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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, (S3xD4):3S3, D4.S3:4S3, (C4xS3).8D6, C3:6(Q8:3D6), (S3xC6).34D4, (C3xD4).12D6, C6.152(S3xD4), 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(C22xS3), (C3xC12).11C23, C32:4C8:7C22, (C3xDic3).14D4, (C3xDic6):8C22, Dic6:S3:10C2, (S3xC12).16C22, (C3xD12).14C22, (D4xC32).7C22, Dic3.11(C3:D4), (C3xS3xD4):3C2, C4.11(C2xS32), (C3xC3:C8):7C22, (C3xD4.S3):3C2, C6.48(C2xC3:D4), C2.26(S3xC3:D4), (C3xC6).126(C2xD4), SmallGroup(288,582)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — D12.7D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — S3xC12 — D6.6D6 — D12.7D6
Lower central
C32 — C3xC6 — C3xC12 — 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, C2xC4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C3:C8, C24, Dic6, C4xS3, C4xS3, D12, D12, C3:D4, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C22xC6, C8:C22, C3xDic3, C3xDic3, C3xC12, S3xC6, S3xC6, C2xC3:S3, C62, C8:S3, D24, C4.Dic3, D4:S3, D4.S3, D4.S3, Q8:2S3, C3xSD16, C4oD12, S3xD4, Q8:3S3, C6xD4, C3xC3:C8, C32:4C8, C6.D6, C3:D12, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C12:S3, D4xC32, S3xC2xC6, Q8:3D6, D12:6C22, D6.Dic3, C3:D24, Dic6:S3, C3xD4.S3, C32:7D8, D6.6D6, C3xS3xD4, D12.7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C8:C22, S32, S3xD4, C2xC3:D4, C2xS32, Q8:3D6, D12:6C22, S3xC3: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:C22S32S3xD4C2xS32Q8:3D6D12:6C22S3xC3:D4D12.7D6
kernelD12.7D6D6.Dic3C3:D24Dic6:S3C3xD4.S3C32:7D8D6.6D6C3xS3xD4D4.S3S3xD4C3xDic3S3xC6C3:C8Dic6C4xS3D12C3xD4Dic3D6C32D4C6C4C3C3C2C1
# 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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