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## G = Dic6.D6order 288 = 25·32

### 5th 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.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — Dic3.D6 — Dic6.D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic6.D6
G = < a,b,c,d | a12=c6=1, b2=a6, d2=a3, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a9c-1 >

Subgroups: 570 in 140 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, C3×C3⋊C8, C6.D6, C322Q8, C3×Dic6, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, D4.D6, C12.31D6, C323Q16, C3×D4.S3, Dic3.D6, C12.D6, Dic6.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S32, S3×D4, C2×S32, D4.D6, Dic3⋊D6, Dic6.D6

Character table of Dic6.D6

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 12E 24A 24B 24C 24D size 1 1 4 18 2 2 4 2 12 12 18 36 2 2 4 8 8 8 8 12 12 4 4 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 2 2 2 2 -2 0 0 -2 0 2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 2 -1 -1 2 2 0 0 0 2 -1 -1 1 1 -2 1 0 -2 2 -1 -1 -1 0 0 0 1 1 orthogonal lifted from D6 ρ11 2 2 2 0 -1 2 -1 2 0 -2 0 0 -1 2 -1 -1 -1 -1 2 -2 0 -1 2 -1 0 1 1 1 0 0 orthogonal lifted from D6 ρ12 2 2 -2 0 -1 2 -1 2 0 2 0 0 -1 2 -1 1 1 1 -2 -2 0 -1 2 -1 0 -1 1 1 0 0 orthogonal lifted from D6 ρ13 2 2 -2 0 2 -1 -1 2 -2 0 0 0 2 -1 -1 1 1 -2 1 0 2 2 -1 -1 1 0 0 0 -1 -1 orthogonal lifted from D6 ρ14 2 2 -2 0 -1 2 -1 2 0 -2 0 0 -1 2 -1 1 1 1 -2 2 0 -1 2 -1 0 1 -1 -1 0 0 orthogonal lifted from D6 ρ15 2 2 2 0 -1 2 -1 2 0 2 0 0 -1 2 -1 -1 -1 -1 2 2 0 -1 2 -1 0 -1 -1 -1 0 0 orthogonal lifted from S3 ρ16 2 2 2 0 2 -1 -1 2 -2 0 0 0 2 -1 -1 -1 -1 2 -1 0 -2 2 -1 -1 1 0 0 0 1 1 orthogonal lifted from D6 ρ17 2 2 0 -2 2 2 2 -2 0 0 2 0 2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 0 2 -1 -1 2 2 0 0 0 2 -1 -1 -1 -1 2 -1 0 2 2 -1 -1 -1 0 0 0 -1 -1 orthogonal lifted from S3 ρ19 4 4 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 -3 3 0 0 0 0 2 2 -1 0 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ20 4 4 0 0 -2 -2 1 -4 0 0 0 0 -2 -2 1 3 -3 0 0 0 0 2 2 -1 0 0 0 0 0 0 orthogonal lifted from Dic3⋊D6 ρ21 4 4 4 0 -2 -2 1 4 0 0 0 0 -2 -2 1 1 1 -2 -2 0 0 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ22 4 4 -4 0 -2 -2 1 4 0 0 0 0 -2 -2 1 -1 -1 2 2 0 0 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ23 4 4 0 0 -2 4 -2 -4 0 0 0 0 -2 4 -2 0 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 0 0 4 -2 -2 -4 0 0 0 0 4 -2 -2 0 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 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 symplectic lifted from D4.D6, Schur index 2 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 symplectic lifted from D4.D6, Schur index 2 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 symplectic lifted from D4.D6, Schur index 2 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 symplectic lifted from D4.D6, Schur index 2 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 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of Dic6.D6 in GL8(𝔽73)

 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 67 0 0 72 0 0 0 0 0 6 1 1 0 0 0 0 0 0 0 0 72 72 2 2 0 0 0 0 1 0 71 0 0 0 0 0 72 72 1 1 0 0 0 0 1 0 72 0
,
 67 67 71 72 0 0 0 0 0 0 72 1 0 0 0 0 36 36 6 0 0 0 0 0 36 37 6 0 0 0 0 0 0 0 0 0 67 0 26 0 0 0 0 0 6 6 47 47 0 0 0 0 7 0 6 0 0 0 0 0 66 66 67 67
,
 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 8 2 1 1 0 0 0 0 2 69 72 0 0 0 0 0 0 0 0 0 65 57 46 19 0 0 0 0 16 8 54 27 0 0 0 0 15 30 8 16 0 0 0 0 43 58 57 65
,
 6 6 2 1 0 0 0 0 6 6 1 2 0 0 0 0 24 25 67 67 0 0 0 0 25 24 67 67 0 0 0 0 0 0 0 0 68 63 5 10 0 0 0 0 10 5 63 68 0 0 0 0 34 68 0 0 0 0 0 0 5 39 0 0

G:=sub<GL(8,GF(73))| [1,1,67,0,0,0,0,0,72,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,1,72,1,0,0,0,0,72,0,72,0,0,0,0,0,2,71,1,72,0,0,0,0,2,0,1,0],[67,0,36,36,0,0,0,0,67,0,36,37,0,0,0,0,71,72,6,6,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,67,6,7,66,0,0,0,0,0,6,0,66,0,0,0,0,26,47,6,67,0,0,0,0,0,47,0,67],[72,72,8,2,0,0,0,0,1,0,2,69,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,65,16,15,43,0,0,0,0,57,8,30,58,0,0,0,0,46,54,8,57,0,0,0,0,19,27,16,65],[6,6,24,25,0,0,0,0,6,6,25,24,0,0,0,0,2,1,67,67,0,0,0,0,1,2,67,67,0,0,0,0,0,0,0,0,68,10,34,5,0,0,0,0,63,5,68,39,0,0,0,0,5,63,0,0,0,0,0,0,10,68,0,0] >;

Dic6.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6.D_6
% in TeX

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

G:=SmallGroup(288,579);
// by ID

G=gap.SmallGroup(288,579);
# by ID

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

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

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