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G = D12:5D6  order 288 = 25·32

5th semidirect product of D12 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12:5D6, Dic6:5D6, C3:C8:9D6, D4:S3:6S3, D4.10S32, D4.S3:6S3, C6.60(S3xD4), C3:3(Q8:3D6), C3:4(D8:S3), (C3xD4).14D6, D12:S3:5C2, C3:D24:12C2, (C3xD12):8C22, C3:Dic3.21D4, C32:5SD16:8C2, C12.31D6:1C2, C32:13(C8:C22), (C3xC12).14C23, C12.14(C22xS3), (C3xDic6):9C22, C2.20(Dic3:D6), C12:S3.9C22, (D4xC32).10C22, C4.14(C2xS32), (D4xC3:S3):2C2, (C3xD4:S3):8C2, (C3xC3:C8):13C22, (C3xD4.S3):4C2, (C2xC3:S3).58D4, (C3xC6).129(C2xD4), (C4xC3:S3).16C22, SmallGroup(288,585)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — D12:5D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C3xD12 — D12:S3 — D12:5D6
Lower central
C32 — C3xC6 — C3xC12 — D12:5D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D12:5D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 866 in 163 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C3xD4, C3xD4, C3xQ8, C22xS3, C8:C22, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C62, C8:S3, C24:C2, D24, D4:S3, D4:S3, D4.S3, Q8:2S3, C3xD8, C3xSD16, S3xD4, D4:2S3, Q8:3S3, C3xC3:C8, S3xDic3, C3:D12, C3xDic6, C3xD12, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C22xC3:S3, D8:S3, Q8:3D6, C12.31D6, C3:D24, C32:5SD16, C3xD4:S3, C3xD4.S3, D12:S3, D4xC3:S3, D12:5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8:C22, S32, S3xD4, C2xS32, D8:S3, Q8:3D6, Dic3:D6, D12:5D6

Character table of D12:5D6

 class 12A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H8A8B12A12B12C12D24A24B24C24D
 size 1141218362242121822488882412124482412121212
ρ1111111111111111111111111111111    trivial
ρ2111-1111111-111111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ311-11-111111-1-1111-1-1-1-111-1111-1-111-1    linear of order 2
ρ411-1-1-1111111-1111-1-1-1-1-1-1111111-1-11    linear of order 2
ρ51111-1-11111-1-111111111-11111-11-1-11    linear of order 2
ρ6111-1-1-111111-11111111-11-11111-111-1    linear of order 2
ρ711-1-11-11111-11111-1-1-1-1-111111-11111    linear of order 2
ρ811-111-1111111111-1-1-1-11-1-11111-1-1-1-1    linear of order 2
ρ92220002-1-12202-1-12-1-1-10202-1-1-10-1-10    orthogonal lifted from S3
ρ10222-200-12-1200-12-1-1-1-1210-2-12-101001    orthogonal lifted from D6
ρ1122-20002-1-12-202-1-1-21110202-1-110-1-10    orthogonal lifted from D6
ρ1222-2-200-12-1200-12-1111-2102-12-10-100-1    orthogonal lifted from D6
ρ132200-20222-2022220000000-2-2-200000    orthogonal lifted from D4
ρ1422-20002-1-12202-1-1-21110-202-1-1-10110    orthogonal lifted from D6
ρ15222200-12-1200-12-1-1-1-12-102-12-10-100-1    orthogonal lifted from S3
ρ16220020222-20-22220000000-2-2-200000    orthogonal lifted from D4
ρ1722-2200-12-1200-12-1111-2-10-2-12-101001    orthogonal lifted from D6
ρ182220002-1-12-202-1-12-1-1-10-202-1-110110    orthogonal lifted from D6
ρ19440000-2-21-400-2-2103-3000022-100000    orthogonal lifted from Dic3:D6
ρ20440000-2-21-400-2-210-33000022-100000    orthogonal lifted from Dic3:D6
ρ21440000-24-2-400-24-200000002-4200000    orthogonal lifted from S3xD4
ρ224-40000444000-4-4-4000000000000000    orthogonal lifted from C8:C22
ρ234400004-2-2-4004-2-20000000-42200000    orthogonal lifted from S3xD4
ρ2444-4000-2-21400-2-212-1-12000-2-2100000    orthogonal lifted from C2xS32
ρ25444000-2-21400-2-21-211-2000-2-2100000    orthogonal lifted from S32
ρ264-400004-2-2000-422000000000000-√6√60    orthogonal lifted from Q8:3D6
ρ274-400004-2-2000-422000000000000√6-√60    orthogonal lifted from Q8:3D6
ρ284-40000-24-20002-4200000000000-√-600√-6    complex lifted from D8:S3
ρ294-40000-24-20002-4200000000000√-600-√-6    complex lifted from D8:S3
ρ308-80000-4-4200044-2000000000000000    orthogonal faithful

Permutation representations of D12:5D6
►On 24 points - transitive group 24T669
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)

(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 20 21 16 17 24)(14 15 22 23 18 19)

(1 5)(2 4)(6 12)(7 11)(8 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)
 
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19), (1,9,5)(2,4,6,8,10,12)(3,11,7)(13,20,21,16,17,24)(14,15,22,23,18,19), (1,5)(2,4)(6,12)(7,11)(8,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;
 
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19), (1,9,5)(2,4,6,8,10,12)(3,11,7)(13,20,21,16,17,24)(14,15,22,23,18,19), (1,5)(2,4)(6,12)(7,11)(8,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );
 
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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)], [(1,9,5),(2,4,6,8,10,12),(3,11,7),(13,20,21,16,17,24),(14,15,22,23,18,19)], [(1,5),(2,4),(6,12),(7,11),(8,10),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)]])
 
G:=TransitiveGroup(24,669);
 

Matrix representation of D12:5D6 ►in GL8(Z)

000-10000
001-10000
01000000
-11000000
00000001
000000-11
00000-100
00001-100
,
0000-1100
00000100
000000-11
00000001
-11000000
01000000
00-110000
00010000
,
0-1000000
1-1000000
00010000
00-110000
000000-11
000000-10
0000-1100
0000-1000
,
-11000000
01000000
001-10000
000-10000
000000-10
000000-11
0000-1000
0000-1100

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

D12:5D6 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_5D_6
 
% in TeX
 
G:=Group("D12:5D6");
 
// GroupNames label
 
G:=SmallGroup(288,585);
 
// by ID
 
G=gap.SmallGroup(288,585);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,675,346,185,80,1356,9414]);
 
// 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^7,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D12:5D6 in TeX

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