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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(S3×D4), C3⋊3(Q8⋊3D6), C3⋊4(D8⋊S3), (C3×D4).14D6, D12⋊S3⋊5C2, C3⋊D24⋊12C2, (C3×D12)⋊8C22, C3⋊Dic3.21D4, C32⋊5SD16⋊8C2, C12.31D6⋊1C2, C32⋊13(C8⋊C22), (C3×C12).14C23, C12.14(C22×S3), (C3×Dic6)⋊9C22, C2.20(Dic3⋊D6), C12⋊S3.9C22, (D4×C32).10C22, C4.14(C2×S32), (D4×C3⋊S3)⋊2C2, (C3×D4⋊S3)⋊8C2, (C3×C3⋊C8)⋊13C22, (C3×D4.S3)⋊4C2, (C2×C3⋊S3).58D4, (C3×C6).129(C2×D4), (C4×C3⋊S3).16C22, SmallGroup(288,585)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12⋊5D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12⋊5D6
Lower central
C32 — C3×C6 — C3×C12 — 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, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C8⋊S3, C24⋊C2, D24, D4⋊S3, D4⋊S3, D4.S3, Q8⋊2S3, C3×D8, C3×SD16, S3×D4, D4⋊2S3, Q8⋊3S3, C3×C3⋊C8, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C22×C3⋊S3, D8⋊S3, Q8⋊3D6, C12.31D6, C3⋊D24, C32⋊5SD16, C3×D4⋊S3, C3×D4.S3, D12⋊S3, D4×C3⋊S3, D12⋊5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S32, S3×D4, C2×S32, 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 S3×D4
ρ224-40000444000-4-4-4000000000000000    orthogonal lifted from C8⋊C22
ρ234400004-2-2-4004-2-20000000-42200000    orthogonal lifted from S3×D4
ρ2444-4000-2-21400-2-212-1-12000-2-2100000    orthogonal lifted from C2×S32
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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