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## G = Dic6.29D6order 288 = 25·32

### 3rd non-split extension by Dic6 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6.29D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊5SD16 — Dic6.29D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.29D6
 Upper central C1 — C2 — C2×C4

Generators and relations for Dic6.29D6
G = < a,b,c,d | a12=c6=1, b2=a6, d2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 570 in 140 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×2], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×6], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×2], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×4], C4×S3 [×4], D12 [×3], C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C3×Q8 [×3], C8.C22, C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3, C62, C24⋊C2 [×2], Dic12 [×2], C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C3×M4(2), C2×Dic6, C4○D12 [×3], C6×Q8, C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C8.D6, Q8.11D6, C325SD16 [×2], C323Q16 [×2], C3×C4.Dic3, C6×Dic6, C12.59D6, Dic6.29D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C8.C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C8.D6, Q8.11D6, C2×C3⋊D12, Dic6.29D6

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D12 C3⋊D4 D12 C3⋊D4 C8.C22 S32 C3⋊D12 C2×S32 C3⋊D12 C8.D6 Q8.11D6 Dic6.29D6 kernel Dic6.29D6 C32⋊5SD16 C32⋊3Q16 C3×C4.Dic3 C6×Dic6 C12.59D6 C4.Dic3 C2×Dic6 C3×C12 C62 C3⋊C8 Dic6 C2×C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 2 2 4

Matrix representation of Dic6.29D6 in GL4(𝔽73) generated by

 14 7 0 0 66 7 0 0 0 0 14 7 0 0 66 7
,
 0 0 72 0 0 0 1 1 1 0 0 0 72 72 0 0
,
 13 43 0 0 30 43 0 0 0 0 43 30 0 0 43 13
,
 0 0 30 43 0 0 30 60 0 27 0 0 46 46 0 0
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,14,66,0,0,7,7],[0,0,1,72,0,0,0,72,72,1,0,0,0,1,0,0],[13,30,0,0,43,43,0,0,0,0,43,43,0,0,30,13],[0,0,0,46,0,0,27,46,30,30,0,0,43,60,0,0] >;

Dic6.29D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{29}D_6
% in TeX

G:=Group("Dic6.29D6");
// GroupNames label

G:=SmallGroup(288,481);
// by ID

G=gap.SmallGroup(288,481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,219,675,80,1356,9414]);
// 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,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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