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G = D1213D6order 288 = 25·32

7th semidirect product of D12 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D1213D6, Dic613D6, C62.5C23, C3242+ 1+4, D48S32, (C4×S3)⋊4D6, (S3×D4)⋊5S3, C3⋊D45D6, (C3×D4)⋊11D6, Dic3⋊D66C2, D42S35S3, (S3×D12)⋊10C2, C33(D4○D12), (C2×Dic3)⋊7D6, (C22×S3)⋊7D6, C33(D46D6), (S3×C12)⋊8C22, D6.D67C2, D6.3D65C2, C6.21(S3×C23), (C3×C6).21C24, D12⋊S310C2, D6.6D610C2, (C3×D12)⋊15C22, (S3×C6).12C23, (C3×C12).33C23, C12.33(C22×S3), (C6×Dic3)⋊7C22, (S3×Dic3)⋊3C22, D6.12(C22×S3), C6.D63C22, C327D45C22, C12⋊S311C22, C3⋊D1216C22, D6⋊S315C22, (C3×Dic6)⋊15C22, C322Q814C22, (D4×C32)⋊13C22, C3⋊Dic3.23C23, Dic3.22(C22×S3), (C3×Dic3).21C23, C4.33(C2×S32), (D4×C3⋊S3)⋊8C2, (C3×S3×D4)⋊10C2, (C2×S32)⋊5C22, (S3×C3⋊D4)⋊5C2, C22.5(C2×S32), (C4×C3⋊S3)⋊5C22, (S3×C2×C6)⋊11C22, C2.23(C22×S32), (C3×C3⋊D4)⋊5C22, (C2×C6).6(C22×S3), (C3×D42S3)⋊10C2, (C2×C3⋊D12)⋊20C2, (C2×C3⋊S3).26C23, (C22×C3⋊S3)⋊8C22, SmallGroup(288,962)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D1213D6
C1C3C32C3×C6S3×C6C2×S32S3×C3⋊D4 — D1213D6
C32C3×C6 — D1213D6
C1C2D4

Generators and relations for D1213D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a7, dad=a5, cbc-1=a6b, dbd=a10b, dcd=c-1 >

Subgroups: 1506 in 359 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4, C4 [×5], C22 [×2], C22 [×13], S3 [×14], C6 [×2], C6 [×11], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], C32, Dic3 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×5], D6 [×2], D6 [×2], D6 [×23], C2×C6 [×4], C2×C6 [×8], C2×D4 [×9], C4○D4 [×6], C3×S3 [×4], C3⋊S3 [×3], C3×C6, C3×C6 [×2], Dic6, Dic6 [×2], C4×S3 [×2], C4×S3 [×9], D12, D12 [×11], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C3⋊D4 [×16], C2×C12 [×4], C3×D4 [×2], C3×D4 [×6], C3×Q8, C22×S3 [×2], C22×S3 [×10], C22×C6 [×2], 2+ 1+4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62 [×2], C2×D12 [×3], C4○D12 [×5], S3×D4, S3×D4 [×10], D42S3, D42S3 [×3], Q83S3 [×2], C2×C3⋊D4 [×4], C6×D4, C3×C4○D4, S3×Dic3 [×2], C6.D6 [×2], D6⋊S3, C3⋊D12 [×2], C3⋊D12 [×6], C322Q8, C3×Dic6, S3×C12 [×2], C3×D12, C6×Dic3 [×2], C3×C3⋊D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C2×S32 [×2], S3×C2×C6 [×2], C22×C3⋊S3 [×2], D46D6, D4○D12, D12⋊S3, D6.D6, D6.6D6, S3×D12, D6.3D6 [×2], C2×C3⋊D12 [×2], S3×C3⋊D4 [×2], Dic3⋊D6 [×2], C3×S3×D4, C3×D42S3, D4×C3⋊S3, D1213D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2+ 1+4, S32, S3×C23 [×2], C2×S32 [×3], D46D6, D4○D12, C22×S32, D1213D6

Permutation representations of D1213D6
On 24 points - transitive group 24T607
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 5 9)(2 12 10 8 6 4)(3 7 11)(13 21 17)(14 16 18 20 22 24)(15 23 19)
(1 9)(3 7)(4 12)(6 10)(13 23)(14 16)(15 21)(17 19)(18 24)(20 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), (1,5,9)(2,12,10,8,6,4)(3,7,11)(13,21,17)(14,16,18,20,22,24)(15,23,19), (1,9)(3,7)(4,12)(6,10)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)>;

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

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

G:=TransitiveGroup(24,607);

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C4D4E4F6A6B6C···6G6H6I6J6K6L6M6N12A12B12C12D12E12F12G12H
order12222222222333444444666···666666661212121212121212
size112266661818182242666618224···4668812121244668121212

42 irreducible representations

dim1111111111112222222224444448
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3S3D6D6D6D6D6D6D62+ 1+4S32C2×S32C2×S32D46D6D4○D12D1213D6
kernelD1213D6D12⋊S3D6.D6D6.6D6S3×D12D6.3D6C2×C3⋊D12S3×C3⋊D4Dic3⋊D6C3×S3×D4C3×D42S3D4×C3⋊S3S3×D4D42S3Dic6C4×S3D12C2×Dic3C3⋊D4C3×D4C22×S3C32D4C4C22C3C3C1
# reps1111122221111112124221112221

Matrix representation of D1213D6 in GL8(ℤ)

000-10000
001-10000
01000000
-11000000
0000000-1
0000001-1
00000100
0000-1100
,
0000001-1
0000000-1
00001-100
00000-100
001-10000
000-10000
1-1000000
0-1000000
,
000-10000
001-10000
0-1000000
1-1000000
0000001-1
00000010
00001-100
00001000
,
-11000000
01000000
00-110000
00010000
0000-1000
0000-1100
000000-10
000000-11

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],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-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,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1] >;

D1213D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{13}D_6
% in TeX

G:=Group("D12:13D6");
// GroupNames label

G:=SmallGroup(288,962);
// by ID

G=gap.SmallGroup(288,962);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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