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## G = Dic6.10D6order 288 = 25·32

### 10th non-split extension by Dic6 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6.10D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — Dic3.D6 — Dic6.10D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.10D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Dic6.10D6
G = < a,b,c,d | a12=1, b2=c6=a6, d2=a9, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 634 in 140 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×7], C6 [×2], C6, C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], C32, Dic3 [×5], C12 [×2], C12 [×7], D6 [×7], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], C4×S3 [×9], D12 [×7], C3×Q8 [×2], C3×Q8 [×3], C8.C22, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C8⋊S3 [×2], C24⋊C2 [×2], Q82S3 [×2], C3⋊Q16 [×2], C3×Q16 [×2], S3×Q8 [×2], Q83S3 [×3], C3×C3⋊C8 [×2], C6.D6, C322Q8, C3×Dic6 [×2], C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, Q16⋊S3 [×2], C12.31D6, C325SD16 [×2], C3×C3⋊Q16 [×2], Dic3.D6, C12.26D6, Dic6.10D6
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, Q16⋊S3 [×2], Dic3⋊D6, Dic6.10D6

Character table of Dic6.10D6

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 24A 24B 24C 24D size 1 1 18 36 2 2 4 2 4 12 12 18 2 2 4 12 12 4 4 8 8 8 8 8 24 24 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 -1 1 1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 0 0 -1 2 -1 2 2 0 2 0 -1 2 -1 2 0 -1 2 -1 -1 -1 -1 2 0 -1 -1 -1 0 0 orthogonal lifted from S3 ρ10 2 2 0 0 -1 2 -1 2 -2 0 -2 0 -1 2 -1 2 0 -1 2 1 1 -1 1 -2 0 1 -1 -1 0 0 orthogonal lifted from D6 ρ11 2 2 0 0 2 -1 -1 2 -2 2 0 0 2 -1 -1 0 -2 2 -1 1 1 -1 -2 1 -1 0 0 0 1 1 orthogonal lifted from D6 ρ12 2 2 0 0 2 -1 -1 2 -2 -2 0 0 2 -1 -1 0 2 2 -1 1 1 -1 -2 1 1 0 0 0 -1 -1 orthogonal lifted from D6 ρ13 2 2 -2 0 2 2 2 -2 0 0 0 2 2 2 2 0 0 -2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 0 0 -1 2 -1 2 2 0 -2 0 -1 2 -1 -2 0 -1 2 -1 -1 -1 -1 2 0 1 1 1 0 0 orthogonal lifted from D6 ρ15 2 2 0 0 -1 2 -1 2 -2 0 2 0 -1 2 -1 -2 0 -1 2 1 1 -1 1 -2 0 -1 1 1 0 0 orthogonal lifted from D6 ρ16 2 2 0 0 2 -1 -1 2 2 2 0 0 2 -1 -1 0 2 2 -1 -1 -1 -1 2 -1 -1 0 0 0 -1 -1 orthogonal lifted from S3 ρ17 2 2 0 0 2 -1 -1 2 2 -2 0 0 2 -1 -1 0 -2 2 -1 -1 -1 -1 2 -1 1 0 0 0 1 1 orthogonal lifted from D6 ρ18 2 2 2 0 2 2 2 -2 0 0 0 -2 2 2 2 0 0 -2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 4 4 0 0 4 -2 -2 -4 0 0 0 0 4 -2 -2 0 0 -4 2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ20 4 4 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 0 0 2 2 -3 3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ21 4 4 0 0 -2 -2 1 4 -4 0 0 0 -2 -2 1 0 0 -2 -2 -1 -1 1 2 2 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ22 4 4 0 0 -2 4 -2 -4 0 0 0 0 -2 4 -2 0 0 2 -4 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ23 4 4 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 0 0 2 2 3 -3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ24 4 4 0 0 -2 -2 1 4 4 0 0 0 -2 -2 1 0 0 -2 -2 1 1 1 -2 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ25 4 -4 0 0 4 4 4 0 0 0 0 0 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ26 4 -4 0 0 4 -2 -2 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -√-6 √-6 complex lifted from Q16⋊S3 ρ27 4 -4 0 0 4 -2 -2 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 √-6 -√-6 complex lifted from Q16⋊S3 ρ28 4 -4 0 0 -2 4 -2 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 √-6 -√-6 0 0 complex lifted from Q16⋊S3 ρ29 4 -4 0 0 -2 4 -2 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 -√-6 √-6 0 0 complex lifted from Q16⋊S3 ρ30 8 -8 0 0 -4 -4 2 0 0 0 0 0 4 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of Dic6.10D6
On 48 points
Generators in S48
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 35 7 29)(2 34 8 28)(3 33 9 27)(4 32 10 26)(5 31 11 25)(6 30 12 36)(13 39 19 45)(14 38 20 44)(15 37 21 43)(16 48 22 42)(17 47 23 41)(18 46 24 40)
(1 16 3 18 5 20 7 22 9 24 11 14)(2 23 4 13 6 15 8 17 10 19 12 21)(25 41 35 39 33 37 31 47 29 45 27 43)(26 48 36 46 34 44 32 42 30 40 28 38)
(1 41 10 38 7 47 4 44)(2 42 11 39 8 48 5 45)(3 43 12 40 9 37 6 46)(13 25 22 34 19 31 16 28)(14 26 23 35 20 32 17 29)(15 27 24 36 21 33 18 30)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,30,12,36)(13,39,19,45)(14,38,20,44)(15,37,21,43)(16,48,22,42)(17,47,23,41)(18,46,24,40), (1,16,3,18,5,20,7,22,9,24,11,14)(2,23,4,13,6,15,8,17,10,19,12,21)(25,41,35,39,33,37,31,47,29,45,27,43)(26,48,36,46,34,44,32,42,30,40,28,38), (1,41,10,38,7,47,4,44)(2,42,11,39,8,48,5,45)(3,43,12,40,9,37,6,46)(13,25,22,34,19,31,16,28)(14,26,23,35,20,32,17,29)(15,27,24,36,21,33,18,30)>;

