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G = Dic3⋊D12order 288 = 25·32

1st semidirect product of Dic3 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: Dic32D12, C62.56C23, (S3×C6)⋊2D4, (C2×D12)⋊3S3, (C6×D12)⋊22C2, D62(C3⋊D4), D6⋊Dic39C2, (C3×Dic3)⋊1D4, C6.13(C2×D12), C2.17(S3×D12), C6.139(S3×D4), (C2×C12).19D6, C34(C12⋊D4), Dic3⋊C417S3, C324(C4⋊D4), C6.11D129C2, C6.8(D42S3), (C2×Dic3).22D6, (C22×S3).35D6, C31(C23.14D6), (C6×C12).181C22, C6.32(Q83S3), C2.15(D12⋊S3), (C6×Dic3).35C22, (C2×C4).22S32, (C2×S3×Dic3)⋊1C2, (C3×C6).45(C2×D4), C2.13(S3×C3⋊D4), C6.33(C2×C3⋊D4), (C2×C3⋊D12)⋊1C2, C22.103(C2×S32), (S3×C2×C6).17C22, (C3×Dic3⋊C4)⋊15C2, (C3×C6).33(C4○D4), (C2×C6).75(C22×S3), (C22×C3⋊S3).14C22, (C2×C3⋊Dic3).40C22, SmallGroup(288,534)

Series: Derived Chief Lower central Upper central

C1C62 — Dic3⋊D12
C1C3C32C3×C6C62C6×Dic3C2×S3×Dic3 — Dic3⋊D12
C32C62 — Dic3⋊D12
C1C22C2×C4

Generators and relations for Dic3⋊D12
 G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 922 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×7], C6 [×6], C6 [×6], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×2], Dic3 [×4], C12 [×6], D6 [×2], D6 [×15], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×3], C3⋊S3, C3×C6 [×3], C4×S3 [×2], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×5], C2×C3⋊S3 [×3], C62, Dic3⋊C4, D6⋊C4 [×4], C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12 [×2], C22×Dic3, C2×C3⋊D4 [×2], C6×D4, S3×Dic3 [×2], C3⋊D12 [×4], C3×D12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, C12⋊D4, C23.14D6, D6⋊Dic3, C3×Dic3⋊C4, C6.11D12, C2×S3×Dic3, C2×C3⋊D12 [×2], C6×D12, Dic3⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12, S3×D4 [×2], D42S3, Q83S3, C2×C3⋊D4, C2×S32, C12⋊D4, C23.14D6, D12⋊S3, S3×D12, S3×C3⋊D4, Dic3⋊D12

Smallest permutation representation of Dic3⋊D12
On 48 points
Generators in S48
(1 42 5 46 9 38)(2 43 6 47 10 39)(3 44 7 48 11 40)(4 45 8 37 12 41)(13 25 21 33 17 29)(14 26 22 34 18 30)(15 27 23 35 19 31)(16 28 24 36 20 32)
(1 20 46 28)(2 29 47 21)(3 22 48 30)(4 31 37 23)(5 24 38 32)(6 33 39 13)(7 14 40 34)(8 35 41 15)(9 16 42 36)(10 25 43 17)(11 18 44 26)(12 27 45 19)
(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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11116612362244661218182···2444121212124···412121212

42 irreducible representations

dim1111111222222222244444444
type+++++++++++++++++-+++
imageC1C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C3⋊D4S32S3×D4D42S3Q83S3C2×S32D12⋊S3S3×D12S3×C3⋊D4
kernelDic3⋊D12D6⋊Dic3C3×Dic3⋊C4C6.11D12C2×S3×Dic3C2×C3⋊D12C6×D12Dic3⋊C4C2×D12C3×Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6Dic3D6C2×C4C6C6C6C22C2C2C2
# reps1111121112222224412111222

Matrix representation of Dic3⋊D12 in GL8(ℤ)

10000000
01000000
00-1-10000
00100000
00001000
00000100
000000-10
0000000-1
,
10000000
01000000
00100000
00-1-10000
00001000
00000100
0000000-1
00000010
,
0-1000000
10000000
00-100000
000-10000
00000-100
00001100
00000001
00000010
,
-10000000
01000000
00100000
00010000
00000100
00001000
00000001
00000010

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

Dic3⋊D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes D_{12}
% in TeX

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

G:=SmallGroup(288,534);
// by ID

G=gap.SmallGroup(288,534);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,58,1356,9414]);
// Polycyclic

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

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