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## G = Dic6.22D6order 288 = 25·32

### 9th non-split extension by Dic6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6.22D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.6D6 — Dic6.22D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.22D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Dic6.22D6
G = < a,b,c,d | a12=1, b2=c6=d2=a6, bab-1=a-1, cac-1=dad-1=a7, cbc-1=a9b, dbd-1=a3b, dcd-1=c5 >

Subgroups: 522 in 130 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×4], C6 [×2], C6 [×2], C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×7], D6, D6 [×3], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6 [×2], Dic6, C4×S3, C4×S3 [×3], D12 [×4], C3⋊D4, C2×C12 [×2], C3×Q8 [×2], C3×Q8 [×4], C8.C22, C3×Dic3, C3×Dic3 [×2], C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, C4.Dic3, Q82S3 [×4], C3⋊Q16, C3⋊Q16 [×2], C3×Q16, C4○D12, S3×Q8, Q83S3, C6×Q8, C3×C3⋊C8, C324C8, C6.D6, C3⋊D12, C3×Dic6 [×2], C3×Dic6, S3×C12, S3×C12, C12⋊S3, Q8×C32, Q16⋊S3, Q8.11D6, D6.Dic3, C325SD16, C322Q16, C3×C3⋊Q16, C3211SD16, D6.6D6, C3×S3×Q8, Dic6.22D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q16⋊S3, Q8.11D6, S3×C3⋊D4, Dic6.22D6

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

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 24A 24B order 1 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 8 8 12 12 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 6 36 2 2 4 2 4 6 12 12 2 2 4 6 6 12 36 4 4 4 4 8 8 8 8 12 12 12 24 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 S32 S3×D4 C2×S32 Q16⋊S3 Q8.11D6 S3×C3⋊D4 Dic6.22D6 kernel Dic6.22D6 D6.Dic3 C32⋊5SD16 C32⋊2Q16 C3×C3⋊Q16 C32⋊11SD16 D6.6D6 C3×S3×Q8 C3⋊Q16 S3×Q8 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 C3×Q8 Dic3 D6 C32 Q8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of Dic6.22D6 in GL8(ℤ)

 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
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 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 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0
,
 1 0 -1 1 0 0 0 0 0 1 -1 0 0 0 0 0 -1 1 -1 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 -1 1 -1 0 0 0 0 0 -1 0 0 -1 0 0 0 0 -1 0 1 -1 0 0 0 0 0 -1 1 0
,
 0 0 0 0 -1 0 0 -1 0 0 0 0 -1 1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 -1 1 0 1 -1 0 0 0 0 0 1 0 -1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 -1 1 0 0 0 0 0

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

Dic6.22D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{22}D_6
% in TeX

G:=Group("Dic6.22D6");
// GroupNames label

G:=SmallGroup(288,596);
// by ID

G=gap.SmallGroup(288,596);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,100,346,185,80,1356,9414]);
// Polycyclic

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

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