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G = D12.7D6  order 288 = 25·32

7th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.7D6, Dic6⋊10D6, C3⋊C8⋊8D6, D4.5S32, (S3×D4)⋊3S3, D4.S3⋊4S3, (C4×S3).8D6, C3⋊6(Q8⋊3D6), (S3×C6).34D4, (C3×D4).12D6, C6.152(S3×D4), C32⋊7D8⋊3C2, C3⋊D24⋊11C2, D6.6D6⋊4C2, D6.Dic3⋊2C2, C12⋊S3⋊5C22, C3⋊3(D12⋊6C22), D6.14(C3⋊D4), C32⋊12(C8⋊C22), C12.11(C22×S3), (C3×C12).11C23, C32⋊4C8⋊7C22, (C3×Dic3).14D4, (C3×Dic6)⋊8C22, Dic6⋊S3⋊10C2, (S3×C12).16C22, (C3×D12).14C22, (D4×C32).7C22, Dic3.11(C3⋊D4), (C3×S3×D4)⋊3C2, C4.11(C2×S32), (C3×C3⋊C8)⋊7C22, (C3×D4.S3)⋊3C2, C6.48(C2×C3⋊D4), C2.26(S3×C3⋊D4), (C3×C6).126(C2×D4), SmallGroup(288,582)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.7D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — D12.7D6
Lower central
C32 — C3×C6 — C3×C12 — D12.7D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D12.7D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a4b, dbd=a7b, dcd=a6c-1 >

Subgroups: 642 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, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C8⋊S3, D24, C4.Dic3, D4⋊S3, D4.S3, D4.S3, Q8⋊2S3, C3×SD16, C4○D12, S3×D4, Q8⋊3S3, C6×D4, C3×C3⋊C8, C32⋊4C8, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, D4×C32, S3×C2×C6, Q8⋊3D6, D12⋊6C22, D6.Dic3, C3⋊D24, Dic6⋊S3, C3×D4.S3, C32⋊7D8, D6.6D6, C3×S3×D4, D12.7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q8⋊3D6, D12⋊6C22, S3×C3⋊D4, D12.7D6

Smallest permutation representation of D12.7D6
►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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)

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

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

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L8A8B12A12B12C12D12E24A24B
order1222223334446666666666668812121212122424
size11461236224261222444668881212123644812241212

33 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3⋊D4C3⋊D4C8⋊C22S32S3×D4C2×S32Q8⋊3D6D12⋊6C22S3×C3⋊D4D12.7D6
kernelD12.7D6D6.Dic3C3⋊D24Dic6⋊S3C3×D4.S3C32⋊7D8D6.6D6C3×S3×D4D4.S3S3×D4C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×D4Dic3D6C32D4C6C4C3C3C2C1
# reps111111111111111122211112221

Matrix representation of D12.7D6 ►in GL8(ℤ)

0-1000000
1-1000000
-10-1-10000
01100000
00000100
0000-1000
0000-1-112
000001-1-1
,
11120000
11210000
-10-1-10000
0-1-1-10000
0000-1-1-10
0000-2012
0000210-2
0000-2-1-11
,
11120000
11210000
0-1-1-10000
-1-1-1-10000
000010-2-2
0000210-2
0000-2012
000020-2-3
,
-10000000
-11000000
0-1-1-10000
10010000
0000-1000
00000100
0000-1-112
0000010-1

G:=sub<GL(8,Integers())| [0,1,-1,0,0,0,0,0,-1,-1,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,2,-1],[1,1,-1,0,0,0,0,0,1,1,0,-1,0,0,0,0,1,2,-1,-1,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,-1,-2,2,-2,0,0,0,0,-1,0,1,-1,0,0,0,0,-1,1,0,-1,0,0,0,0,0,2,-2,1],[1,1,0,-1,0,0,0,0,1,1,-1,-1,0,0,0,0,1,2,-1,-1,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,1,2,-2,2,0,0,0,0,0,1,0,0,0,0,0,0,-2,0,1,-2,0,0,0,0,-2,-2,2,-3],[-1,-1,0,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,1,-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1] >;
 

D12.7D6 in GAP, Magma, Sage, TeX 

D_{12}._7D_6
 
% in TeX
 
G:=Group("D12.7D6");
 
// GroupNames label
 
G:=SmallGroup(288,582);
 
// by ID
 
G=gap.SmallGroup(288,582);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b,d*b*d=a^7*b,d*c*d=a^6*c^-1>;
 
// generators/relations
 

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