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G = He3⋊5SD16  order 432 = 24·33

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

non-abelian, supersoluble, monomial

Aliases: He3⋊5SD16, C12.11S32, (C3×C12).6D6, (C3×C6).3D12, He3⋊3C8⋊3C2, C32⋊4C8⋊3S3, He3⋊4Q8⋊1C2, C12⋊S3.2S3, (C2×He3).11D4, He3⋊4D4.2C2, C4.3(C32⋊D6), C32⋊3(C24⋊C2), C2.6(He3⋊3D4), C32⋊3(D4.S3), (C4×He3).6C22, C6.31(C3⋊D12), C3.3(D12.S3), (C3×C6).6(C3⋊D4), SmallGroup(432,85)

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 He3⋊5SD16
 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, Q8⋊2S3, C32⋊C6, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C3×D12, C12⋊S3, He3⋊3C4, C4×He3, C2×C32⋊C6, Dic6⋊S3, C32⋊5SD16, He3⋊3C8, He3⋊4D4, He3⋊4Q8, He3⋊5SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, D12, C3⋊D4, S32, C24⋊C2, D4.S3, C3⋊D12, C32⋊D6, D12.S3, He3⋊3D4, He3⋊5SD16

Character table of He3⋊5SD16

 class 12A2B3A3B3C3D4A4B6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G12H24A24B24C24D
 size 1136266122362661236361818466121212363618181818
ρ111111111111111111111111111111    trivial
ρ211-11111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ311111111-1111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ411-111111-11111-1-111111111-1-11111    linear of order 2
ρ522-22-12-1202-12-11100222-1-1-1000000    orthogonal lifted from D6
ρ62222-12-1202-12-1-1-100222-1-1-1000000    orthogonal lifted from S3
ρ722022-1-12022-1-100222-1-12-1-100-1-1-1-1    orthogonal lifted from S3
ρ822022-1-12022-1-100-2-22-1-12-1-1001111    orthogonal lifted from D6
ρ92202222-2022220000-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ1022022-1-1-2022-1-10000-211-21100-√3-√3√3√3    orthogonal lifted from D12
ρ1122022-1-1-2022-1-10000-211-21100√3√3-√3-√3    orthogonal lifted from D12
ρ122-20222200-2-2-2-200-√-2√-200000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ132-20222200-2-2-2-200√-2-√-200000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ142202-12-1-202-12-1√-3-√-300-2-2-2111000000    complex lifted from C3⋊D4
ρ152202-12-1-202-12-1-√-3√-300-2-2-2111000000    complex lifted from C3⋊D4
ρ162-2022-1-100-2-21100-√-2√-20-√3√30-√3√300-ζ83ζ3+ζ8ζ3+ζ8-ζ87ζ3+ζ85ζ3+ζ85-ζ87ζ32+ζ85ζ32+ζ85-ζ83ζ32+ζ8ζ32+ζ8    complex lifted from C24⋊C2
ρ172-2022-1-100-2-21100√-2-√-20-√3√30-√3√300-ζ87ζ3+ζ85ζ3+ζ85-ζ83ζ3+ζ8ζ3+ζ8-ζ83ζ32+ζ8ζ32+ζ8-ζ87ζ32+ζ85ζ32+ζ85    complex lifted from C24⋊C2
ρ182-2022-1-100-2-21100√-2-√-20√3-√30√3-√300-ζ87ζ32+ζ85ζ32+ζ85-ζ83ζ32+ζ8ζ32+ζ8-ζ83ζ3+ζ8ζ3+ζ8-ζ87ζ3+ζ85ζ3+ζ85    complex lifted from C24⋊C2
ρ192-2022-1-100-2-21100-√-2√-20√3-√30√3-√300-ζ83ζ32+ζ8ζ32+ζ8-ζ87ζ32+ζ85ζ32+ζ85-ζ87ζ3+ζ85ζ3+ζ85-ζ83ζ3+ζ8ζ3+ζ8    complex lifted from C24⋊C2
ρ204404-2-21-404-2-210000-4222-1-1000000    orthogonal lifted from C3⋊D12
ρ214404-2-21404-2-2100004-2-2-211000000    orthogonal lifted from S32
ρ224-404-24-200-42-420000000000000000    symplectic lifted from D4.S3, Schur index 2
ρ234-404-2-2100-422-1000002√3-2√30-√3√3000000    symplectic lifted from D12.S3, Schur index 2
ρ244-404-2-2100-422-100000-2√32√30√3-√3000000    symplectic lifted from D12.S3, Schur index 2
ρ25660-30006-2-30000000-300000110000    orthogonal lifted from C32⋊D6
ρ26660-300062-30000000-300000-1-10000    orthogonal lifted from C32⋊D6
ρ27660-3000-60-30000000300000-√-3√-30000    complex lifted from He3⋊3D4
ρ28660-3000-60-30000000300000√-3-√-30000    complex lifted from He3⋊3D4
ρ2912-120-60000060000000000000000000    orthogonal faithful

Smallest permutation representation of He3⋊5SD16
►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 He3⋊5SD16 ►in GL10(𝔽73)

727200000000
1000000000
007272000000
0010000000
0000100000
0000010000
00000072100
00000072000
000072720727272
0000001010
,
1000000000
0100000000
0010000000
0001000000
00007210000
00007200000
00000072100
00000072000
0000101001
00000720727272
,
8000000000
0800000000
00640000000
00064000000
0000001000
0000000100
00000000721
0000727272727172
0000000010
0000100010
,
003611000000
006225000000
376200000000
114800000000
00000720000
00007200000
0000111121
00000000172
00000000720
000000072720
,
1000000000
727200000000
00720000000
0011000000
0000010000
0000100000
0000000100
0000001000
0000000010
0000727272727272

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] >;
 

He3⋊5SD16 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
 

Export

Character table of He3⋊5SD16 in TeX

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