Copied to
clipboard

## G = C12.6Dic6order 288 = 25·32

### 6th non-split extension by C12 of Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C12.6Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C12 — C3×C4⋊Dic3 — C12.6Dic6
 Lower central C32 — C3×C6 — C3×C12 — C12.6Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C12.6Dic6
G = < a,b,c | a12=b12=1, c2=a6b6, bab-1=a-1, cac-1=a7, cbc-1=a9b-1 >

Subgroups: 242 in 83 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C324C8, C6×Dic3, C6×C12, C12.Q8, C3×C4⋊Dic3, C2×C324C8, C12.6Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, SD16, Dic6, C4×S3, C3⋊D4, C4.Q8, S32, Dic3⋊C4, D4.S3, Q82S3, C6.D6, D6⋊S3, C322Q8, C12.Q8, Dic6⋊S3, C62.C22, C12.6Dic6

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

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

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

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

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 4 4 4 4 4 4 4 type + + + + - + + - + - + + - - image C1 C2 C2 C4 S3 Q8 D4 D6 SD16 Dic6 C4×S3 C3⋊D4 S32 D4.S3 Q8⋊2S3 C6.D6 C32⋊2Q8 D6⋊S3 Dic6⋊S3 kernel C12.6Dic6 C3×C4⋊Dic3 C2×C32⋊4C8 C32⋊4C8 C4⋊Dic3 C3×C12 C62 C2×C12 C3×C6 C12 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 # reps 1 2 1 4 2 1 1 2 4 4 4 4 1 2 2 1 1 1 4

Matrix representation of C12.6Dic6 in GL6(𝔽73)

 46 3 0 0 0 0 0 27 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
,
 58 3 0 0 0 0 22 15 0 0 0 0 0 0 0 46 0 0 0 0 46 0 0 0 0 0 0 0 30 43 0 0 0 0 30 60
,
 35 21 0 0 0 0 46 38 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 11 13 0 0 0 0 2 62

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,3,27,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[58,22,0,0,0,0,3,15,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,30,30,0,0,0,0,43,60],[35,46,0,0,0,0,21,38,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,11,2,0,0,0,0,13,62] >;

C12.6Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("C12.6Dic6");
// GroupNames label

G:=SmallGroup(288,222);
// by ID

G=gap.SmallGroup(288,222);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,148,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^6*b^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b^-1>;
// generators/relations

׿
×
𝔽