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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,30,12,36)(13,39,19,45)(14,38,20,44)(15,37,21,43)(16,48,22,42)(17,47,23,41)(18,46,24,40), (1,16,3,18,5,20,7,22,9,24,11,14)(2,23,4,13,6,15,8,17,10,19,12,21)(25,41,35,39,33,37,31,47,29,45,27,43)(26,48,36,46,34,44,32,42,30,40,28,38), (1,41,10,38,7,47,4,44)(2,42,11,39,8,48,5,45)(3,43,12,40,9,37,6,46)(13,25,22,34,19,31,16,28)(14,26,23,35,20,32,17,29)(15,27,24,36,21,33,18,30) );

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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,35,7,29),(2,34,8,28),(3,33,9,27),(4,32,10,26),(5,31,11,25),(6,30,12,36),(13,39,19,45),(14,38,20,44),(15,37,21,43),(16,48,22,42),(17,47,23,41),(18,46,24,40)], [(1,16,3,18,5,20,7,22,9,24,11,14),(2,23,4,13,6,15,8,17,10,19,12,21),(25,41,35,39,33,37,31,47,29,45,27,43),(26,48,36,46,34,44,32,42,30,40,28,38)], [(1,41,10,38,7,47,4,44),(2,42,11,39,8,48,5,45),(3,43,12,40,9,37,6,46),(13,25,22,34,19,31,16,28),(14,26,23,35,20,32,17,29),(15,27,24,36,21,33,18,30)])

Matrix representation of Dic6.10D6 in GL8(𝔽73)

 0 72 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 71 0 0 0 0 0 0 1 72 0 0 0 0 0 0 54 2 72 2 0 0 0 0 72 56 72 1
,
 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 61 12 0 0 0 0 0 0 67 12 0 0 0 0 0 0 45 72 0 12 0 0 0 0 8 29 6 0
,
 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 64 3 3 0 0 0 0 0 37 12 0 3 0 0 0 0 33 70 9 70 0 0 0 0 36 12 36 61
,
 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 6 71 0 36 0 0 0 0 19 18 18 0 0 0 0 0 68 53 55 35 0 0 0 0 67 54 1 67

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,1,54,72,0,0,0,0,71,72,2,56,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,61,67,45,8,0,0,0,0,12,12,72,29,0,0,0,0,0,0,0,6,0,0,0,0,0,0,12,0],[0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,64,37,33,36,0,0,0,0,3,12,70,12,0,0,0,0,3,0,9,36,0,0,0,0,0,3,70,61],[0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,6,19,68,67,0,0,0,0,71,18,53,54,0,0,0,0,0,18,55,1,0,0,0,0,36,0,35,67] >;

Dic6.10D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{10}D_6
% in TeX

G:=Group("Dic6.10D6");
// GroupNames label

G:=SmallGroup(288,593);
// by ID

G=gap.SmallGroup(288,593);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,100,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=1,b^2=c^6=a^6,d^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;
// generators/relations

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