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## G = Dic6.24D6order 288 = 25·32

### 11st non-split extension by Dic6 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic6.24D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — S3×Dic6 — Dic6.24D6
 Lower central C32 — C3×C6 — Dic6.24D6
 Upper central C1 — C2 — D4

Generators and relations for Dic6.24D6
G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a7, bc=cb, bd=db, dcd-1=a6c-1 >

Subgroups: 1018 in 312 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, 2- 1+4, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C2×Dic6, C4○D12, D42S3, D42S3, S3×Q8, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C322Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, Q8○D12, S3×Dic6, Dic3.D6, D6.D6, D6.3D6, D6.4D6, C2×C322Q8, C3×D42S3, C12.D6, Dic6.24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8○D12, C22×S32, Dic6.24D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 D6 2- 1+4 S32 C2×S32 C2×S32 Q8○D12 Dic6.24D6 kernel Dic6.24D6 S3×Dic6 Dic3.D6 D6.D6 D6.3D6 D6.4D6 C2×C32⋊2Q8 C3×D4⋊2S3 C12.D6 D4⋊2S3 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 C32 D4 C4 C22 C3 C1 # reps 1 2 1 1 4 2 2 2 1 2 2 2 4 4 2 1 1 1 2 4 1

Matrix representation of Dic6.24D6 in GL6(𝔽13)

 1 1 0 0 0 0 12 0 0 0 0 0 0 0 12 2 7 0 0 0 11 1 0 7 0 0 4 0 1 11 0 0 0 4 2 12
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 7 12 8 0 0 0 1 6 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 0 0 0 0 11 4 0 0 0 0 11 2 11 11 0 0 11 0 2 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 8 5 0 0 0 0 0 0 5 0 0 0 0 0 5 8

G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,11,4,0,0,0,2,1,0,4,0,0,7,0,1,2,0,0,0,7,11,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,7,1,0,0,0,5,12,6,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,11,11,11,0,0,2,4,2,0,0,0,0,0,11,2,0,0,0,0,11,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,5,5,0,0,0,0,0,8] >;

Dic6.24D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{24}D_6
% in TeX

G:=Group("Dic6.24D6");
// GroupNames label

G:=SmallGroup(288,957);
// by ID

G=gap.SmallGroup(288,957);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,1356,9414]);
// 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^7,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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