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## G = Dic6.9D6order 288 = 25·32

### 9th non-split extension by Dic6 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic6.9D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — Dic3.D6 — Dic6.9D6
 Lower central C32 — C3×C6 — C3×C12 — Dic6.9D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Dic6.9D6
G = < a,b,c,d | a12=1, b2=c6=a6, d2=a3, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 546 in 139 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×6], C6 [×2], C6, C8 [×2], C2×C4 [×3], Q8, Q8 [×5], C32, Dic3 [×9], C12 [×2], C12 [×7], D6 [×3], C2×C8, Q16 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×9], C4×S3 [×9], C3×Q8 [×2], C3×Q8 [×3], C2×Q16, C3×Dic3 [×2], C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, S3×C8 [×2], Dic12 [×2], C3⋊Q16 [×2], C3⋊Q16 [×2], C3×Q16 [×2], S3×Q8 [×5], C3×C3⋊C8 [×2], C6.D6, C322Q8, C3×Dic6 [×2], C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, S3×Q16 [×2], C12.29D6, C323Q16 [×2], C3×C3⋊Q16 [×2], Dic3.D6, Q8×C3⋊S3, Dic6.9D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, C22×S3 [×2], C2×Q16, S32, S3×D4 [×2], C2×S32, S3×Q16 [×2], Dic3⋊D6, Dic6.9D6

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

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C ··· 12G 12H 12I 24A 24B 24C 24D order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 9 9 2 2 4 2 4 12 12 18 36 2 2 4 6 6 6 6 4 4 8 ··· 8 24 24 12 12 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 Q16 S32 S3×D4 C2×S32 S3×Q16 Dic3⋊D6 Dic6.9D6 kernel Dic6.9D6 C12.29D6 C32⋊3Q16 C3×C3⋊Q16 Dic3.D6 Q8×C3⋊S3 C3⋊Q16 C3⋊Dic3 C2×C3⋊S3 C3⋊C8 Dic6 C3×Q8 C3⋊S3 Q8 C6 C4 C3 C2 C1 # reps 1 1 2 2 1 1 2 1 1 2 2 2 4 1 2 1 4 2 1

Matrix representation of Dic6.9D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 62 0 0 0 0 62 43 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 61 0 0 0 0 61 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 72 1 0 0 0 0 0 0 16 57 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,62,0,0,0,0,62,43,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,61,0,0,0,0,61,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic6.9D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._9D_6
% in TeX

G:=Group("Dic6.9D6");
// GroupNames label

G:=SmallGroup(288,592);
// by ID

G=gap.SmallGroup(288,592);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,100,675,346,185,80,1356,9414]);
// Polycyclic

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

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