direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C12×C3⋊C8, C12⋊2C24, C122.10C2, C3⋊1(C4×C24), (C3×C12)⋊7C8, C32⋊6(C4×C8), C6.1(C4×C12), C6.6(C2×C24), C4.18(S3×C12), (C6×C12).28C4, (C4×C12).26S3, (C4×C12).12C6, C42.6(C3×S3), C12.109(C4×S3), (C2×C12).15C12, C12.23(C2×C12), (C2×C12).450D6, C62.92(C2×C4), (C3×C6).11C42, C2.1(Dic3×C12), C6.17(C4×Dic3), (C2×C12).30Dic3, C22.7(C6×Dic3), (C6×C12).328C22, C2.1(C6×C3⋊C8), C6.17(C2×C3⋊C8), (C6×C3⋊C8).23C2, (C2×C3⋊C8).11C6, (C2×C4).87(S3×C6), (C3×C6).37(C2×C8), (C2×C6).34(C2×C12), (C2×C4).7(C3×Dic3), (C2×C12).117(C2×C6), (C3×C12).103(C2×C4), (C2×C6).56(C2×Dic3), SmallGroup(288,236)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C12×C3⋊C8 |
Generators and relations for C12×C3⋊C8
G = < a,b,c | a12=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 146 in 103 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C4×C8, C3×C12, C62, C2×C3⋊C8, C4×C12, C4×C12, C2×C24, C3×C3⋊C8, C6×C12, C6×C12, C4×C3⋊C8, C4×C24, C6×C3⋊C8, C122, C12×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C42, C2×C8, C3×S3, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C4×C8, C3×Dic3, S3×C6, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, S3×C12, C6×Dic3, C4×C3⋊C8, C4×C24, C6×C3⋊C8, Dic3×C12, C12×C3⋊C8
►On 96 points
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 57 48 31 87 24 61 81)(2 58 37 32 88 13 62 82)(3 59 38 33 89 14 63 83)(4 60 39 34 90 15 64 84)(5 49 40 35 91 16 65 73)(6 50 41 36 92 17 66 74)(7 51 42 25 93 18 67 75)(8 52 43 26 94 19 68 76)(9 53 44 27 95 20 69 77)(10 54 45 28 96 21 70 78)(11 55 46 29 85 22 71 79)(12 56 47 30 86 23 72 80)
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,57,48,31,87,24,61,81)(2,58,37,32,88,13,62,82)(3,59,38,33,89,14,63,83)(4,60,39,34,90,15,64,84)(5,49,40,35,91,16,65,73)(6,50,41,36,92,17,66,74)(7,51,42,25,93,18,67,75)(8,52,43,26,94,19,68,76)(9,53,44,27,95,20,69,77)(10,54,45,28,96,21,70,78)(11,55,46,29,85,22,71,79)(12,56,47,30,86,23,72,80)>;
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,57,48,31,87,24,61,81)(2,58,37,32,88,13,62,82)(3,59,38,33,89,14,63,83)(4,60,39,34,90,15,64,84)(5,49,40,35,91,16,65,73)(6,50,41,36,92,17,66,74)(7,51,42,25,93,18,67,75)(8,52,43,26,94,19,68,76)(9,53,44,27,95,20,69,77)(10,54,45,28,96,21,70,78)(11,55,46,29,85,22,71,79)(12,56,47,30,86,23,72,80) );
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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(1,57,48,31,87,24,61,81),(2,58,37,32,88,13,62,82),(3,59,38,33,89,14,63,83),(4,60,39,34,90,15,64,84),(5,49,40,35,91,16,65,73),(6,50,41,36,92,17,66,74),(7,51,42,25,93,18,67,75),(8,52,43,26,94,19,68,76),(9,53,44,27,95,20,69,77),(10,54,45,28,96,21,70,78),(11,55,46,29,85,22,71,79),(12,56,47,30,86,23,72,80)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6O | 8A | ··· | 8P | 12A | ··· | 12X | 12Y | ··· | 12BH | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | Dic3 | D6 | C3×S3 | C3⋊C8 | C4×S3 | C3×Dic3 | S3×C6 | C3×C3⋊C8 | S3×C12 |
| kernel | C12×C3⋊C8 | C6×C3⋊C8 | C122 | C4×C3⋊C8 | C3×C3⋊C8 | C6×C12 | C2×C3⋊C8 | C4×C12 | C3×C12 | C3⋊C8 | C2×C12 | C12 | C4×C12 | C2×C12 | C2×C12 | C42 | C12 | C12 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 16 | 8 | 32 | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 |
Matrix representation of C12×C3⋊C8 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 24 |
| 1 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 46 | 8 |
| 51 | 0 | 0 |
| 0 | 43 | 27 |
| 0 | 18 | 30 |
G:=sub<GL(3,GF(73))| [64,0,0,0,24,0,0,0,24],[1,0,0,0,64,46,0,0,8],[51,0,0,0,43,18,0,27,30] >;
C12×C3⋊C8 in GAP, Magma, Sage, TeX
C_{12}\times C_3\rtimes C_8 % in TeX
G:=Group("C12xC3:C8"); // GroupNames label
G:=SmallGroup(288,236);
// by ID
G=gap.SmallGroup(288,236);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,176,136,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations