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G = SD32⋊S3  order 192 = 26·3

2nd semidirect product of SD32 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.4D6, D6.8D8, C16.2D6, SD32⋊2S3, Q16.1D6, Dic24⋊6C2, C48.9C22, C24.18C23, Dic3.10D8, Dic12.3C22, C3⋊C8.4D4, C4.6(S3×D4), D6.C8⋊2C2, D8⋊3S3.C2, D8.S3⋊4C2, (S3×Q16)⋊4C2, C3⋊Q32⋊1C2, (C4×S3).9D4, C6.37(C2×D8), C2.21(S3×D8), (C3×SD32)⋊2C2, C12.12(C2×D4), C3⋊2(Q32⋊C2), C3⋊C16.1C22, (S3×C8).5C22, C8.24(C22×S3), (C3×D8).4C22, (C3×Q16).2C22, SmallGroup(192,474)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — SD32⋊S3
Chief series
C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×Q16 — SD32⋊S3
Lower central
C3 — C6 — C12 — C24 — SD32⋊S3
Upper central
C1 — C2 — C4 — C8 — SD32

Generators and relations for SD32⋊S3
 G = < a,b,c,d | a16=b2=c3=d2=1, bab=a7, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >

Subgroups: 268 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, Dic3, Dic3, C12, C12, D6, C2×C6, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, M5(2), SD32, SD32, Q32, C2×Q16, C4○D8, C3⋊C16, C48, S3×C8, Dic12, D4.S3, C3⋊Q16, C3×D8, C3×Q16, D4⋊2S3, S3×Q8, Q32⋊C2, D6.C8, Dic24, D8.S3, C3⋊Q32, C3×SD32, D8⋊3S3, S3×Q16, SD32⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S3×D4, Q32⋊C2, S3×D8, SD32⋊S3

Character table of SD32⋊S3

 class 12A2B2C34A4B4C4D4E6A6B8A8B8C12A12B16A16B16C16D24A24B48A48B48C48D
 size 1168226824242162212416441212444444
ρ1111111111111111111111111111    trivial
ρ2111-111111-11-111111-1-1-1-111-1-1-1-1    linear of order 2
ρ311-1111-1-11-11111-11-1-1-11111-1-1-1-1    linear of order 2
ρ411-1-111-1-1111-111-11-111-1-1111111    linear of order 2
ρ51111111-1-11111111-1-1-1-1-111-1-1-1-1    linear of order 2
ρ6111-1111-1-1-11-11111-11111111111    linear of order 2
ρ711-1111-11-1-11111-11111-1-1111111    linear of order 2
ρ811-1-111-11-111-111-111-1-11111-1-1-1-1    linear of order 2
ρ9220-2-120200-11220-1-1-2-200-1-11111    orthogonal lifted from D6
ρ10222022200020-2-2-2200000-2-20000    orthogonal lifted from D4
ρ112202-120-200-1-1220-11-2-200-1-11111    orthogonal lifted from D6
ρ12220-2-120-200-11220-112200-1-1-1-1-1-1    orthogonal lifted from D6
ρ132202-120200-1-1220-1-12200-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422-2022-200020-2-22200000-2-20000    orthogonal lifted from D4
ρ1522-202-2200020000-20√2-√2-√2√200-√2-√2√2√2    orthogonal lifted from D8
ρ1622202-2-200020000-20√2-√2√2-√200-√2-√2√2√2    orthogonal lifted from D8
ρ1722202-2-200020000-20-√2√2-√2√200√2√2-√2-√2    orthogonal lifted from D8
ρ1822-202-2200020000-20-√2√2√2-√200√2√2-√2-√2    orthogonal lifted from D8
ρ194400-240000-20-4-40-200000220000    orthogonal lifted from S3×D4
ρ204400-2-40000-20000202√2-2√20000√2√2-√2-√2    orthogonal lifted from S3×D8
ρ214400-2-40000-2000020-2√22√20000-√2-√2√2√2    orthogonal lifted from S3×D8
ρ224-400400000-40-2√22√20000000-2√22√20000    symplectic lifted from Q32⋊C2, Schur index 2
ρ234-400400000-402√2-2√200000002√2-2√20000    symplectic lifted from Q32⋊C2, Schur index 2
ρ244-400-20000020-2√22√20000000√2-√22ζ165ζ3+ζ165+2ζ163ζ3+ζ1632ζ165ζ32+ζ165+2ζ163ζ32+ζ1632ζ1615ζ32+ζ1615+2ζ169ζ32+ζ1692ζ167ζ32+ζ167+2ζ16ζ32+ζ16    symplectic faithful, Schur index 2
ρ254-400-200000202√2-2√20000000-√2√22ζ167ζ32+ζ167+2ζ16ζ32+ζ162ζ1615ζ32+ζ1615+2ζ169ζ32+ζ1692ζ165ζ3+ζ165+2ζ163ζ3+ζ1632ζ165ζ32+ζ165+2ζ163ζ32+ζ163    symplectic faithful, Schur index 2
ρ264-400-20000020-2√22√20000000√2-√22ζ165ζ32+ζ165+2ζ163ζ32+ζ1632ζ165ζ3+ζ165+2ζ163ζ3+ζ1632ζ167ζ32+ζ167+2ζ16ζ32+ζ162ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169    symplectic faithful, Schur index 2
ρ274-400-200000202√2-2√20000000-√2√22ζ1615ζ32+ζ1615+2ζ169ζ32+ζ1692ζ167ζ32+ζ167+2ζ16ζ32+ζ162ζ165ζ32+ζ165+2ζ163ζ32+ζ1632ζ165ζ3+ζ165+2ζ163ζ3+ζ163    symplectic faithful, Schur index 2

Smallest permutation representation of SD32⋊S3
►On 96 points
Generators in S96
(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 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)(49 57)(50 64)(51 55)(52 62)(54 60)(56 58)(59 63)(65 69)(66 76)(68 74)(70 72)(71 79)(73 77)(78 80)(81 83)(82 90)(84 88)(85 95)(87 93)(89 91)(92 96)

(1 94 39)(2 95 40)(3 96 41)(4 81 42)(5 82 43)(6 83 44)(7 84 45)(8 85 46)(9 86 47)(10 87 48)(11 88 33)(12 89 34)(13 90 35)(14 91 36)(15 92 37)(16 93 38)(17 57 79)(18 58 80)(19 59 65)(20 60 66)(21 61 67)(22 62 68)(23 63 69)(24 64 70)(25 49 71)(26 50 72)(27 51 73)(28 52 74)(29 53 75)(30 54 76)(31 55 77)(32 56 78)

(1 25)(2 18)(3 27)(4 20)(5 29)(6 22)(7 31)(8 24)(9 17)(10 26)(11 19)(12 28)(13 21)(14 30)(15 23)(16 32)(33 59)(34 52)(35 61)(36 54)(37 63)(38 56)(39 49)(40 58)(41 51)(42 60)(43 53)(44 62)(45 55)(46 64)(47 57)(48 50)(65 88)(66 81)(67 90)(68 83)(69 92)(70 85)(71 94)(72 87)(73 96)(74 89)(75 82)(76 91)(77 84)(78 93)(79 86)(80 95)
 
G:=sub<Sym(96)| (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)(49,57)(50,64)(51,55)(52,62)(54,60)(56,58)(59,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96), (1,94,39)(2,95,40)(3,96,41)(4,81,42)(5,82,43)(6,83,44)(7,84,45)(8,85,46)(9,86,47)(10,87,48)(11,88,33)(12,89,34)(13,90,35)(14,91,36)(15,92,37)(16,93,38)(17,57,79)(18,58,80)(19,59,65)(20,60,66)(21,61,67)(22,62,68)(23,63,69)(24,64,70)(25,49,71)(26,50,72)(27,51,73)(28,52,74)(29,53,75)(30,54,76)(31,55,77)(32,56,78), (1,25)(2,18)(3,27)(4,20)(5,29)(6,22)(7,31)(8,24)(9,17)(10,26)(11,19)(12,28)(13,21)(14,30)(15,23)(16,32)(33,59)(34,52)(35,61)(36,54)(37,63)(38,56)(39,49)(40,58)(41,51)(42,60)(43,53)(44,62)(45,55)(46,64)(47,57)(48,50)(65,88)(66,81)(67,90)(68,83)(69,92)(70,85)(71,94)(72,87)(73,96)(74,89)(75,82)(76,91)(77,84)(78,93)(79,86)(80,95)>;
 
