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G = Dic33D12order 288 = 25·32

2nd semidirect product of Dic3 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: Dic33D12, C62.80C23, D6⋊C45S3, C6.23(S3×D4), Dic3⋊C48S3, (C3×Dic3)⋊2D4, (C2×C12).27D6, C2.25(S3×D12), C6.24(C2×D12), C31(Dic3⋊D4), C31(C12⋊D4), C328(C4⋊D4), (C6×C12).8C22, C6.17(C4○D12), C6.D128C2, (C2×Dic3).76D6, (C22×S3).15D6, C2.11(Dic3⋊D6), C6.18(Q83S3), C2.19(D6.6D6), (C6×Dic3).69C22, (C2×C4).33S32, (C2×C3⋊S3)⋊1D4, (C3×D6⋊C4)⋊7C2, (C2×C12⋊S3)⋊1C2, C22.118(C2×S32), (C2×C6.D6)⋊1C2, (C3×C6).106(C2×D4), (S3×C2×C6).30C22, (C2×C3⋊D12)⋊16C2, (C3×Dic3⋊C4)⋊22C2, (C3×C6).50(C4○D4), (C2×C6).99(C22×S3), (C22×C3⋊S3).24C22, SmallGroup(288,558)

Series: Derived Chief Lower central Upper central

C1C62 — Dic33D12
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — Dic33D12
C32C62 — Dic33D12
C1C22C2×C4

Generators and relations for Dic33D12
 G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 1090 in 215 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×11], C6 [×6], C6 [×4], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×2], Dic3 [×2], C12 [×8], D6 [×25], C2×C6 [×2], C2×C6 [×4], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3, C3⋊S3 [×3], C3×C6 [×3], C4×S3 [×4], D12 [×12], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×6], C22×C6, C4⋊D4, C3×Dic3 [×2], C3×Dic3 [×2], C3×C12, S3×C6 [×3], C2×C3⋊S3 [×2], C2×C3⋊S3 [×5], C62, Dic3⋊C4, D6⋊C4, D6⋊C4 [×2], C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4 [×2], C2×D12 [×5], C2×C3⋊D4 [×2], C6.D6 [×2], C3⋊D12 [×4], C6×Dic3 [×3], C12⋊S3 [×2], C6×C12, S3×C2×C6, C22×C3⋊S3 [×2], Dic3⋊D4, C12⋊D4, C6.D12, C3×Dic3⋊C4, C3×D6⋊C4, C2×C6.D6, C2×C3⋊D12 [×2], C2×C12⋊S3, Dic33D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12, C4○D12, S3×D4 [×3], Q83S3, C2×S32, Dic3⋊D4, C12⋊D4, D6.6D6, S3×D12, Dic3⋊D6, Dic33D12

Smallest permutation representation of Dic33D12
On 48 points
Generators in S48
(1 46 9 42 5 38)(2 47 10 43 6 39)(3 48 11 44 7 40)(4 37 12 45 8 41)(13 31 17 35 21 27)(14 32 18 36 22 28)(15 33 19 25 23 29)(16 34 20 26 24 30)
(1 16 42 26)(2 27 43 17)(3 18 44 28)(4 29 45 19)(5 20 46 30)(6 31 47 21)(7 22 48 32)(8 33 37 23)(9 24 38 34)(10 35 39 13)(11 14 40 36)(12 25 41 15)
(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)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(25 29)(26 28)(30 36)(31 35)(32 34)(38 48)(39 47)(40 46)(41 45)(42 44)

G:=sub<Sym(48)| (1,46,9,42,5,38)(2,47,10,43,6,39)(3,48,11,44,7,40)(4,37,12,45,8,41)(13,31,17,35,21,27)(14,32,18,36,22,28)(15,33,19,25,23,29)(16,34,20,26,24,30), (1,16,42,26)(2,27,43,17)(3,18,44,28)(4,29,45,19)(5,20,46,30)(6,31,47,21)(7,22,48,32)(8,33,37,23)(9,24,38,34)(10,35,39,13)(11,14,40,36)(12,25,41,15), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,29)(26,28)(30,36)(31,35)(32,34)(38,48)(39,47)(40,46)(41,45)(42,44)>;

G:=Group( (1,46,9,42,5,38)(2,47,10,43,6,39)(3,48,11,44,7,40)(4,37,12,45,8,41)(13,31,17,35,21,27)(14,32,18,36,22,28)(15,33,19,25,23,29)(16,34,20,26,24,30), (1,16,42,26)(2,27,43,17)(3,18,44,28)(4,29,45,19)(5,20,46,30)(6,31,47,21)(7,22,48,32)(8,33,37,23)(9,24,38,34)(10,35,39,13)(11,14,40,36)(12,25,41,15), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)(25,29)(26,28)(30,36)(31,35)(32,34)(38,48)(39,47)(40,46)(41,45)(42,44) );

G=PermutationGroup([(1,46,9,42,5,38),(2,47,10,43,6,39),(3,48,11,44,7,40),(4,37,12,45,8,41),(13,31,17,35,21,27),(14,32,18,36,22,28),(15,33,19,25,23,29),(16,34,20,26,24,30)], [(1,16,42,26),(2,27,43,17),(3,18,44,28),(4,29,45,19),(5,20,46,30),(6,31,47,21),(7,22,48,32),(8,33,37,23),(9,24,38,34),(10,35,39,13),(11,14,40,36),(12,25,41,15)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,21),(14,20),(15,19),(16,18),(22,24),(25,29),(26,28),(30,36),(31,35),(32,34),(38,48),(39,47),(40,46),(41,45),(42,44)])

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K12A···12H12I···12N
order122222223334444446···66666612···1212···12
size11111218183622446666122···244412124···412···12

42 irreducible representations

dim111111122222222224444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C4○D12S32S3×D4Q83S3C2×S32D6.6D6S3×D12Dic3⋊D6
kernelDic33D12C6.D12C3×Dic3⋊C4C3×D6⋊C4C2×C6.D6C2×C3⋊D12C2×C12⋊S3Dic3⋊C4D6⋊C4C3×Dic3C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6Dic3C6C2×C4C6C6C22C2C2C2
# reps111112111223212441311222

Matrix representation of Dic33D12 in GL8(𝔽13)

10000000
01000000
000120000
001120000
00001000
00000100
000000120
000000012
,
120000000
012000000
000120000
001200000
000012000
000001200
000000107
00000063
,
112000000
10000000
001200000
000120000
00000100
000012000
00000001
00000010
,
120000000
121000000
00010000
00100000
000012000
00000100
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3],[1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

Dic33D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_3D_{12}
% in TeX

G:=Group("Dic3:3D12");
// GroupNames label

G:=SmallGroup(288,558);
// by ID

G=gap.SmallGroup(288,558);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,135,142,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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