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G = Dic35D8order 192 = 26·3

2nd semidirect product of Dic3 and D8 acting through Inn(Dic3)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D246C4, Dic35D8, C33(C4×D8), C87(C4×S3), C244(C2×C4), C2.3(S3×D8), D127(C2×C4), C6.52(C4×D4), C6.26(C2×D8), C2.D813S3, C4⋊C4.166D6, (C8×Dic3)⋊3C2, (C2×D24).9C2, (C2×C8).225D6, Dic35D47C2, C6.73(C4○D8), C6.D819C2, C22.87(S3×D4), C12.35(C4○D4), C12.46(C22×C4), (C2×C24).77C22, C4.7(Q83S3), C2.3(D24⋊C2), (C2×C12).288C23, (C2×Dic3).207D4, (C2×D12).80C22, C2.12(Dic35D4), (C4×Dic3).231C22, C4.43(S3×C2×C4), (C3×C2.D8)⋊2C2, (C2×C6).293(C2×D4), (C3×C4⋊C4).81C22, (C2×C3⋊C8).229C22, (C2×C4).391(C22×S3), SmallGroup(192,431)

Series: Derived Chief Lower central Upper central

C1C12 — Dic35D8
C1C3C6C12C2×C12C4×Dic3Dic35D4 — Dic35D8
C3C6C12 — Dic35D8
C1C22C2×C4C2.D8

Generators and relations for Dic35D8
 G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 448 in 134 conjugacy classes, 51 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×3], C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×6], C23 [×2], Dic3 [×2], Dic3, C12 [×2], C12 [×2], D6 [×8], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], C3⋊C8, C24 [×2], C4×S3 [×4], D12 [×4], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, D24 [×4], C2×C3⋊C8, C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4 [×2], C2×C24, S3×C2×C4 [×2], C2×D12 [×2], C4×D8, C6.D8 [×2], C8×Dic3, C3×C2.D8, Dic35D4 [×2], C2×D24, Dic35D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], D8 [×2], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, C2×D8, C4○D8, S3×C2×C4, S3×D4, Q83S3, C4×D8, Dic35D4, S3×D8, D24⋊C2, Dic35D8

Smallest permutation representation of Dic35D8
On 96 points
Generators in S96
(1 88 77 51 29 12)(2 81 78 52 30 13)(3 82 79 53 31 14)(4 83 80 54 32 15)(5 84 73 55 25 16)(6 85 74 56 26 9)(7 86 75 49 27 10)(8 87 76 50 28 11)(17 89 33 60 42 67)(18 90 34 61 43 68)(19 91 35 62 44 69)(20 92 36 63 45 70)(21 93 37 64 46 71)(22 94 38 57 47 72)(23 95 39 58 48 65)(24 96 40 59 41 66)
(1 18 51 61)(2 19 52 62)(3 20 53 63)(4 21 54 64)(5 22 55 57)(6 23 56 58)(7 24 49 59)(8 17 50 60)(9 95 74 48)(10 96 75 41)(11 89 76 42)(12 90 77 43)(13 91 78 44)(14 92 79 45)(15 93 80 46)(16 94 73 47)(25 38 84 72)(26 39 85 65)(27 40 86 66)(28 33 87 67)(29 34 88 68)(30 35 81 69)(31 36 82 70)(32 37 83 71)
(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 5)(2 4)(6 8)(9 87)(10 86)(11 85)(12 84)(13 83)(14 82)(15 81)(16 88)(17 23)(18 22)(19 21)(25 77)(26 76)(27 75)(28 74)(29 73)(30 80)(31 79)(32 78)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(50 56)(51 55)(52 54)(57 61)(58 60)(62 64)(65 89)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A24B24C24D
order1222222234444444444446668888888812121212121224242424
size1111121212122223333444466222222266664488884444

42 irreducible representations

dim1111111222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C4S3D4D6D6D8C4○D4C4×S3C4○D8Q83S3S3×D4S3×D8D24⋊C2
kernelDic35D8C6.D8C8×Dic3C3×C2.D8Dic35D4C2×D24D24C2.D8C2×Dic3C4⋊C4C2×C8Dic3C12C8C6C4C22C2C2
# reps1211218122142441122

Matrix representation of Dic35D8 in GL5(𝔽73)

720000
01000
00100
000721
000720
,
460000
072000
007200
00001
00010
,
10000
0412500
035000
00010
00001
,
720000
072000
025100
00001
00010

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[46,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,41,35,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,25,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

Dic35D8 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_5D_8
% in TeX

G:=Group("Dic3:5D8");
// GroupNames label

G:=SmallGroup(192,431);
// by ID

G=gap.SmallGroup(192,431);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,120,135,100,570,297,136,6278]);
// Polycyclic

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

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