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## G = He3⋊2SD16order 432 = 24·33

### The semidirect product of He3 and SD16 acting via SD16/C2=D4

Aliases: He32SD16, C6.1S3≀C2, He32D4.C2, He32C82C2, He32Q81C2, C2.3(He3⋊D4), (C2×He3).1D4, C3.(C322SD16), He33C4.1C22, SmallGroup(432,234)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊3C4 — He3⋊2SD16
 Chief series C1 — C3 — He3 — C2×He3 — He3⋊3C4 — He3⋊2D4 — He3⋊2SD16
 Lower central He3 — C2×He3 — He3⋊3C4 — He3⋊2SD16
 Upper central C1 — C2

Generators and relations for He32SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, eae=ab=ba, cac-1=ab-1, dad-1=cb=bc, bd=db, ebe=b-1, dcd-1=a-1b, ece=c-1, ede=d3 >

Subgroups: 539 in 58 conjugacy classes, 11 normal (all characteristic)
C1, C2, C2, C3, C3 [×2], C4 [×2], C22, S3 [×2], C6, C6 [×3], C8, D4, Q8, C32 [×2], Dic3 [×4], C12 [×2], D6 [×2], C2×C6, SD16, C3×S3, C3⋊S3, C3×C6 [×2], C24, Dic6 [×2], D12, C3⋊D4, He3, C3×Dic3 [×3], C3⋊Dic3, S3×C6, C2×C3⋊S3, C24⋊C2, C32⋊C6, C2×He3, C3⋊D12, C322Q8, C32⋊C12, He33C4, C2×C32⋊C6, He32C8, He32Q8, He32D4, He32SD16
Quotients: C1, C2 [×3], C22, D4, SD16, S3≀C2, C322SD16, He3⋊D4, He32SD16

Character table of He32SD16

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 36 2 12 12 18 36 2 12 12 36 36 18 18 18 18 36 36 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 2 2 2 -2 0 2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 √-2 -√-2 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ7 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 -√-2 √-2 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ8 4 4 0 4 1 -2 0 -2 4 -2 1 0 0 0 0 0 0 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ9 4 4 0 4 1 -2 0 2 4 -2 1 0 0 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ10 4 4 2 4 -2 1 0 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 4 -2 1 0 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 4 1 -2 0 0 -4 2 -1 0 0 0 0 0 0 √3 -√3 0 0 0 0 symplectic lifted from C32⋊2SD16, Schur index 2 ρ13 4 -4 0 4 1 -2 0 0 -4 2 -1 0 0 0 0 0 0 -√3 √3 0 0 0 0 symplectic lifted from C32⋊2SD16, Schur index 2 ρ14 4 -4 0 4 -2 1 0 0 -4 -1 2 -√-3 √-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ15 4 -4 0 4 -2 1 0 0 -4 -1 2 √-3 -√-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ16 6 6 0 -3 0 0 -2 0 -3 0 0 0 0 2 2 1 1 0 0 -1 -1 -1 -1 orthogonal lifted from He3⋊D4 ρ17 6 6 0 -3 0 0 -2 0 -3 0 0 0 0 -2 -2 1 1 0 0 1 1 1 1 orthogonal lifted from He3⋊D4 ρ18 6 6 0 -3 0 0 2 0 -3 0 0 0 0 0 0 -1 -1 0 0 √3 -√3 -√3 √3 orthogonal lifted from He3⋊D4 ρ19 6 6 0 -3 0 0 2 0 -3 0 0 0 0 0 0 -1 -1 0 0 -√3 √3 √3 -√3 orthogonal lifted from He3⋊D4 ρ20 6 -6 0 -3 0 0 0 0 3 0 0 0 0 -√-2 √-2 √3 -√3 0 0 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 complex faithful ρ21 6 -6 0 -3 0 0 0 0 3 0 0 0 0 -√-2 √-2 -√3 √3 0 0 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ32+ζ87-ζ85ζ32 complex faithful ρ22 6 -6 0 -3 0 0 0 0 3 0 0 0 0 √-2 -√-2 -√3 √3 0 0 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 complex faithful ρ23 6 -6 0 -3 0 0 0 0 3 0 0 0 0 √-2 -√-2 √3 -√3 0 0 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ3+ζ83-ζ8ζ3 complex faithful

Smallest permutation representation of He32SD16
On 72 points
Generators in S72
(1 22 52)(2 37 64)(3 24 54)(4 12 26)(5 18 56)(6 33 60)(7 20 50)(8 16 30)(9 69 63)(10 53 48)(13 65 59)(14 49 44)(17 58 42)(19 28 66)(21 62 46)(23 32 70)(25 38 41)(29 34 45)(35 51 68)(39 55 72)
(1 22 52)(2 23 53)(3 24 54)(4 17 55)(5 18 56)(6 19 49)(7 20 50)(8 21 51)(9 63 69)(10 64 70)(11 57 71)(12 58 72)(13 59 65)(14 60 66)(15 61 67)(16 62 68)(25 41 38)(26 42 39)(27 43 40)(28 44 33)(29 45 34)(30 46 35)(31 47 36)(32 48 37)
(1 47 69)(3 71 41)(5 43 65)(7 67 45)(9 22 36)(10 70 64)(11 38 24)(12 58 72)(13 18 40)(14 66 60)(15 34 20)(16 62 68)(25 54 57)(26 39 42)(27 59 56)(28 44 33)(29 50 61)(30 35 46)(31 63 52)(32 48 37)
(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 4)(3 7)(6 8)(9 31)(10 26)(11 29)(12 32)(13 27)(14 30)(15 25)(16 28)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(33 62)(34 57)(35 60)(36 63)(37 58)(38 61)(39 64)(40 59)(41 67)(42 70)(43 65)(44 68)(45 71)(46 66)(47 69)(48 72)

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

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

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

Matrix representation of He32SD16 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 40 4 40 4 40 4 69 44 69 44 69 44 40 4 4 29 29 40 69 44 44 33 33 69 40 4 29 40 4 29 69 44 33 69 44 33
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 1 72 0 0 0 0 0 72 0 0

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[40,69,40,69,40,69,4,44,4,44,4,44,40,69,4,44,29,33,4,44,29,33,40,69,40,69,29,33,4,44,4,44,40,69,29,33],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,72,72,0,0] >;

He32SD16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("He3:2SD16");
// GroupNames label

G:=SmallGroup(432,234);
// by ID

G=gap.SmallGroup(432,234);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,254,135,58,1124,851,298,348,1027,537,14118,7069]);
// 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=a*b^-1,d*a*d^-1=c*b=b*c,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a^-1*b,e*c*e=c^-1,e*d*e=d^3>;
// generators/relations

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