metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12)⋊1C8, (C2×C12).228D4, (C22×C12).2C4, (C22×C4).20D6, C6.14(C22⋊C8), C6.18(C23⋊C4), (C2×C6).23M4(2), (C22×C4).6Dic3, C6.6(C4.10D4), C23.27(C2×Dic3), C12.55D4.12C2, C2.4(C12.55D4), C2.1(C12.10D4), C2.3(C23.7D6), C22.5(C4.Dic3), (C22×C12).325C22, C3⋊2(C22.M4(2)), C22.25(C6.D4), (C2×C4)⋊(C3⋊C8), (C2×C4⋊C4).1S3, C22.3(C2×C3⋊C8), (C6×C4⋊C4).25C2, (C2×C6).30(C2×C8), (C2×C4).160(C3⋊D4), (C2×C6).86(C22⋊C4), (C22×C6).124(C2×C4), SmallGroup(192,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C12)⋊C8
G = < a,b,c | a2=b12=c8=1, ab=ba, cac-1=ab6, cbc-1=ab-1 >
Subgroups: 168 in 78 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C22⋊C8, C2×C4⋊C4, C2×C3⋊C8, C3×C4⋊C4, C22×C12, C22.M4(2), C12.55D4, C6×C4⋊C4, (C2×C12)⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C23⋊C4, C4.10D4, C2×C3⋊C8, C4.Dic3, C6.D4, C22.M4(2), C12.55D4, C23.7D6, C12.10D4, (C2×C12)⋊C8
►On 96 points
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 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 62 19 82 37 31 51 91)(2 61 14 75 38 30 58 96)(3 72 21 80 39 29 53 89)(4 71 16 73 40 28 60 94)(5 70 23 78 41 27 55 87)(6 69 18 83 42 26 50 92)(7 68 13 76 43 25 57 85)(8 67 20 81 44 36 52 90)(9 66 15 74 45 35 59 95)(10 65 22 79 46 34 54 88)(11 64 17 84 47 33 49 93)(12 63 24 77 48 32 56 86)
G:=sub<Sym(96)| (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,62,19,82,37,31,51,91)(2,61,14,75,38,30,58,96)(3,72,21,80,39,29,53,89)(4,71,16,73,40,28,60,94)(5,70,23,78,41,27,55,87)(6,69,18,83,42,26,50,92)(7,68,13,76,43,25,57,85)(8,67,20,81,44,36,52,90)(9,66,15,74,45,35,59,95)(10,65,22,79,46,34,54,88)(11,64,17,84,47,33,49,93)(12,63,24,77,48,32,56,86)>;
G:=Group( (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,62,19,82,37,31,51,91)(2,61,14,75,38,30,58,96)(3,72,21,80,39,29,53,89)(4,71,16,73,40,28,60,94)(5,70,23,78,41,27,55,87)(6,69,18,83,42,26,50,92)(7,68,13,76,43,25,57,85)(8,67,20,81,44,36,52,90)(9,66,15,74,45,35,59,95)(10,65,22,79,46,34,54,88)(11,64,17,84,47,33,49,93)(12,63,24,77,48,32,56,86) );
G=PermutationGroup([[(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(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)], [(1,62,19,82,37,31,51,91),(2,61,14,75,38,30,58,96),(3,72,21,80,39,29,53,89),(4,71,16,73,40,28,60,94),(5,70,23,78,41,27,55,87),(6,69,18,83,42,26,50,92),(7,68,13,76,43,25,57,85),(8,67,20,81,44,36,52,90),(9,66,15,74,45,35,59,95),(10,65,22,79,46,34,54,88),(11,64,17,84,47,33,49,93),(12,63,24,77,48,32,56,86)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Dic3 | D6 | M4(2) | C3⋊C8 | C3⋊D4 | C4.Dic3 | C23⋊C4 | C4.10D4 | C23.7D6 | C12.10D4 |
| kernel | (C2×C12)⋊C8 | C12.55D4 | C6×C4⋊C4 | C22×C12 | C2×C12 | C2×C4⋊C4 | C2×C12 | C22×C4 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of (C2×C12)⋊C8 ►in GL8(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 27 | 46 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 71 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 46 | 62 | 71 |
| 0 | 0 | 0 | 0 | 41 | 0 | 61 | 11 |
| 51 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 27 | 46 | 1 | 71 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 27 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,27,0,0,0,0,0,1,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[72,71,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,54,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,61,1,27,41,0,0,0,0,1,12,46,0,0,0,0,0,0,0,62,61,0,0,0,0,0,0,71,11],[51,0,0,0,0,0,0,0,22,22,0,0,0,0,0,0,0,0,25,45,0,0,0,0,0,0,64,48,0,0,0,0,0,0,0,0,0,27,0,1,0,0,0,0,0,46,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,71,0,27] >;
(C2×C12)⋊C8 in GAP, Magma, Sage, TeX
(C_2\times C_{12})\rtimes C_8 % in TeX
G:=Group("(C2xC12):C8"); // GroupNames label
G:=SmallGroup(192,87);
// by ID
G=gap.SmallGroup(192,87);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,387,100,1123,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^12=c^8=1,a*b=b*a,c*a*c^-1=a*b^6,c*b*c^-1=a*b^-1>;
// generators/relations