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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, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, 2- 1+4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, S3×Q8, S3×Q8, Q83S3, C6×Q8, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C4×C3⋊S3, C12⋊S3, Q8×C32, Q8.15D6, Dic3.D6, D6.D6, D6.6D6, C3×S3×Q8, C12.26D6, Dic6.26D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8.15D6, 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 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 27 19 33)(14 26 20 32)(15 25 21 31)(16 36 22 30)(17 35 23 29)(18 34 24 28)
(1 26 9 30 5 34)(2 31 10 35 6 27)(3 36 11 28 7 32)(4 29 12 33 8 25)(13 47 21 39 17 43)(14 40 22 44 18 48)(15 45 23 37 19 41)(16 38 24 42 20 46)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,27,19,33)(14,26,20,32)(15,25,21,31)(16,36,22,30)(17,35,23,29)(18,34,24,28), (1,26,9,30,5,34)(2,31,10,35,6,27)(3,36,11,28,7,32)(4,29,12,33,8,25)(13,47,21,39,17,43)(14,40,22,44,18,48)(15,45,23,37,19,41)(16,38,24,42,20,46), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,27,19,33)(14,26,20,32)(15,25,21,31)(16,36,22,30)(17,35,23,29)(18,34,24,28), (1,26,9,30,5,34)(2,31,10,35,6,27)(3,36,11,28,7,32)(4,29,12,33,8,25)(13,47,21,39,17,43)(14,40,22,44,18,48)(15,45,23,37,19,41)(16,38,24,42,20,46), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(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)], [(1,42,7,48),(2,41,8,47),(3,40,9,46),(4,39,10,45),(5,38,11,44),(6,37,12,43),(13,27,19,33),(14,26,20,32),(15,25,21,31),(16,36,22,30),(17,35,23,29),(18,34,24,28)], [(1,26,9,30,5,34),(2,31,10,35,6,27),(3,36,11,28,7,32),(4,29,12,33,8,25),(13,47,21,39,17,43),(14,40,22,44,18,48),(15,45,23,37,19,41),(16,38,24,42,20,46)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(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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