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G = Dic612D6order 288 = 25·32

6th semidirect product of Dic6 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: Dic612D6, C62.3C23, D46S32, (C4×S3)⋊8D6, (C3×D4)⋊9D6, C3⋊D43D6, D42S36S3, Dic3⋊D65C2, (S3×C12)⋊6C22, (C2×Dic3)⋊14D6, D6.3D64C2, D6.6D69C2, (C3×C6).19C24, C6.19(S3×C23), C3⋊D125C22, (S3×C6).10C23, C12.31(C22×S3), (C3×C12).31C23, (C6×Dic3)⋊6C22, D6.11(C22×S3), C322Q84C22, C6.D69C22, C327D43C22, C12⋊S310C22, Dic3.D612C2, (C3×Dic6)⋊14C22, (S3×Dic3)⋊14C22, (D4×C32)⋊11C22, C3⋊Dic3.21C23, Dic3.9(C22×S3), (C3×Dic3).13C23, (C4×S32)⋊6C2, C4.31(C2×S32), (D4×C3⋊S3)⋊7C2, C34(S3×C4○D4), C22.3(C2×S32), C328(C2×C4○D4), C3⋊S32(C4○D4), C2.21(C22×S32), (C3×D42S3)⋊9C2, (C2×S32).12C22, (C2×C6.D6)⋊4C2, (C3×C3⋊D4)⋊3C22, (C2×C6).4(C22×S3), (C4×C3⋊S3).43C22, (C2×C3⋊S3).24C23, (C22×C3⋊S3).58C22, SmallGroup(288,960)

Series: Derived Chief Lower central Upper central

C1C3×C6 — Dic612D6
C1C3C32C3×C6S3×C6C2×S32C4×S32 — Dic612D6
C32C3×C6 — Dic612D6
C1C2D4

Generators and relations for Dic612D6
 G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=a-1, cac-1=a7, dad=a5, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 1346 in 355 conjugacy classes, 110 normal (18 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4, C4 [×7], C22 [×2], C22 [×11], S3 [×14], C6 [×2], C6 [×9], C2×C4 [×16], D4, D4 [×11], Q8 [×4], C23 [×3], C32, Dic3 [×6], Dic3 [×3], C12 [×2], C12 [×7], D6 [×2], D6 [×25], C2×C6 [×4], C2×C6 [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×2], Dic6 [×4], C4×S3 [×2], C4×S3 [×19], D12 [×7], C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×4], C3⋊D4 [×10], C2×C12 [×6], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C22×S3 [×8], C2×C4○D4, C3×Dic3 [×6], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62 [×2], S3×C2×C4 [×6], C4○D12 [×6], S3×D4 [×7], D42S3 [×2], D42S3 [×4], S3×Q8 [×2], Q83S3 [×2], C3×C4○D4 [×2], S3×Dic3 [×2], C6.D6, C6.D6 [×6], C3⋊D12 [×4], C322Q8 [×2], C3×Dic6 [×2], S3×C12 [×2], C6×Dic3 [×4], C3×C3⋊D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C2×S32, C22×C3⋊S3 [×2], S3×C4○D4 [×2], Dic3.D6, D6.6D6 [×2], C4×S32, D6.3D6 [×4], C2×C6.D6 [×2], Dic3⋊D6 [×2], C3×D42S3 [×2], D4×C3⋊S3, Dic612D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, S3×C23 [×2], C2×S32 [×3], S3×C4○D4 [×2], C22×S32, Dic612D6

Permutation representations of Dic612D6
On 24 points - transitive group 24T608
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 13 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)
(1 2 9 10 5 6)(3 4 11 12 7 8)(13 18 17 22 21 14)(15 20 19 24 23 16)
(1 3)(2 8)(4 6)(5 11)(7 9)(10 12)(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), (1,13,7,19)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14), (1,2,9,10,5,6)(3,4,11,12,7,8)(13,18,17,22,21,14)(15,20,19,24,23,16), (1,3)(2,8)(4,6)(5,11)(7,9)(10,12)(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), (1,13,7,19)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14), (1,2,9,10,5,6)(3,4,11,12,7,8)(13,18,17,22,21,14)(15,20,19,24,23,16), (1,3)(2,8)(4,6)(5,11)(7,9)(10,12)(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)], [(1,13,7,19),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14)], [(1,2,9,10,5,6),(3,4,11,12,7,8),(13,18,17,22,21,14),(15,20,19,24,23,16)], [(1,3),(2,8),(4,6),(5,11),(7,9),(10,12),(13,23),(14,16),(15,21),(17,19),(18,24),(20,22)])

G:=TransitiveGroup(24,608);

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C···6G6H6I6J6K12A12B12C12D12E12F12G12H12I12J12K
order12222222223334444444444666···666661212121212121212121212
size11226699181822423333666618224···4881212446666812121212

45 irreducible representations

dim111111111222222244448
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D6D6D6D6D6C4○D4S32C2×S32C2×S32S3×C4○D4Dic612D6
kernelDic612D6Dic3.D6D6.6D6C4×S32D6.3D6C2×C6.D6Dic3⋊D6C3×D42S3D4×C3⋊S3D42S3Dic6C4×S3C2×Dic3C3⋊D4C3×D4C3⋊S3D4C4C22C3C1
# reps112142221222442411241

Matrix representation of Dic612D6 in GL6(𝔽13)

1250000
1010000
000100
0012100
000010
000001
,
500000
280000
0001200
0012000
000010
000001
,
180000
0120000
0012000
0001200
0000121
0000120
,
100000
010000
000100
001000
0000120
0000121

G:=sub<GL(6,GF(13))| [12,10,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,2,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

Dic612D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{12}D_6
% in TeX

G:=Group("Dic6:12D6");
// GroupNames label

G:=SmallGroup(288,960);
// by ID

G=gap.SmallGroup(288,960);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,185,1356,9414]);
// Polycyclic

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

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