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## G = C12.15Dic6order 288 = 25·32

### 2nd non-split extension by C12 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.15Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C6×C3⋊C8 — C12.15Dic6
 Lower central C32 — C3×C6 — C12.15Dic6
 Upper central C1 — C2×C4

Generators and relations for C12.15Dic6
G = < a,b,c | a12=1, b12=a6, c2=a6b6, bab-1=a5, ac=ca, cbc-1=a3b11 >

Subgroups: 266 in 91 conjugacy classes, 42 normal (20 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×10], C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12, C4⋊C8, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8 [×2], C4×Dic3 [×3], C2×C24 [×2], C3×C3⋊C8 [×2], C2×C3⋊Dic3 [×2], C6×C12, Dic3⋊C8 [×2], C6×C3⋊C8 [×2], C4×C3⋊Dic3, C12.15Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D4, Q8, D6 [×2], C4⋊C4, C2×C8, M4(2), Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C4⋊C8, S32, S3×C8 [×2], C8⋊S3 [×2], Dic3⋊C4 [×2], C6.D6, D6⋊S3, C322Q8, Dic3⋊C8 [×2], C12.29D6, C12.31D6, C62.C22, C12.15Dic6

Smallest permutation representation of C12.15Dic6
On 96 points
Generators in S96
(1 40 5 44 9 48 13 28 17 32 21 36)(2 25 22 45 18 41 14 37 10 33 6 29)(3 42 7 46 11 26 15 30 19 34 23 38)(4 27 24 47 20 43 16 39 12 35 8 31)(49 76 69 96 65 92 61 88 57 84 53 80)(50 93 54 73 58 77 62 81 66 85 70 89)(51 78 71 74 67 94 63 90 59 86 55 82)(52 95 56 75 60 79 64 83 68 87 72 91)
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 90 19 84 13 78 7 96)(2 66 20 60 14 54 8 72)(3 76 21 94 15 88 9 82)(4 52 22 70 16 64 10 58)(5 86 23 80 17 74 11 92)(6 62 24 56 18 50 12 68)(25 85 43 79 37 73 31 91)(26 61 44 55 38 49 32 67)(27 95 45 89 39 83 33 77)(28 71 46 65 40 59 34 53)(29 81 47 75 41 93 35 87)(30 57 48 51 42 69 36 63)

G:=sub<Sym(96)| (1,40,5,44,9,48,13,28,17,32,21,36)(2,25,22,45,18,41,14,37,10,33,6,29)(3,42,7,46,11,26,15,30,19,34,23,38)(4,27,24,47,20,43,16,39,12,35,8,31)(49,76,69,96,65,92,61,88,57,84,53,80)(50,93,54,73,58,77,62,81,66,85,70,89)(51,78,71,74,67,94,63,90,59,86,55,82)(52,95,56,75,60,79,64,83,68,87,72,91), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,90,19,84,13,78,7,96)(2,66,20,60,14,54,8,72)(3,76,21,94,15,88,9,82)(4,52,22,70,16,64,10,58)(5,86,23,80,17,74,11,92)(6,62,24,56,18,50,12,68)(25,85,43,79,37,73,31,91)(26,61,44,55,38,49,32,67)(27,95,45,89,39,83,33,77)(28,71,46,65,40,59,34,53)(29,81,47,75,41,93,35,87)(30,57,48,51,42,69,36,63)>;

G:=Group( (1,40,5,44,9,48,13,28,17,32,21,36)(2,25,22,45,18,41,14,37,10,33,6,29)(3,42,7,46,11,26,15,30,19,34,23,38)(4,27,24,47,20,43,16,39,12,35,8,31)(49,76,69,96,65,92,61,88,57,84,53,80)(50,93,54,73,58,77,62,81,66,85,70,89)(51,78,71,74,67,94,63,90,59,86,55,82)(52,95,56,75,60,79,64,83,68,87,72,91), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,90,19,84,13,78,7,96)(2,66,20,60,14,54,8,72)(3,76,21,94,15,88,9,82)(4,52,22,70,16,64,10,58)(5,86,23,80,17,74,11,92)(6,62,24,56,18,50,12,68)(25,85,43,79,37,73,31,91)(26,61,44,55,38,49,32,67)(27,95,45,89,39,83,33,77)(28,71,46,65,40,59,34,53)(29,81,47,75,41,93,35,87)(30,57,48,51,42,69,36,63) );

G=PermutationGroup([(1,40,5,44,9,48,13,28,17,32,21,36),(2,25,22,45,18,41,14,37,10,33,6,29),(3,42,7,46,11,26,15,30,19,34,23,38),(4,27,24,47,20,43,16,39,12,35,8,31),(49,76,69,96,65,92,61,88,57,84,53,80),(50,93,54,73,58,77,62,81,66,85,70,89),(51,78,71,74,67,94,63,90,59,86,55,82),(52,95,56,75,60,79,64,83,68,87,72,91)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,90,19,84,13,78,7,96),(2,66,20,60,14,54,8,72),(3,76,21,94,15,88,9,82),(4,52,22,70,16,64,10,58),(5,86,23,80,17,74,11,92),(6,62,24,56,18,50,12,68),(25,85,43,79,37,73,31,91),(26,61,44,55,38,49,32,67),(27,95,45,89,39,83,33,77),(28,71,46,65,40,59,34,53),(29,81,47,75,41,93,35,87),(30,57,48,51,42,69,36,63)])

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 8A ··· 8H 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 1 1 1 1 18 18 18 18 2 ··· 2 4 4 4 6 ··· 6 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - + - + - - + image C1 C2 C2 C4 C8 S3 D4 Q8 D6 M4(2) Dic6 C3⋊D4 C4×S3 S3×C8 C8⋊S3 S32 D6⋊S3 C32⋊2Q8 C6.D6 C12.29D6 C12.31D6 kernel C12.15Dic6 C6×C3⋊C8 C4×C3⋊Dic3 C2×C3⋊Dic3 C3⋊Dic3 C2×C3⋊C8 C3×C12 C3×C12 C2×C12 C3×C6 C12 C12 C2×C6 C6 C6 C2×C4 C4 C4 C22 C2 C2 # reps 1 2 1 4 8 2 1 1 2 2 4 4 4 8 8 1 1 1 1 2 2

Matrix representation of C12.15Dic6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 22 0 0 0 0 51 22 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 23 28 0 0 0 0 28 50 0 0 0 0 0 0 0 51 0 0 0 0 51 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,51,0,0,0,0,22,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[23,28,0,0,0,0,28,50,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C12.15Dic6 in GAP, Magma, Sage, TeX

C_{12}._{15}{\rm Dic}_6
% in TeX

G:=Group("C12.15Dic6");
// GroupNames label

G:=SmallGroup(288,220);
// by ID

G=gap.SmallGroup(288,220);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^6*b^6,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^3*b^11>;
// generators/relations

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