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G = He3⋊3SD16  order 432 = 24·33

1st semidirect product of He3 and SD16 acting via SD16/C4=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊3SD16, C12.81S32, (C3×C12).1D6, He3⋊4C8⋊2C2, He3⋊3Q8⋊1C2, (C2×He3).6D4, C32⋊4Q8⋊1S3, C12⋊S3.1S3, He3⋊4D4.1C2, C4.8(C32⋊D6), C6.1(D6⋊S3), C2.3(He3⋊2D4), C32⋊2(D4.S3), (C4×He3).1C22, C32⋊2(Q8⋊2S3), C3.1(Dic6⋊S3), (C3×C6).1(C3⋊D4), SmallGroup(432,78)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C4×He3 — He3⋊3SD16
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4D4 — He3⋊3SD16
Lower central
He3 — C2×He3 — C4×He3 — He3⋊3SD16
Upper central
C1 — C2 — C4

Generators and relations for He3⋊3SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 511 in 78 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⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, D4.S3, Q8⋊2S3, C32⋊C6, C2×He3, C3×C3⋊C8, C3×Dic6, C3×D12, C32⋊4Q8, C12⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, D12.S3, C32⋊5SD16, He3⋊4C8, He3⋊3Q8, He3⋊4D4, He3⋊3SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊D4, S32, D4.S3, Q8⋊2S3, D6⋊S3, C32⋊D6, Dic6⋊S3, He3⋊2D4, He3⋊3SD16

Character table of He3⋊3SD16

 class 12A2B3A3B3C3D4A4B6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G12H24A24B24C24D
 size 11362661223626612363618182212121212363618181818
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
ρ522022-1-12-22-12-10000222-1-1-1110000    orthogonal lifted from D6
ρ622-22-12-12022-1-1110022-12-1-1000000    orthogonal lifted from D6
ρ72222-12-12022-1-1-1-10022-12-1-1000000    orthogonal lifted from S3
ρ82202222-2022220000-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ922022-1-1222-12-10000222-1-1-1-1-10000    orthogonal lifted from S3
ρ102202-12-1-2022-1-1-√-3√-300-2-21-211000000    complex lifted from C3⋊D4
ρ112202-12-1-2022-1-1√-3-√-300-2-21-211000000    complex lifted from C3⋊D4
ρ1222022-1-1-202-12-10000-2-2-2111√-3-√-30000    complex lifted from C3⋊D4
ρ1322022-1-1-202-12-10000-2-2-2111-√-3√-30000    complex lifted from C3⋊D4
ρ142-20222200-2-2-2-200-√-2√-200000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ152-20222200-2-2-2-200√-2-√-200000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ164404-2-21404-2-21000044-2-211000000    orthogonal lifted from S32
ρ174-4044-2-200-42-420000000000000000    orthogonal lifted from Q8⋊2S3
ρ184404-2-21-404-2-210000-4-422-1-1000000    symplectic lifted from D6⋊S3, Schur index 2
ρ194-404-24-200-4-4220000000000000000    symplectic lifted from D4.S3, Schur index 2
ρ204-404-2-2100-422-100000000-3i3i000000    complex lifted from Dic6⋊S3
ρ214-404-2-2100-422-1000000003i-3i000000    complex lifted from Dic6⋊S3
ρ22660-300060-300000-2-2-3-30000001111    orthogonal lifted from C32⋊D6
ρ23660-300060-30000022-3-3000000-1-1-1-1    orthogonal lifted from C32⋊D6
ρ24660-3000-60-3000000033000000√3-√3-√3√3    orthogonal lifted from He3⋊2D4
ρ25660-3000-60-3000000033000000-√3√3√3-√3    orthogonal lifted from He3⋊2D4
ρ266-60-300000300000√-2-√-2-3√33√3000000ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3    complex faithful
ρ276-60-300000300000-√-2√-23√3-3√3000000ζ87ζ32+ζ87-ζ85ζ32ζ87ζ3+ζ87-ζ85ζ3ζ83ζ3+ζ83-ζ8ζ3ζ83ζ32+ζ83-ζ8ζ32    complex faithful
ρ286-60-300000300000-√-2√-2-3√33√3000000ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3    complex faithful
ρ296-60-300000300000√-2-√-23√3-3√3000000ζ83ζ32+ζ83-ζ8ζ32ζ83ζ3+ζ83-ζ8ζ3ζ87ζ3+ζ87-ζ85ζ3ζ87ζ32+ζ87-ζ85ζ32    complex faithful

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

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

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

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

Matrix representation of He3⋊3SD16 ►in GL10(𝔽73)

59272719000000
46135446000000
23271346000000
46502759000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
00007210000
00007200000
00000072100
00000072000
00000000721
00000000720
,
07200000000
17200000000
00072000000
00172000000
0000100000
0000010000
00000007200
00000017200
00000000721
00000000720
,
005111000000
006222000000
11315111000000
42626222000000
00007200000
00000720000
00000000720
00000000072
00000072000
00000007200
,
1000000000
0100000000
10720000000
01072000000
0000010000
0000100000
0000000001
0000000010
0000000100
0000001000

G:=sub<GL(10,GF(73))| [59,46,23,46,0,0,0,0,0,0,27,13,27,50,0,0,0,0,0,0,27,54,13,27,0,0,0,0,0,0,19,46,46,59,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[0,0,11,42,0,0,0,0,0,0,0,0,31,62,0,0,0,0,0,0,51,62,51,62,0,0,0,0,0,0,11,22,11,22,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0],[1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

He3⋊3SD16 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_3{\rm SD}_{16}
 
% in TeX
 
G:=Group("He3:3SD16");
 
// GroupNames label
 
G:=SmallGroup(432,78);
 
// by ID
 
G=gap.SmallGroup(432,78);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,571,4037,537,14118,7069]);
 
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,b*d=d*b,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⋊3SD16 in TeX

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