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G = Dic6⋊3D6  order 288 = 25·32

3rd semidirect product of Dic6 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: Dic6⋊3D6, D12.21D6, D4⋊2S32, C3⋊C8⋊5D6, D4⋊S3⋊1S3, (C3×D4)⋊6D6, (S3×D12)⋊5C2, (C4×S3).5D6, (S3×C6).8D4, D4⋊2S3⋊1S3, C3⋊7(D8⋊S3), C6.147(S3×D4), C32⋊7D8⋊2C2, C3⋊2(D4⋊D6), D6.6(C3⋊D4), D6.Dic3⋊5C2, C32⋊9(C8⋊C22), (C3×C12).2C23, C12.2(C22×S3), Dic6⋊S3⋊7C2, C32⋊5SD16⋊7C2, C32⋊4C8⋊4C22, (C3×Dic3).34D4, (C3×Dic6)⋊5C22, (D4×C32)⋊2C22, (S3×C12).11C22, C12⋊S3.6C22, (C3×D12).10C22, Dic3.15(C3⋊D4), C4.2(C2×S32), (C3×D4⋊S3)⋊5C2, (C3×C3⋊C8)⋊10C22, C2.21(S3×C3⋊D4), C6.43(C2×C3⋊D4), (C3×D4⋊2S3)⋊1C2, (C3×C6).117(C2×D4), SmallGroup(288,573)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — Dic6⋊3D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — Dic6⋊3D6
Lower central
C32 — C3×C6 — C3×C12 — Dic6⋊3D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for Dic6⋊3D6
 G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a5, cbc-1=a6b, dbd=a9b, dcd=c-1 >

Subgroups: 706 in 147 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, C8⋊S3, C24⋊C2, C4.Dic3, D4⋊S3, D4⋊S3, D4.S3, Q8⋊2S3, C3×D8, C2×D12, S3×D4, D4⋊2S3, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C12⋊S3, D4×C32, C2×S32, D8⋊S3, D4⋊D6, D6.Dic3, Dic6⋊S3, C32⋊5SD16, C3×D4⋊S3, C32⋊7D8, S3×D12, C3×D4⋊2S3, Dic6⋊3D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, D8⋊S3, D4⋊D6, S3×C3⋊D4, Dic6⋊3D6

Smallest permutation representation of Dic6⋊3D6
►On 48 points
Generators in S48
(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 41 7 47)(2 40 8 46)(3 39 9 45)(4 38 10 44)(5 37 11 43)(6 48 12 42)(13 30 19 36)(14 29 20 35)(15 28 21 34)(16 27 22 33)(17 26 23 32)(18 25 24 31)

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

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

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I6J8A8B12A12B12C12D12E12F12G24A24B
order122222333444666666666688121212121212122424
size11461236224261222444888122412364466812121212

33 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3⋊D4C3⋊D4C8⋊C22S32S3×D4C2×S32D8⋊S3D4⋊D6S3×C3⋊D4Dic6⋊3D6
kernelDic6⋊3D6D6.Dic3Dic6⋊S3C32⋊5SD16C3×D4⋊S3C32⋊7D8S3×D12C3×D4⋊2S3D4⋊S3D4⋊2S3C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×D4Dic3D6C32D4C6C4C3C3C2C1
# reps111111111111111122211112221

Matrix representation of Dic6⋊3D6 ►in GL8(𝔽73)

721000000
720000000
10010000
07272720000
00000100
000072000
0000727212
0000017272
,
11210000
001720000
007200000
0727200000
00002350570
000034341632
000039232727
000034112362
,
11120000
11210000
07272720000
727272720000
000039232727
00002339027
00002350570
000039625011
,
720000000
721000000
07272720000
10010000
00001000
000007200
0000727212
000010072

G:=sub<GL(8,GF(73))| [72,72,1,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[1,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,2,1,72,72,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,23,34,39,34,0,0,0,0,50,34,23,11,0,0,0,0,57,16,27,23,0,0,0,0,0,32,27,62],[1,1,0,72,0,0,0,0,1,1,72,72,0,0,0,0,1,2,72,72,0,0,0,0,2,1,72,72,0,0,0,0,0,0,0,0,39,23,23,39,0,0,0,0,23,39,50,62,0,0,0,0,27,0,57,50,0,0,0,0,27,27,0,11],[72,72,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,72,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,72] >;
 

Dic6⋊3D6 in GAP, Magma, Sage, TeX 

{\rm Dic}_6\rtimes_3D_6
 
% in TeX
 
G:=Group("Dic6:3D6");
 
// GroupNames label
 
G:=SmallGroup(288,573);
 
// by ID
 
G=gap.SmallGroup(288,573);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=c^6=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^6*b,d*b*d=a^9*b,d*c*d=c^-1>;
 
// generators/relations
 

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