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G = He3⋊3D8  order 432 = 24·33

2nd semidirect product of He3 and D8 acting via D8/C4=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊3D8, C32⋊D24, C12.9S32, (C3×C6).1D12, (C3×C12).4D6, C12⋊S3⋊2S3, He3⋊3C8⋊1C2, He3⋊4D4⋊2C2, He3⋊5D4⋊1C2, C32⋊4C8⋊1S3, (C2×He3).9D4, C32⋊2(D4⋊S3), C4.1(C32⋊D6), C3.3(C3⋊D24), C2.4(He3⋊3D4), (C4×He3).4C22, C6.29(C3⋊D12), (C3×C6).4(C3⋊D4), SmallGroup(432,83)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C4×He3 — He3⋊3D8
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4D4 — He3⋊3D8
Lower central
He3 — C2×He3 — C4×He3 — He3⋊3D8
Upper central
C1 — C2 — C4

Generators and relations for He3⋊3D8
 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=d-1 >

Subgroups: 699 in 93 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D24, D4⋊S3, C32⋊C6, He3⋊C2, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×D12, C12⋊S3, C4×He3, C2×C32⋊C6, C2×He3⋊C2, C32⋊2D8, C3⋊D24, He3⋊3C8, He3⋊4D4, He3⋊5D4, He3⋊3D8
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, C3⋊D4, S32, D24, D4⋊S3, C3⋊D12, C32⋊D6, C3⋊D24, He3⋊3D4, He3⋊3D8

Character table of He3⋊3D8

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H8A8B12A12B12C12D12E12F24A24B24C24D
 size 1136362661222661236363636181846612121218181818
ρ111111111111111111111111111111    trivial
ρ211-1-1111111111-1-1-1-1111111111111    linear of order 2
ρ3111-11111111111-11-1-1-1111111-1-1-1-1    linear of order 2
ρ411-11111111111-11-11-1-1111111-1-1-1-1    linear of order 2
ρ5220022-1-1222-1-10000-2-22-1-1-12-11111    orthogonal lifted from D6
ρ6220-22-12-122-12-1010100222-1-1-10000    orthogonal lifted from D6
ρ722022-12-122-12-10-10-100222-1-1-10000    orthogonal lifted from S3
ρ8220022-1-1222-1-10000222-1-1-12-1-1-1-1-1    orthogonal lifted from S3
ρ922002222-22222000000-2-2-2-2-2-20000    orthogonal lifted from D4
ρ10220022-1-1-222-1-1000000-2111-21√3√3-√3-√3    orthogonal lifted from D12
ρ11220022-1-1-222-1-1000000-2111-21-√3-√3√3√3    orthogonal lifted from D12
ρ122-20022220-2-2-2-20000-√2√2000000√2-√2√2-√2    orthogonal lifted from D8
ρ132-20022220-2-2-2-20000√2-√2000000-√2√2-√2√2    orthogonal lifted from D8
ρ142-20022-1-10-2-2110000√2-√20-√3√3-√30√3ζ83ζ3+ζ8ζ3+ζ8ζ83ζ32+ζ83+ζ8ζ32ζ87ζ3+ζ87+ζ85ζ3ζ87ζ32+ζ85ζ32+ζ85    orthogonal lifted from D24
ρ152-20022-1-10-2-2110000-√2√20-√3√3-√30√3ζ83ζ32+ζ83+ζ8ζ32ζ83ζ3+ζ8ζ3+ζ8ζ87ζ32+ζ85ζ32+ζ85ζ87ζ3+ζ87+ζ85ζ3    orthogonal lifted from D24
ρ162-20022-1-10-2-2110000√2-√20√3-√3√30-√3ζ87ζ3+ζ87+ζ85ζ3ζ87ζ32+ζ85ζ32+ζ85ζ83ζ3+ζ8ζ3+ζ8ζ83ζ32+ζ83+ζ8ζ32    orthogonal lifted from D24
ρ172-20022-1-10-2-2110000-√2√20√3-√3√30-√3ζ87ζ32+ζ85ζ32+ζ85ζ87ζ3+ζ87+ζ85ζ3ζ83ζ32+ζ83+ζ8ζ32ζ83ζ3+ζ8ζ3+ζ8    orthogonal lifted from D24
ρ1822002-12-1-22-12-10-√-30√-300-2-2-21110000    complex lifted from C3⋊D4
ρ1922002-12-1-22-12-10√-30-√-300-2-2-21110000    complex lifted from C3⋊D4
ρ2044004-2-2144-2-210000004-2-21-210000    orthogonal lifted from S32
ρ214-4004-24-20-42-420000000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ2244004-2-21-44-2-21000000-422-12-10000    orthogonal lifted from C3⋊D12
ρ234-4004-2-210-422-10000000-2√32√3√30-√30000    orthogonal lifted from C3⋊D24
ρ244-4004-2-210-422-100000002√3-2√3-√30√30000    orthogonal lifted from C3⋊D24
ρ2566-20-30006-3000101000-3000000000    orthogonal lifted from C32⋊D6
ρ266620-30006-3000-10-1000-3000000000    orthogonal lifted from C32⋊D6
ρ276600-3000-6-3000-√-30√-30003000000000    complex lifted from He3⋊3D4
ρ286600-3000-6-3000√-30-√-30003000000000    complex lifted from He3⋊3D4
ρ2912-1200-6000060000000000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of He3⋊3D8
►On 72 points
Generators in S72
(1 25 47)(2 26 48)(3 27 41)(4 28 42)(5 29 43)(6 30 44)(7 31 45)(8 32 46)(9 33 51)(10 34 52)(11 35 53)(12 36 54)(13 37 55)(14 38 56)(15 39 49)(16 40 50)(17 68 57)(18 69 58)(19 70 59)(20 71 60)(21 72 61)(22 65 62)(23 66 63)(24 67 64)

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

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

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

Matrix representation of He3⋊3D8 ►in GL10(𝔽73)

1000000000
0100000000
0010000000
0001000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
0000010000
000072720000
0000000100
000000727200
0000000001
000000007272
,
727200000000
1000000000
007272000000
0010000000
000000007272
0000000010
0000010000
000072720000
0000001000
0000000100
,
570160000000
16165757000000
570570000000
16161616000000
0000691459555955
000018469146914
0000595569145955
000069141846914
0000595559556914
000069146914184
,
00720000000
0011000000
72000000000
1100000000
00007200000
00000720000
00000000720
00000000072
00000072000
00000007200

G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0],[57,16,57,16,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,57,57,16,0,0,0,0,0,0,0,57,0,16,0,0,0,0,0,0,0,0,0,0,69,18,59,69,59,69,0,0,0,0,14,4,55,14,55,14,0,0,0,0,59,69,69,18,59,69,0,0,0,0,55,14,14,4,55,14,0,0,0,0,59,69,59,69,69,18,0,0,0,0,55,14,55,14,14,4],[0,0,72,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0] >;
 

He3⋊3D8 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_3D_8
 
% in TeX
 
G:=Group("He3:3D8");
 
// GroupNames label
 
G:=SmallGroup(432,83);
 
// by ID
 
G=gap.SmallGroup(432,83);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,571,4037,537,14118,7069]);
 
// 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^-1>;
 
// generators/relations
 

Export

Character table of He3⋊3D8 in TeX

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