 45)(23 53 46)(24 54 47)
(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)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 54)(50 53)(51 52)(55 56)(57 60)(58 59)(61 64)(62 63)(65 70)(66 69)(67 68)(71 72)

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

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

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

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 2 3 3 6 6 6 2 2 3 3 4 4 6 6 6 12 ··· 12 36 36 18 18 4 6 6 12 12 12 18 18 18 18

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 He3⋊6D8 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×C3⋊D4 D4⋊S3 C3×D4⋊S3 C32⋊C6 C2×C32⋊C6 He3⋊6D4 kernel He3⋊6D8 He3⋊3C8 He3⋊4D4 D4×He3 C32⋊7D8 C32⋊4C8 C12⋊S3 D4×C32 C1 D4×C32 C2×He3 C3×C12 He3 C3×D4 C3×C6 C3×C6 C12 C32 C6 C32 C3 D4 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of He36D8 in GL10(𝔽73)

 64 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 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 0 0 72 1 0 0 0 0 9 9 65 65 71 72 0 0 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 8 0
,
 1 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 64 0 8 0 0 1 0 0 0 0 0 9 0 65 72 72
,
 1 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 9 9 65 65 72 71 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 64 0 8 0 0 0 0 1 0 0 64 0 8
,
 0 0 41 14 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 32 59 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 0 0 0 0 0 3 17 49 63 14 7 0 0 0 0 0 7 0 7 7 66 0 0 0 0 56 70 17 10 66 59 0 0 0 0 63 70 10 17 59 66 0 0 0 0 17 24 66 49 53 63 0 0 0 0 7 7 3 0 46 49
,
 0 0 32 59 0 0 0 0 0 0 0 0 47 41 0 0 0 0 0 0 32 59 0 0 0 0 0 0 0 0 47 41 0 0 0 0 0 0 0 0 0 0 0 0 70 56 24 10 59 66 0 0 0 0 0 66 0 66 66 7 0 0 0 0 17 3 56 63 7 14 0 0 0 0 10 3 63 56 14 7 0 0 0 0 56 49 7 24 20 10 0 0 0 0 66 66 70 0 27 24

G:=sub<GL(10,GF(73))| [64,0,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,65,0,0,0,0,0,0,0,1,0,65,0,0,0,0,0,0,0,0,72,71,8,8,0,0,0,0,0,0,1,72,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,64,0,0,0,0,0,1,0,0,0,0,9,0,0,0,0,0,0,72,72,8,0,0,0,0,0,0,0,1,0,0,65,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,65,0,0,0,0,0,0,0,72,72,65,0,64,64,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,71,72,8,8],[0,0,32,47,0,0,0,0,0,0,0,0,59,0,0,0,0,0,0,0,41,26,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,56,63,17,7,0,0,0,0,17,7,70,70,24,7,0,0,0,0,49,0,17,10,66,3,0,0,0,0,63,7,10,17,49,0,0,0,0,0,14,7,66,59,53,46,0,0,0,0,7,66,59,66,63,49],[0,0,32,47,0,0,0,0,0,0,0,0,59,41,0,0,0,0,0,0,32,47,0,0,0,0,0,0,0,0,59,41,0,0,0,0,0,0,0,0,0,0,0,0,70,0,17,10,56,66,0,0,0,0,56,66,3,3,49,66,0,0,0,0,24,0,56,63,7,70,0,0,0,0,10,66,63,56,24,0,0,0,0,0,59,66,7,14,20,27,0,0,0,0,66,7,14,7,10,24] >;

He36D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_6D_8
% in TeX

G:=Group("He3:6D8");
// GroupNames label

G:=SmallGroup(432,153);
// by ID

G=gap.SmallGroup(432,153);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,1011,514,80,4037,2035,14118]);
// 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^-1>;
// generators/relations

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