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G = D125D6order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D125D6, Dic65D6, C3⋊C89D6, D4⋊S36S3, D4.10S32, D4.S36S3, C6.60(S3×D4), C33(Q83D6), C34(D8⋊S3), (C3×D4).14D6, D12⋊S35C2, C3⋊D2412C2, (C3×D12)⋊8C22, C3⋊Dic3.21D4, C325SD168C2, C12.31D61C2, C3213(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

C1C3×C12 — D125D6
C1C3C32C3×C6C3×C12C3×D12D12⋊S3 — D125D6
C32C3×C6C3×C12 — D125D6
C1C2C4D4

Generators and relations for D125D6
 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 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×9], C6 [×2], C6 [×6], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3 [×4], C12 [×2], C12 [×2], D6 [×16], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×4], D12, D12 [×4], C2×Dic3, C3⋊D4 [×5], C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×S3 [×4], C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3 [×3], C62, C8⋊S3 [×2], C24⋊C2, D24, D4⋊S3, D4⋊S3, D4.S3, Q82S3, C3×D8, C3×SD16, S3×D4 [×3], D42S3, Q83S3, C3×C3⋊C8 [×2], S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C22×C3⋊S3, D8⋊S3, Q83D6, C12.31D6, C3⋊D24, C325SD16, C3×D4⋊S3, C3×D4.S3, D12⋊S3, D4×C3⋊S3, D125D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8⋊C22, S32, S3×D4 [×2], C2×S32, D8⋊S3, Q83D6, Dic3⋊D6, D125D6

Character table of D125D6

 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
ρ264-400004-2-2000-422000000000000-660    orthogonal lifted from Q83D6
ρ274-400004-2-2000-4220000000000006-60    orthogonal lifted from Q83D6
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 D125D6
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 15)(2 14)(3 13)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)
(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 14 21 22 17 18)(15 16 23 24 19 20)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

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

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

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

G:=TransitiveGroup(24,669);

Matrix representation of D125D6 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] >;

D125D6 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 D125D6 in TeX

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