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## G = C12.8Dic6order 288 = 25·32

### 8th non-split extension by C12 of Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C12.8Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C3×C4⋊Dic3 — C12.8Dic6
 Lower central C32 — C3×C6 — C3×C12 — C12.8Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C12.8Dic6
G = < a,b,c | a12=b12=1, c2=b6, bab-1=a-1, cac-1=a7, cbc-1=a3b-1 >

Subgroups: 242 in 83 conjugacy classes, 40 normal (18 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6 [×3], C3⋊C8 [×6], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C2.D8, C3×Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×3], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C324C8 [×2], C6×Dic3 [×2], C6×C12, C6.Q16 [×2], C3×C4⋊Dic3 [×2], C2×C324C8, C12.8Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, D6 [×2], C4⋊C4, D8, Q16, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C2.D8, S32, Dic3⋊C4 [×2], D4⋊S3 [×2], C3⋊Q16 [×2], C6.D6, D6⋊S3, C322Q8, C6.Q16 [×2], C322D8, C322Q16, C62.C22, C12.8Dic6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + - + + + - - + + - + - - image C1 C2 C2 C4 S3 Q8 D4 D6 D8 Q16 Dic6 C4×S3 C3⋊D4 S32 D4⋊S3 C3⋊Q16 C6.D6 C32⋊2Q8 D6⋊S3 C32⋊2D8 C32⋊2Q16 kernel C12.8Dic6 C3×C4⋊Dic3 C2×C32⋊4C8 C32⋊4C8 C4⋊Dic3 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 2 1 4 2 1 1 2 2 2 4 4 4 1 2 2 1 1 1 2 2

Matrix representation of C12.8Dic6 in GL6(𝔽73)

 1 71 0 0 0 0 1 72 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 21 25 0 0 0 0 70 52 0 0 0 0 0 0 0 27 0 0 0 0 27 0 0 0 0 0 0 0 13 30 0 0 0 0 43 43
,
 35 61 0 0 0 0 29 38 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 23 68 0 0 0 0 18 50

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,71,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[21,70,0,0,0,0,25,52,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,13,43,0,0,0,0,30,43],[35,29,0,0,0,0,61,38,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,23,18,0,0,0,0,68,50] >;

C12.8Dic6 in GAP, Magma, Sage, TeX

C_{12}._8{\rm Dic}_6
% in TeX

G:=Group("C12.8Dic6");
// GroupNames label

G:=SmallGroup(288,224);
// by ID

G=gap.SmallGroup(288,224);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b^-1>;
// generators/relations

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