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G = SD32⋊S3order 192 = 26·3

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

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, D42S3, S3×Q8, Q32⋊C2, D6.C8, Dic24, D8.S3, C3⋊Q32, C3×SD32, D83S3, 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 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 12A 12B 16A 16B 16C 16D 24A 24B 48A 48B 48C 48D size 1 1 6 8 2 2 6 8 24 24 2 16 2 2 12 4 16 4 4 12 12 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 -1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 0 -2 -1 2 0 2 0 0 -1 1 2 2 0 -1 -1 -2 -2 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 0 2 2 2 0 0 0 2 0 -2 -2 -2 2 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 0 2 -1 2 0 -2 0 0 -1 -1 2 2 0 -1 1 -2 -2 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 0 -2 -1 2 0 -2 0 0 -1 1 2 2 0 -1 1 2 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ13 2 2 0 2 -1 2 0 2 0 0 -1 -1 2 2 0 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 0 2 2 -2 0 0 0 2 0 -2 -2 2 2 0 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 0 2 -2 2 0 0 0 2 0 0 0 0 -2 0 √2 -√2 -√2 √2 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ16 2 2 2 0 2 -2 -2 0 0 0 2 0 0 0 0 -2 0 √2 -√2 √2 -√2 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ17 2 2 2 0 2 -2 -2 0 0 0 2 0 0 0 0 -2 0 -√2 √2 -√2 √2 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ18 2 2 -2 0 2 -2 2 0 0 0 2 0 0 0 0 -2 0 -√2 √2 √2 -√2 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ19 4 4 0 0 -2 4 0 0 0 0 -2 0 -4 -4 0 -2 0 0 0 0 0 2 2 0 0 0 0 orthogonal lifted from S3×D4 ρ20 4 4 0 0 -2 -4 0 0 0 0 -2 0 0 0 0 2 0 2√2 -2√2 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from S3×D8 ρ21 4 4 0 0 -2 -4 0 0 0 0 -2 0 0 0 0 2 0 -2√2 2√2 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from S3×D8 ρ22 4 -4 0 0 4 0 0 0 0 0 -4 0 -2√2 2√2 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ23 4 -4 0 0 4 0 0 0 0 0 -4 0 2√2 -2√2 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ24 4 -4 0 0 -2 0 0 0 0 0 2 0 -2√2 2√2 0 0 0 0 0 0 0 √2 -√2 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 symplectic faithful, Schur index 2 ρ25 4 -4 0 0 -2 0 0 0 0 0 2 0 2√2 -2√2 0 0 0 0 0 0 0 -√2 √2 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 symplectic faithful, Schur index 2 ρ26 4 -4 0 0 -2 0 0 0 0 0 2 0 -2√2 2√2 0 0 0 0 0 0 0 √2 -√2 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 -2 0 0 0 0 0 2 0 2√2 -2√2 0 0 0 0 0 0 0 -√2 √2 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 2ζ165ζ3+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

 0 5 3 6 3 2 3 3 6 2 2 6 5 4 5 3
,
 5 3 1 6 1 0 0 5 0 0 6 0 6 1 4 3
,
 0 0 5 2 5 3 0 3 2 5 0 3 5 5 1 2
,
 3 2 2 2 3 0 1 0 5 2 1 1 2 2 2 3
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

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