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## G = Dic6.26D6order 288 = 25·32

### 13rd non-split extension by Dic6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic6.26D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — D6⋊S3 — D6.D6 — Dic6.26D6
 Lower central C32 — C3×C6 — Dic6.26D6
 Upper central C1 — C2 — Q8

Generators and relations for Dic6.26D6
G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a5, bc=cb, dbd-1=a6b, dcd-1=a6c-1 >

Subgroups: 1082 in 312 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×3], C4 [×7], C22 [×5], S3 [×11], C6 [×2], C6 [×3], C2×C4 [×15], D4 [×10], Q8, Q8 [×9], C32, Dic3 [×6], Dic3 [×3], C12 [×6], C12 [×9], D6 [×2], D6 [×9], C2×C6 [×2], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×2], C3⋊S3 [×3], C3×C6, Dic6 [×6], Dic6 [×6], C4×S3 [×6], C4×S3 [×21], D12 [×15], C3⋊D4 [×8], C2×C12 [×6], C3×Q8 [×2], C3×Q8 [×7], 2- 1+4, C3×Dic3 [×6], C3⋊Dic3, C3×C12 [×3], S3×C6 [×2], C2×C3⋊S3 [×3], C4○D12 [×12], S3×Q8 [×2], S3×Q8 [×6], Q83S3 [×9], C6×Q8 [×2], C6.D6 [×6], D6⋊S3, C3⋊D12 [×6], C322Q8 [×3], C3×Dic6 [×6], S3×C12 [×6], C4×C3⋊S3 [×3], C12⋊S3 [×3], Q8×C32, Q8.15D6 [×2], Dic3.D6 [×3], D6.D6 [×3], D6.6D6 [×6], C3×S3×Q8 [×2], C12.26D6, Dic6.26D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2- 1+4, S32, S3×C23 [×2], C2×S32 [×3], Q8.15D6 [×2], C22×S32, Dic6.26D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 2- 1+4 S32 C2×S32 Q8.15D6 Dic6.26D6 kernel Dic6.26D6 Dic3.D6 D6.D6 D6.6D6 C3×S3×Q8 C12.26D6 S3×Q8 Dic6 C4×S3 C3×Q8 C32 Q8 C4 C3 C1 # reps 1 3 3 6 2 1 2 6 6 2 1 1 3 4 1

Matrix representation of Dic6.26D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 1 0 0 0 0 12 1 0 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 5 0 0 0 0 0 5 0 0 0 0 0 0 5 8 0 0 0 0 0 8
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 4 11 0 0 11 11 0 0 0 0 9 2 0 0
,
 12 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 12 1 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,1,0,0,0,0,12,12,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,5,5,0,0,0,0,0,0,5,0,0,0,0,0,8,8],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,9,0,0,0,0,11,2,0,0,2,4,0,0,0,0,2,11,0,0],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,12,12,0,0] >;

Dic6.26D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{26}D_6
% in TeX

G:=Group("Dic6.26D6");
// GroupNames label

G:=SmallGroup(288,964);
// by ID

G=gap.SmallGroup(288,964);
# by ID

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

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

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