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G = Dic33SD16order 192 = 26·3

2nd semidirect product of Dic3 and SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic33SD16, (C3×D4).7D4, (C2×C8).143D6, (C2×Q8).74D6, Dic3⋊C832C2, (C2×D4).143D6, C12.170(C2×D4), C35(D4.D4), D4.2(C3⋊D4), (C2×SD16).3S3, (C6×SD16).6C2, (D4×Dic3).8C2, C2.26(S3×SD16), C6.43(C2×SD16), Dic3⋊Q83C2, C12.96(C4○D4), C4.9(D42S3), Q82Dic325C2, C2.Dic1233C2, (C6×D4).88C22, C22.260(S3×D4), (C6×Q8).69C22, C6.112(C4⋊D4), (C2×C24).290C22, (C2×C12).439C23, (C2×Dic3).182D4, C2.26(D4.D6), C6.45(C8.C22), C4⋊Dic3.169C22, (C4×Dic3).46C22, C2.24(C23.14D6), (C2×Dic6).123C22, C4.38(C2×C3⋊D4), (C2×C6).351(C2×D4), (C2×D4.S3).8C2, (C2×C3⋊C8).151C22, (C2×C4).528(C22×S3), SmallGroup(192,721)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Dic33SD16
C1C3C6C12C2×C12C4×Dic3D4×Dic3 — Dic33SD16
C3C6C2×C12 — Dic33SD16
C1C22C2×C4C2×SD16

Generators and relations for Dic33SD16
 G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c3 >

Subgroups: 328 in 120 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C2×SD16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C6.D4, C2×C24, C3×SD16, C2×Dic6, C22×Dic3, C6×D4, C6×Q8, D4.D4, Dic3⋊C8, C2.Dic12, Q82Dic3, C2×D4.S3, D4×Dic3, Dic3⋊Q8, C6×SD16, Dic33SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8.C22, S3×D4, D42S3, C2×C3⋊D4, D4.D4, S3×SD16, D4.D6, C23.14D6, Dic33SD16

Smallest permutation representation of Dic33SD16
On 96 points
Generators in S96
(1 53 69 19 43 77)(2 54 70 20 44 78)(3 55 71 21 45 79)(4 56 72 22 46 80)(5 49 65 23 47 73)(6 50 66 24 48 74)(7 51 67 17 41 75)(8 52 68 18 42 76)(9 26 95 62 81 36)(10 27 96 63 82 37)(11 28 89 64 83 38)(12 29 90 57 84 39)(13 30 91 58 85 40)(14 31 92 59 86 33)(15 32 93 60 87 34)(16 25 94 61 88 35)
(1 12 19 57)(2 58 20 13)(3 14 21 59)(4 60 22 15)(5 16 23 61)(6 62 24 9)(7 10 17 63)(8 64 18 11)(25 65 88 73)(26 74 81 66)(27 67 82 75)(28 76 83 68)(29 69 84 77)(30 78 85 70)(31 71 86 79)(32 80 87 72)(33 45 92 55)(34 56 93 46)(35 47 94 49)(36 50 95 48)(37 41 96 51)(38 52 89 42)(39 43 90 53)(40 54 91 44)
(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 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 21)(18 24)(20 22)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(74 76)(75 79)(78 80)(81 83)(82 86)(85 87)(89 95)(91 93)(92 96)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
order12222234444444446666688881212121224242424
size111144222668121212242228844121244884444

33 irreducible representations

dim1111111122222222244444
type++++++++++++++--+-
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6SD16C4○D4C3⋊D4C8.C22D42S3S3×D4S3×SD16D4.D6
kernelDic33SD16Dic3⋊C8C2.Dic12Q82Dic3C2×D4.S3D4×Dic3Dic3⋊Q8C6×SD16C2×SD16C2×Dic3C3×D4C2×C8C2×D4C2×Q8Dic3C12D4C6C4C22C2C2
# reps1111111112211142411122

Matrix representation of Dic33SD16 in GL6(𝔽73)

7200000
0720000
001000
000100
0000721
0000720
,
7590000
14660000
0072000
0007200
00005132
00001022
,
010000
100000
00121200
0067000
0000720
0000072
,
100000
010000
00727100
000100
000010
000001

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,51,10,0,0,0,0,32,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,67,0,0,0,0,12,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,71,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic33SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_3{\rm SD}_{16}
% in TeX

G:=Group("Dic3:3SD16");
// GroupNames label

G:=SmallGroup(192,721);
// by ID

G=gap.SmallGroup(192,721);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,135,184,570,297,136,6278]);
// 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,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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