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## G = He3⋊D8order 432 = 24·33

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

Aliases: He3⋊D8, C6.2S3≀C2, C3.(C32⋊D8), He32C81C2, He32D41C2, C2.4(He3⋊D4), (C2×He3).2D4, He33C4.2C22, SmallGroup(432,235)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊3C4 — He3⋊D8
 Chief series C1 — C3 — He3 — C2×He3 — He3⋊3C4 — He3⋊2D4 — He3⋊D8
 Lower central He3 — C2×He3 — He3⋊3C4 — He3⋊D8
 Upper central C1 — C2

Generators and relations for He3⋊D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, eae=ab=ba, cac-1=dcd-1=ab-1, dad-1=bc-1, bc=cb, bd=db, ebe=b-1, ece=c-1, ede=d-1 >

Subgroups: 679 in 65 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, Dic3, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C24, D12, C3⋊D4, He3, C3×Dic3, S3×C6, C2×C3⋊S3, D24, C32⋊C6, C2×He3, C3⋊D12, He33C4, C2×C32⋊C6, He32C8, He32D4, He3⋊D8
Quotients: C1, C2, C22, D4, D8, S3≀C2, C32⋊D8, He3⋊D4, He3⋊D8

Character table of He3⋊D8

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 24A 24B 24C 24D size 1 1 36 36 2 12 12 18 2 12 12 36 36 36 36 18 18 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 -√2 √2 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ7 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 √2 -√2 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ8 4 4 0 2 4 -2 1 0 4 -2 1 0 0 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ9 4 4 0 -2 4 -2 1 0 4 -2 1 0 0 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ10 4 4 2 0 4 1 -2 0 4 1 -2 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ11 4 4 -2 0 4 1 -2 0 4 1 -2 1 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ12 4 -4 0 0 4 1 -2 0 -4 -1 2 -√-3 √-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ13 4 -4 0 0 4 -2 1 0 -4 2 -1 0 0 √-3 -√-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ14 4 -4 0 0 4 -2 1 0 -4 2 -1 0 0 -√-3 √-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ15 4 -4 0 0 4 1 -2 0 -4 -1 2 √-3 -√-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ16 6 6 0 0 -3 0 0 -2 -3 0 0 0 0 0 0 2 2 1 1 -1 -1 -1 -1 orthogonal lifted from He3⋊D4 ρ17 6 6 0 0 -3 0 0 -2 -3 0 0 0 0 0 0 -2 -2 1 1 1 1 1 1 orthogonal lifted from He3⋊D4 ρ18 6 6 0 0 -3 0 0 2 -3 0 0 0 0 0 0 0 0 -1 -1 -√3 √3 √3 -√3 orthogonal lifted from He3⋊D4 ρ19 6 6 0 0 -3 0 0 2 -3 0 0 0 0 0 0 0 0 -1 -1 √3 -√3 -√3 √3 orthogonal lifted from He3⋊D4 ρ20 6 -6 0 0 -3 0 0 0 3 0 0 0 0 0 0 -√2 √2 -√3 √3 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ32+ζ8ζ32+ζ8 orthogonal faithful ρ21 6 -6 0 0 -3 0 0 0 3 0 0 0 0 0 0 √2 -√2 -√3 √3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ3+ζ83+ζ8ζ3 orthogonal faithful ρ22 6 -6 0 0 -3 0 0 0 3 0 0 0 0 0 0 -√2 √2 √3 -√3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ32+ζ87+ζ85ζ32 orthogonal faithful ρ23 6 -6 0 0 -3 0 0 0 3 0 0 0 0 0 0 √2 -√2 √3 -√3 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ3+ζ85ζ3+ζ85 orthogonal faithful

Smallest permutation representation of He3⋊D8
On 72 points
Generators in S72
(1 57 13)(2 54 17)(3 59 15)(4 69 37)(5 61 9)(6 50 21)(7 63 11)(8 65 33)(10 26 71)(12 44 52)(14 30 67)(16 48 56)(18 68 47)(19 32 60)(22 72 43)(23 28 64)(25 38 49)(29 34 53)(35 46 58)(39 42 62)
(1 57 13)(2 58 14)(3 59 15)(4 60 16)(5 61 9)(6 62 10)(7 63 11)(8 64 12)(17 46 67)(18 47 68)(19 48 69)(20 41 70)(21 42 71)(22 43 72)(23 44 65)(24 45 66)(25 49 38)(26 50 39)(27 51 40)(28 52 33)(29 53 34)(30 54 35)(31 55 36)(32 56 37)
(1 45 29)(3 31 47)(5 41 25)(7 27 43)(9 20 38)(11 40 22)(13 24 34)(15 36 18)(17 67 46)(19 48 69)(21 71 42)(23 44 65)(26 50 39)(28 33 52)(30 54 35)(32 37 56)(49 61 70)(51 72 63)(53 57 66)(55 68 59)
(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 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 52)(18 51)(19 50)(20 49)(21 56)(22 55)(23 54)(24 53)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)

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

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

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

Matrix representation of He3⋊D8 in GL10(𝔽73)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 30 57 72 1 0 0 0 0 0 0 10 6 72 0 0 0 0 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 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 0 0 72 72 0 0 0 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72
,
 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 41 1 0 0 0 0 0 0 0 46 36 0 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 48 64 56 34 0 0 0 0 0 0 72 20 39 17 0 0 0 0 0 0 35 57 57 56 0 0 0 0 0 0 50 22 7 21 0 0 0 0 0 0 0 0 0 0 24 48 24 48 24 48 0 0 0 0 25 49 25 49 25 49 0 0 0 0 24 48 25 49 24 49 0 0 0 0 25 49 24 49 24 48 0 0 0 0 24 48 24 49 25 49 0 0 0 0 25 49 24 48 24 49
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 37 59 43 60 0 0 0 0 0 0 9 52 13 30 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 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

G:=sub<GL(10,GF(73))| [1,0,30,10,0,0,0,0,0,0,0,1,57,6,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[72,72,1,46,0,0,0,0,0,0,1,0,41,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,72,35,50,0,0,0,0,0,0,64,20,57,22,0,0,0,0,0,0,56,39,57,7,0,0,0,0,0,0,34,17,56,21,0,0,0,0,0,0,0,0,0,0,24,25,24,25,24,25,0,0,0,0,48,49,48,49,48,49,0,0,0,0,24,25,25,24,24,24,0,0,0,0,48,49,49,49,49,48,0,0,0,0,24,25,24,24,25,24,0,0,0,0,48,49,49,48,49,49],[0,1,37,9,0,0,0,0,0,0,1,0,59,52,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0] >;

He3⋊D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes D_8
% in TeX

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

G:=SmallGroup(432,235);
// by ID

G=gap.SmallGroup(432,235);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,254,135,58,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,e*a*e=a*b=b*a,c*a*c^-1=d*c*d^-1=a*b^-1,d*a*d^-1=b*c^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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