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## G = He3⋊5SD16order 432 = 24·33

### 3rd semidirect product of He3 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3 — He3⋊5SD16
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4D4 — He3⋊5SD16
 Lower central He3 — C2×He3 — C4×He3 — He3⋊5SD16
 Upper central C1 — C2 — C4

Generators and relations for He35SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 527 in 82 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×Q8, He3, C3×Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, D4.S3, Q82S3, C32⋊C6, C2×He3, C3×C3⋊C8, C324C8, C3×Dic6, C3×D12, C12⋊S3, He33C4, C4×He3, C2×C32⋊C6, Dic6⋊S3, C325SD16, He33C8, He34D4, He34Q8, He35SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, D12, C3⋊D4, S32, C24⋊C2, D4.S3, C3⋊D12, C32⋊D6, D12.S3, He33D4, He35SD16

Character table of He35SD16

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B 24C 24D size 1 1 36 2 6 6 12 2 36 2 6 6 12 36 36 18 18 4 6 6 12 12 12 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 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 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 -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 -1 -1 1 1 1 1 linear of order 2 ρ5 2 2 -2 2 -1 2 -1 2 0 2 -1 2 -1 1 1 0 0 2 2 2 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 2 2 -1 2 -1 2 0 2 -1 2 -1 -1 -1 0 0 2 2 2 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ7 2 2 0 2 2 -1 -1 2 0 2 2 -1 -1 0 0 2 2 2 -1 -1 2 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 2 2 -1 -1 2 0 2 2 -1 -1 0 0 -2 -2 2 -1 -1 2 -1 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ9 2 2 0 2 2 2 2 -2 0 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 2 2 -1 -1 -2 0 2 2 -1 -1 0 0 0 0 -2 1 1 -2 1 1 0 0 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ11 2 2 0 2 2 -1 -1 -2 0 2 2 -1 -1 0 0 0 0 -2 1 1 -2 1 1 0 0 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ12 2 -2 0 2 2 2 2 0 0 -2 -2 -2 -2 0 0 -√-2 √-2 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ13 2 -2 0 2 2 2 2 0 0 -2 -2 -2 -2 0 0 √-2 -√-2 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ14 2 2 0 2 -1 2 -1 -2 0 2 -1 2 -1 √-3 -√-3 0 0 -2 -2 -2 1 1 1 0 0 0 0 0 0 complex lifted from C3⋊D4 ρ15 2 2 0 2 -1 2 -1 -2 0 2 -1 2 -1 -√-3 √-3 0 0 -2 -2 -2 1 1 1 0 0 0 0 0 0 complex lifted from C3⋊D4 ρ16 2 -2 0 2 2 -1 -1 0 0 -2 -2 1 1 0 0 -√-2 √-2 0 -√3 √3 0 -√3 √3 0 0 -ζ83ζ3+ζ8ζ3+ζ8 -ζ87ζ3+ζ85ζ3+ζ85 -ζ87ζ32+ζ85ζ32+ζ85 -ζ83ζ32+ζ8ζ32+ζ8 complex lifted from C24⋊C2 ρ17 2 -2 0 2 2 -1 -1 0 0 -2 -2 1 1 0 0 √-2 -√-2 0 -√3 √3 0 -√3 √3 0 0 -ζ87ζ3+ζ85ζ3+ζ85 -ζ83ζ3+ζ8ζ3+ζ8 -ζ83ζ32+ζ8ζ32+ζ8 -ζ87ζ32+ζ85ζ32+ζ85 complex lifted from C24⋊C2 ρ18 2 -2 0 2 2 -1 -1 0 0 -2 -2 1 1 0 0 √-2 -√-2 0 √3 -√3 0 √3 -√3 0 0 -ζ87ζ32+ζ85ζ32+ζ85 -ζ83ζ32+ζ8ζ32+ζ8 -ζ83ζ3+ζ8ζ3+ζ8 -ζ87ζ3+ζ85ζ3+ζ85 complex lifted from C24⋊C2 ρ19 2 -2 0 2 2 -1 -1 0 0 -2 -2 1 1 0 0 -√-2 √-2 0 √3 -√3 0 √3 -√3 0 0 -ζ83ζ32+ζ8ζ32+ζ8 -ζ87ζ32+ζ85ζ32+ζ85 -ζ87ζ3+ζ85ζ3+ζ85 -ζ83ζ3+ζ8ζ3+ζ8 complex lifted from C24⋊C2 ρ20 4 4 0 4 -2 -2 1 -4 0 4 -2 -2 1 0 0 0 0 -4 2 2 2 -1 -1 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ21 4 4 0 4 -2 -2 1 4 0 4 -2 -2 1 0 0 0 0 4 -2 -2 -2 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ22 4 -4 0 4 -2 4 -2 0 0 -4 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ23 4 -4 0 4 -2 -2 1 0 0 -4 2 2 -1 0 0 0 0 0 2√3 -2√3 0 -√3 √3 0 0 0 0 0 