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G = D12⋊2Dic3  order 288 = 25·32

2nd semidirect product of D12 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D12⋊2Dic3, C12.79D12, C62.24D4, Dic6⋊2Dic3, C32⋊4C4≀C2, (C3×D12)⋊4C4, C12.19(C4×S3), (C4×Dic3)⋊1S3, C4○D12.1S3, (C3×Dic6)⋊4C4, C4.3(S3×Dic3), C6.39(D6⋊C4), (Dic3×C12)⋊4C2, (C3×C12).110D4, (C2×C12).286D6, C3⋊3(C42⋊4S3), C12.58D6⋊3C2, C12.78(C3⋊D4), (C6×C12).31C22, C12.24(C2×Dic3), C3⋊2(Q8⋊3Dic3), C4.28(C3⋊D12), C2.10(D6⋊Dic3), C6.9(C6.D4), C22.2(D6⋊S3), (C2×C4).56S32, (C3×C12).29(C2×C4), (C3×C4○D12).2C2, (C2×C6).50(C3⋊D4), (C3×C6).36(C22⋊C4), SmallGroup(288,217)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12⋊2Dic3
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C3×C4○D12 — D12⋊2Dic3
Lower central
C32 — C3×C6 — C3×C12 — D12⋊2Dic3
Upper central
C1 — C4 — C2×C4

Generators and relations for D12⋊2Dic3
 G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 306 in 98 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4≀C2, C3×Dic3, C3×C12, S3×C6, C62, C4.Dic3, C4×Dic3, C4×C12, C4○D12, C3×C4○D4, C32⋊4C8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C42⋊4S3, Q8⋊3Dic3, Dic3×C12, C12.58D6, C3×C4○D12, D12⋊2Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, C4≀C2, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C42⋊4S3, Q8⋊3Dic3, D6⋊Dic3, D12⋊2Dic3

Smallest permutation representation of D12⋊2Dic3
►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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G6H6I6J8A8B12A···12F12G···12K12L···12S12T12U
order12223334444444466666666668812···1212···1212···121212
size1121222411266661222224444121236362···24···46···61212

48 irreducible representations

dim1111112222222222222444444
type++++++++--+++-+-
imageC1C2C2C2C4C4S3S3D4D4Dic3Dic3D6C4×S3D12C3⋊D4C3⋊D4C4≀C2C42⋊4S3S32S3×Dic3C3⋊D12D6⋊S3Q8⋊3Dic3D12⋊2Dic3
kernelD12⋊2Dic3Dic3×C12C12.58D6C3×C4○D12C3×Dic6C3×D12C4×Dic3C4○D12C3×C12C62Dic6D12C2×C12C12C12C12C2×C6C32C3C2×C4C4C4C22C3C1
# reps1111221111112222448111124

Matrix representation of D12⋊2Dic3 ►in GL6(𝔽73)

2700000
0460000
0072100
0072000
000010
000001
,
0460000
2700000
001000
0017200
0000720
0000072
,
100000
0720000
0072000
0007200
000011
0000720
,
100000
0270000
0027000
0002700
0000270
00004646

G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,27,0,0,0,0,46,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,46,0,0,0,0,0,46] >;
 

D12⋊2Dic3 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_2{\rm Dic}_3
 
% in TeX
 
G:=Group("D12:2Dic3");
 
// GroupNames label
 
G:=SmallGroup(288,217);
 
// by ID
 
G=gap.SmallGroup(288,217);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,675,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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