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G = He32SD16order 432 = 24·33

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

non-abelian, soluble

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

C1C3He3He33C4 — He32SD16
C1C3He3C2×He3He33C4He32D4 — He32SD16
He3C2×He3He33C4 — He32SD16
C1C2

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 12A2B3A3B3C4A4B6A6B6C6D6E8A8B12A12B12C12D24A24B24C24D
 size 113621212183621212363618181818363618181818
ρ111111111111111111111111    trivial
ρ21111111-111111-1-111-1-1-1-1-1-1    linear of order 2
ρ311-111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ411-11111-1111-1-11111-1-11111    linear of order 2
ρ5220222-202220000-2-2000000    orthogonal lifted from D4
ρ62-2022200-2-2-200-2--20000-2--2-2--2    complex lifted from SD16
ρ72-2022200-2-2-200--2-20000--2-2--2-2    complex lifted from SD16
ρ844041-20-24-21000000110000    orthogonal lifted from S3≀C2
ρ944041-2024-21000000-1-10000    orthogonal lifted from S3≀C2
ρ104424-210041-2-1-10000000000    orthogonal lifted from S3≀C2
ρ1144-24-210041-2110000000000    orthogonal lifted from S3≀C2
ρ124-4041-200-42-10000003-30000    symplectic lifted from C322SD16, Schur index 2
ρ134-4041-200-42-1000000-330000    symplectic lifted from C322SD16, Schur index 2
ρ144-404-2100-4-12--3-30000000000    complex lifted from C322SD16
ρ154-404-2100-4-12-3--30000000000    complex lifted from C322SD16
ρ16660-300-20-30000221100-1-1-1-1    orthogonal lifted from He3⋊D4
ρ17660-300-20-30000-2-211001111    orthogonal lifted from He3⋊D4
ρ18660-30020-3000000-1-1003-3-33    orthogonal lifted from He3⋊D4
ρ19660-30020-3000000-1-100-333-3    orthogonal lifted from He3⋊D4
ρ206-60-3000030000--2-23-300ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3    complex faithful
ρ216-60-3000030000--2-2-3300ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ328785ζ32    complex faithful
ρ226-60-3000030000-2--2-3300ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32    complex faithful
ρ236-60-3000030000-2--23-300ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ3838ζ3    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)

7210000
7200000
0007200
0017200
000010
000001
,
7210000
7200000
0072100
0072000
0000721
0000720
,
001000
000100
000010
000001
100000
010000
,
404404404
694469446944
4044292940
694444333369
4042940429
694433694433
,
1720000
0720000
0000172
0000072
0017200
0007200

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

Export

Character table of He32SD16 in TeX

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