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G = Dic3.SD16order 192 = 26·3

1st non-split extension by Dic3 of SD16 acting via SD16/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3.1D8, Dic3.1SD16, C12⋊Q82C2, C4⋊C4.2D6, C2.7(S3×D8), C6.20(C2×D8), D4⋊C41S3, (C2×D4).17D6, C6.D81C2, (C2×C8).199D6, C32(C4.4D8), C2.8(S3×SD16), (C8×Dic3)⋊19C2, C12.1(C4○D4), C123D4.2C2, C2.D2420C2, D4⋊Dic31C2, C6.19(C2×SD16), C4.20(C4○D12), (C2×Dic3).86D4, (C6×D4).21C22, C22.162(S3×D4), C4.46(D42S3), (C2×C12).200C23, (C2×C24).221C22, C6.22(C4.4D4), (C2×D12).44C22, C4⋊Dic3.60C22, (C4×Dic3).222C22, C2.12(C23.11D6), (C2×C6).213(C2×D4), (C3×C4⋊C4).5C22, (C3×D4⋊C4)⋊21C2, (C2×C3⋊C8).209C22, (C2×C4).307(C22×S3), SmallGroup(192,319)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Dic3.SD16
C1C3C6C12C2×C12C4×Dic3C123D4 — Dic3.SD16
C3C6C2×C12 — Dic3.SD16
C1C22C2×C4D4⋊C4

Generators and relations for Dic3.SD16
 G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=a3c3 >

Subgroups: 424 in 118 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×C8, D4⋊C4, D4⋊C4, C41D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×D12, C2×C3⋊D4, C6×D4, C4.4D8, C6.D8, C8×Dic3, C2.D24, D4⋊Dic3, C3×D4⋊C4, C12⋊Q8, C123D4, Dic3.SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, C22×S3, C4.4D4, C2×D8, C2×SD16, C4○D12, S3×D4, D42S3, C4.4D8, C23.11D6, S3×D8, S3×SD16, Dic3.SD16

Smallest permutation representation of Dic3.SD16
On 96 points
Generators in S96
(1 80 84 63 24 37)(2 73 85 64 17 38)(3 74 86 57 18 39)(4 75 87 58 19 40)(5 76 88 59 20 33)(6 77 81 60 21 34)(7 78 82 61 22 35)(8 79 83 62 23 36)(9 89 67 49 26 46)(10 90 68 50 27 47)(11 91 69 51 28 48)(12 92 70 52 29 41)(13 93 71 53 30 42)(14 94 72 54 31 43)(15 95 65 55 32 44)(16 96 66 56 25 45)
(1 16 63 56)(2 9 64 49)(3 10 57 50)(4 11 58 51)(5 12 59 52)(6 13 60 53)(7 14 61 54)(8 15 62 55)(17 67 73 46)(18 68 74 47)(19 69 75 48)(20 70 76 41)(21 71 77 42)(22 72 78 43)(23 65 79 44)(24 66 80 45)(25 37 96 84)(26 38 89 85)(27 39 90 86)(28 40 91 87)(29 33 92 88)(30 34 93 81)(31 35 94 82)(32 36 95 83)
(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)
(1 50)(2 13)(3 56)(4 11)(5 54)(6 9)(7 52)(8 15)(10 63)(12 61)(14 59)(16 57)(17 30)(18 96)(19 28)(20 94)(21 26)(22 92)(23 32)(24 90)(25 74)(27 80)(29 78)(31 76)(33 72)(34 46)(35 70)(36 44)(37 68)(38 42)(39 66)(40 48)(41 82)(43 88)(45 86)(47 84)(49 60)(51 58)(53 64)(55 62)(65 83)(67 81)(69 87)(71 85)(73 93)(75 91)(77 89)(79 95)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D
order12222234444444466666888888881212121224242424
size11118242226666824222882222666644884444

36 irreducible representations

dim111111112222222224444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D6D6D6D8SD16C4○D4C4○D12D42S3S3×D4S3×D8S3×SD16
kernelDic3.SD16C6.D8C8×Dic3C2.D24D4⋊Dic3C3×D4⋊C4C12⋊Q8C123D4D4⋊C4C2×Dic3C4⋊C4C2×C8C2×D4Dic3Dic3C12C4C4C22C2C2
# reps111111111211144441122

Matrix representation of Dic3.SD16 in GL6(𝔽73)

7200000
0720000
000100
00727200
0000720
0000072
,
010000
7200000
0072000
001100
00001665
00002357
,
57570000
16570000
0072000
0007200
0000460
0000046
,
100000
0720000
0072000
0007200
0000578
00003216

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,16,23,0,0,0,0,65,57],[57,16,0,0,0,0,57,57,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,32,0,0,0,0,8,16] >;

Dic3.SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_3.{\rm SD}_{16}
% in TeX

G:=Group("Dic3.SD16");
// GroupNames label

G:=SmallGroup(192,319);
// by ID

G=gap.SmallGroup(192,319);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,422,135,100,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=a^3*c^3>;
// generators/relations

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