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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 [×5], C3 [×2], C3, C4, C4 [×9], C22 [×2], C22 [×3], S3 [×5], C6 [×2], C6 [×9], C2×C4 [×15], D4, D4 [×9], Q8 [×10], C32, Dic3 [×6], Dic3 [×9], C12 [×2], C12 [×7], D6 [×2], D6 [×3], C2×C6 [×4], C2×C6 [×4], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×2], C3⋊S3, C3×C6, C3×C6 [×2], Dic6 [×2], Dic6 [×17], C4×S3 [×2], C4×S3 [×11], D12 [×2], C2×Dic3 [×4], C2×Dic3 [×10], C3⋊D4 [×4], C3⋊D4 [×10], C2×C12 [×6], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], 2- 1+4, C3×Dic3 [×6], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, S3×C6 [×2], C2×C3⋊S3, C62 [×2], C2×Dic6 [×6], C4○D12 [×6], D42S3 [×2], D42S3 [×11], S3×Q8 [×4], C3×C4○D4 [×2], S3×Dic3 [×4], C6.D6 [×2], D6⋊S3, C3⋊D12 [×2], C322Q8, C322Q8 [×6], C3×Dic6 [×2], S3×C12 [×2], C6×Dic3 [×4], C3×C3⋊D4 [×4], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32, Q8○D12 [×2], S3×Dic6 [×2], Dic3.D6, D6.D6, D6.3D6 [×4], D6.4D6 [×2], C2×C322Q8 [×2], C3×D42S3 [×2], C12.D6, Dic6.24D6
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○D12 [×2], 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 46 7 40)(2 45 8 39)(3 44 9 38)(4 43 10 37)(5 42 11 48)(6 41 12 47)(13 29 19 35)(14 28 20 34)(15 27 21 33)(16 26 22 32)(17 25 23 31)(18 36 24 30)
(1 23 9 19 5 15)(2 18 10 14 6 22)(3 13 11 21 7 17)(4 20 12 16 8 24)(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 44 7 38)(2 39 8 45)(3 46 9 40)(4 41 10 47)(5 48 11 42)(6 43 12 37)(13 33 19 27)(14 28 20 34)(15 35 21 29)(16 30 22 36)(17 25 23 31)(18 32 24 26)

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

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

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

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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