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G = He3⋊4SD16  order 432 = 24·33

2nd semidirect product of He3 and SD16 acting via SD16/C4=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊4SD16, C12.10S32, (C3×C12).5D6, (C3×C6).2D12, He3⋊3C8⋊2C2, C32⋊4C8⋊2S3, He3⋊3Q8⋊2C2, C32⋊4Q8⋊2S3, (C2×He3).10D4, He3⋊5D4.1C2, C4.2(C32⋊D6), C32⋊2(C24⋊C2), C2.5(He3⋊3D4), (C4×He3).5C22, C6.30(C3⋊D12), C32⋊3(Q8⋊2S3), C3.3(C32⋊5SD16), (C3×C6).5(C3⋊D4), SmallGroup(432,84)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C4×He3 — He3⋊4SD16
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊3Q8 — He3⋊4SD16
Lower central
He3 — C2×He3 — C4×He3 — He3⋊4SD16
Upper central
C1 — C2 — C4

Generators and relations for He3⋊4SD16
 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=b-1, be=eb, dcd-1=ece=c-1, ede=d3 >

Subgroups: 559 in 86 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×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C24⋊C2, D4.S3, Q8⋊2S3, He3⋊C2, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C3×D12, C32⋊4Q8, C32⋊C12, C4×He3, C2×He3⋊C2, Dic6⋊S3, D12.S3, He3⋊3C8, He3⋊3Q8, He3⋊5D4, He3⋊4SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, D12, C3⋊D4, S32, C24⋊C2, Q8⋊2S3, C3⋊D12, C32⋊D6, C32⋊5SD16, He3⋊3D4, He3⋊4SD16

Character table of He3⋊4SD16

 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
ρ522022-1-1202-12-100222-1-1-1-1200-1-1-1-1    orthogonal lifted from S3
ρ62202-12-12222-1-10000222-1-1-1-1-10000    orthogonal lifted from S3
ρ722022-1-1202-12-100-2-22-1-1-1-12001111    orthogonal lifted from D6
ρ82202-12-12-222-1-10000222-1-1-1110000    orthogonal lifted from D6
ρ92202222-2022220000-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ1022022-1-1-202-12-10000-21111-200√3-√3-√3√3    orthogonal lifted from D12
ρ1122022-1-1-202-12-10000-21111-200-√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
ρ142202-12-1-2022-1-10000-2-2-2111-√-3√-30000    complex lifted from C3⋊D4
ρ152202-12-1-2022-1-10000-2-2-2111√-3-√-30000    complex lifted from C3⋊D4
ρ162-2022-1-100-21-2100√-2-√-20-√3√3√3-√3000-ζ87ζ32+ζ85ζ32+ζ85-ζ87ζ3+ζ85ζ3+ζ85-ζ83ζ3+ζ8ζ3+ζ8-ζ83ζ32+ζ8ζ32+ζ8    complex lifted from C24⋊C2
ρ172-2022-1-100-21-2100-√-2√-20√3-√3-√3√3000-ζ83ζ3+ζ8ζ3+ζ8-ζ83ζ32+ζ8ζ32+ζ8-ζ87ζ32+ζ85ζ32+ζ85-ζ87ζ3+ζ85ζ3+ζ85    complex lifted from C24⋊C2
ρ182-2022-1-100-21-2100-√-2√-20-√3√3√3-√3000-ζ83ζ32+ζ8ζ32+ζ8-ζ83ζ3+ζ8ζ3+ζ8-ζ87ζ3+ζ85ζ3+ζ85-ζ87ζ32+ζ85ζ32+ζ85    complex lifted from C24⋊C2
ρ192-2022-1-100-21-2100√-2-√-20√3-√3-√3√3000-ζ87ζ3+ζ85ζ3+ζ85-ζ87ζ32+ζ85ζ32+ζ85-ζ83ζ32+ζ8ζ32+ζ8-ζ83ζ3+ζ8ζ3+ζ8    complex lifted from C24⋊C2
ρ204404-2-21404-2-2100004-2-211-2000000    orthogonal lifted from S32
ρ214404-2-21-404-2-210000-422-1-12000000    orthogonal lifted from C3⋊D12
ρ224-404-24-200-4-4220000000000000000    orthogonal lifted from Q8⋊2S3
ρ234-404-2-2100-422-1000002√3-2√3√3-√30000000    orthogonal lifted from C32⋊5SD16
ρ244-404-2-2100-422-100000-2√32√3-√3√30000000    orthogonal lifted from C32⋊5SD16
ρ25662-300060-3000-1-100-300000000000    orthogonal lifted from C32⋊D6
ρ2666-2-300060-30001100-300000000000    orthogonal lifted from C32⋊D6
ρ27660-3000-60-3000-√-3√-300300000000000    complex lifted from He3⋊3D4
ρ28660-3000-60-3000√-3-√-300300000000000    complex lifted from He3⋊3D4
ρ2912-120-60000060000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of He3⋊4SD16
►On 72 points
Generators in S72
(1 68 34)(2 69 35)(3 70 36)(4 71 37)(5 72 38)(6 65 39)(7 66 40)(8 67 33)(9 28 44)(10 29 45)(11 30 46)(12 31 47)(13 32 48)(14 25 41)(15 26 42)(16 27 43)(17 60 54)(18 61 55)(19 62 56)(20 63 49)(21 64 50)(22 57 51)(23 58 52)(24 59 53)

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

(9 50 35)(10 36 51)(11 52 37)(12 38 53)(13 54 39)(14 40 55)(15 56 33)(16 34 49)(41 61 66)(42 67 62)(43 63 68)(44 69 64)(45 57 70)(46 71 58)(47 59 72)(48 65 60)

(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)

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

Matrix representation of He3⋊4SD16 ►in GL10(𝔽73)

72010000000
07201000000
72000000000
07200000000
00000072100
000011717200
00000072010
00000072001
00000072000
00001072000
,
1000000000
0100000000
0010000000
0001000000
00007210000
00007200000
00007200100
000001727200
00007200001
000001007272
,
07200000000
17200000000
00072000000
00172000000
0000100000
0000010000
000011727200
0000001000
0000000001
000011007272
,
036062000000
360620000000
011025000000
110250000000
000068550000
00005050000
000068055500
00005055231800
000068000555
00005055002318
,
05907000000
59070000000
066014000000
660140000000
00007200000
00000720000
00000000720
00000000072
00000072000
00000007200

G:=sub<GL(10,GF(73))| [72,0,72,0,0,0,0,0,0,0,0,72,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,71,72,72,72,72,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,72,0,72,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[0,36,0,11,0,0,0,0,0,0,36,0,11,0,0,0,0,0,0,0,0,62,0,25,0,0,0,0,0,0,62,0,25,0,0,0,0,0,0,0,0,0,0,0,68,50,68,50,68,50,0,0,0,0,55,5,0,55,0,55,0,0,0,0,0,0,55,23,0,0,0,0,0,0,0,0,5,18,0,0,0,0,0,0,0,0,0,0,55,23,0,0,0,0,0,0,0,0,5,18],[0,59,0,66,0,0,0,0,0,0,59,0,66,0,0,0,0,0,0,0,0,7,0,14,0,0,0,0,0,0,7,0,14,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0] >;
 

He3⋊4SD16 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_4{\rm SD}_{16}
 
% in TeX
 
G:=Group("He3:4SD16");
 
// GroupNames label
 
G:=SmallGroup(432,84);
 
// by ID
 
G=gap.SmallGroup(432,84);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,571,4037,537,14118,7069]);
 
// 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=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

Export

Character table of He3⋊4SD16 in TeX

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