0 symplectic lifted from D12.S3, Schur index 2 ρ24 4 -4 0 4 -2 -2 1 0 0 -4 2 2 -1 0 0 0 0 0 -2√3 2√3 0 √3 -√3 0 0 0 0 0 0 symplectic lifted from D12.S3, Schur index 2 ρ25 6 6 0 -3 0 0 0 6 -2 -3 0 0 0 0 0 0 0 -3 0 0 0 0 0 1 1 0 0 0 0 orthogonal lifted from C32⋊D6 ρ26 6 6 0 -3 0 0 0 6 2 -3 0 0 0 0 0 0 0 -3 0 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from C32⋊D6 ρ27 6 6 0 -3 0 0 0 -6 0 -3 0 0 0 0 0 0 0 3 0 0 0 0 0 -√-3 √-3 0 0 0 0 complex lifted from He3⋊3D4 ρ28 6 6 0 -3 0 0 0 -6 0 -3 0 0 0 0 0 0 0 3 0 0 0 0 0 √-3 -√-3 0 0 0 0 complex lifted from He3⋊3D4 ρ29 12 -12 0 -6 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of He35SD16
On 72 points
Generators in S72
(17 36 50)(18 37 51)(19 38 52)(20 39 53)(21 40 54)(22 33 55)(23 34 56)(24 35 49)(41 59 72)(42 60 65)(43 61 66)(44 62 67)(45 63 68)(46 64 69)(47 57 70)(48 58 71)
(1 15 31)(2 32 16)(3 9 25)(4 26 10)(5 11 27)(6 28 12)(7 13 29)(8 30 14)(17 50 36)(18 37 51)(19 52 38)(20 39 53)(21 54 40)(22 33 55)(23 56 34)(24 35 49)(41 72 59)(42 60 65)(43 66 61)(44 62 67)(45 68 63)(46 64 69)(47 70 57)(48 58 71)
(1 68 20)(2 21 69)(3 70 22)(4 23 71)(5 72 24)(6 17 65)(7 66 18)(8 19 67)(9 57 33)(10 34 58)(11 59 35)(12 36 60)(13 61 37)(14 38 62)(15 63 39)(16 40 64)(25 47 55)(26 56 48)(27 41 49)(28 50 42)(29 43 51)(30 52 44)(31 45 53)(32 54 46)
(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 29)(10 32)(11 27)(12 30)(13 25)(14 28)(15 31)(16 26)(17 19)(18 22)(21 23)(33 51)(34 54)(35 49)(36 52)(37 55)(38 50)(39 53)(40 56)(41 59)(42 62)(43 57)(44 60)(45 63)(46 58)(47 61)(48 64)(65 67)(66 70)(69 71)

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

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

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

Matrix representation of He35SD16 in GL10(𝔽73)

 72 72 0 0 0 0 0 0 0 0 1 0 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 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 72 72 0 72 72 72 0 0 0 0 0 0 1 0 1 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 1 0 1 0 0 1 0 0 0 0 0 72 0 72 72 72
,
 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 64 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 72 72 72 72 71 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0
,
 0 0 36 11 0 0 0 0 0 0 0 0 62 25 0 0 0 0 0 0 37 62 0 0 0 0 0 0 0 0 11 48 0 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 1 1 1 2 1 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 72 0
,
 1 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 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 72 72 72 72 72 72

G:=sub<GL(10,GF(73))| [72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,72,72,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,72,72,1,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,72,71,1,1,0,0,0,0,0,0,1,72,0,0],[0,0,37,11,0,0,0,0,0,0,0,0,62,48,0,0,0,0,0,0,36,62,0,0,0,0,0,0,0,0,11,25,0,0,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,2,1,72,72,0,0,0,0,0,0,1,72,0,0],[1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72] >;

He35SD16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("He3:5SD16");
// GroupNames label

G:=SmallGroup(432,85);
// by ID

G=gap.SmallGroup(432,85);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,571,4037,537,14118,7069]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

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