G:=Group( (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)(49,57)(50,64)(51,55)(52,62)(54,60)(56,58)(59,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96), (1,94,39)(2,95,40)(3,96,41)(4,81,42)(5,82,43)(6,83,44)(7,84,45)(8,85,46)(9,86,47)(10,87,48)(11,88,33)(12,89,34)(13,90,35)(14,91,36)(15,92,37)(16,93,38)(17,57,79)(18,58,80)(19,59,65)(20,60,66)(21,61,67)(22,62,68)(23,63,69)(24,64,70)(25,49,71)(26,50,72)(27,51,73)(28,52,74)(29,53,75)(30,54,76)(31,55,77)(32,56,78), (1,25)(2,18)(3,27)(4,20)(5,29)(6,22)(7,31)(8,24)(9,17)(10,26)(11,19)(12,28)(13,21)(14,30)(15,23)(16,32)(33,59)(34,52)(35,61)(36,54)(37,63)(38,56)(39,49)(40,58)(41,51)(42,60)(43,53)(44,62)(45,55)(46,64)(47,57)(48,50)(65,88)(66,81)(67,90)(68,83)(69,92)(70,85)(71,94)(72,87)(73,96)(74,89)(75,82)(76,91)(77,84)(78,93)(79,86)(80,95) );
 
G=PermutationGroup([[(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,45),(34,36),(35,43),(37,41),(38,48),(40,46),(42,44),(49,57),(50,64),(51,55),(52,62),(54,60),(56,58),(59,63),(65,69),(66,76),(68,74),(70,72),(71,79),(73,77),(78,80),(81,83),(82,90),(84,88),(85,95),(87,93),(89,91),(92,96)], [(1,94,39),(2,95,40),(3,96,41),(4,81,42),(5,82,43),(6,83,44),(7,84,45),(8,85,46),(9,86,47),(10,87,48),(11,88,33),(12,89,34),(13,90,35),(14,91,36),(15,92,37),(16,93,38),(17,57,79),(18,58,80),(19,59,65),(20,60,66),(21,61,67),(22,62,68),(23,63,69),(24,64,70),(25,49,71),(26,50,72),(27,51,73),(28,52,74),(29,53,75),(30,54,76),(31,55,77),(32,56,78)], [(1,25),(2,18),(3,27),(4,20),(5,29),(6,22),(7,31),(8,24),(9,17),(10,26),(11,19),(12,28),(13,21),(14,30),(15,23),(16,32),(33,59),(34,52),(35,61),(36,54),(37,63),(38,56),(39,49),(40,58),(41,51),(42,60),(43,53),(44,62),(45,55),(46,64),(47,57),(48,50),(65,88),(66,81),(67,90),(68,83),(69,92),(70,85),(71,94),(72,87),(73,96),(74,89),(75,82),(76,91),(77,84),(78,93),(79,86),(80,95)]])
 

Matrix representation of SD32⋊S3 ►in GL4(𝔽7) generated by

0536
3233
6226
5453
,
5316
1005
0060
6143
,
0052
5303
2503
5512
,
3222
3010
5211
2223
G:=sub<GL(4,GF(7))| [0,3,6,5,5,2,2,4,3,3,2,5,6,3,6,3],[5,1,0,6,3,0,0,1,1,0,6,4,6,5,0,3],[0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[3,3,5,2,2,0,2,2,2,1,1,2,2,0,1,3] >;
 

SD32⋊S3 in GAP, Magma, Sage, TeX 

{\rm SD}_{32}\rtimes S_3
 
% in TeX
 
G:=Group("SD32:S3");
 
// GroupNames label
 
G:=SmallGroup(192,474);
 
// by ID
 
G=gap.SmallGroup(192,474);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,135,346,185,192,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^16=b^2=c^3=d^2=1,b*a*b=a^7,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of SD32⋊S3 in TeX

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