direct product, abelian, monomial, 2-elementary
Aliases: C2×C4×C12, SmallGroup(96,161)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C4×C12 |
| C1 — C2×C4×C12 |
| C1 — C2×C4×C12 |
Generators and relations for C2×C4×C12
G = < a,b,c | a2=b4=c12=1, ab=ba, ac=ca, bc=cb >
Subgroups: 108, all normal (8 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C23, C12, C2×C6, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C4×C12, C22×C12, C2×C4×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C4×C12, C22×C12, C2×C4×C12
►Regular action on 96 points
Generators in S96
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 73)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 72 44 17)(2 61 45 18)(3 62 46 19)(4 63 47 20)(5 64 48 21)(6 65 37 22)(7 66 38 23)(8 67 39 24)(9 68 40 13)(10 69 41 14)(11 70 42 15)(12 71 43 16)(25 60 79 86)(26 49 80 87)(27 50 81 88)(28 51 82 89)(29 52 83 90)(30 53 84 91)(31 54 73 92)(32 55 74 93)(33 56 75 94)(34 57 76 95)(35 58 77 96)(36 59 78 85)
(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)
G:=sub<Sym(96)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,73)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,72,44,17)(2,61,45,18)(3,62,46,19)(4,63,47,20)(5,64,48,21)(6,65,37,22)(7,66,38,23)(8,67,39,24)(9,68,40,13)(10,69,41,14)(11,70,42,15)(12,71,43,16)(25,60,79,86)(26,49,80,87)(27,50,81,88)(28,51,82,89)(29,52,83,90)(30,53,84,91)(31,54,73,92)(32,55,74,93)(33,56,75,94)(34,57,76,95)(35,58,77,96)(36,59,78,85), (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)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,73)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,72,44,17)(2,61,45,18)(3,62,46,19)(4,63,47,20)(5,64,48,21)(6,65,37,22)(7,66,38,23)(8,67,39,24)(9,68,40,13)(10,69,41,14)(11,70,42,15)(12,71,43,16)(25,60,79,86)(26,49,80,87)(27,50,81,88)(28,51,82,89)(29,52,83,90)(30,53,84,91)(31,54,73,92)(32,55,74,93)(33,56,75,94)(34,57,76,95)(35,58,77,96)(36,59,78,85), (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) );
G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,73),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(61,94),(62,95),(63,96),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,72,44,17),(2,61,45,18),(3,62,46,19),(4,63,47,20),(5,64,48,21),(6,65,37,22),(7,66,38,23),(8,67,39,24),(9,68,40,13),(10,69,41,14),(11,70,42,15),(12,71,43,16),(25,60,79,86),(26,49,80,87),(27,50,81,88),(28,51,82,89),(29,52,83,90),(30,53,84,91),(31,54,73,92),(32,55,74,93),(33,56,75,94),(34,57,76,95),(35,58,77,96),(36,59,78,85)], [(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)]])
C2×C4×C12 is a maximal subgroup of
C12.8C42 (C2×C12)⋊3C8 C12⋊7M4(2) C42.285D6 C42.270D6 C12⋊4(C4⋊C4) (C2×Dic6)⋊7C4 C42⋊6Dic3 (C2×C42).6S3 C42⋊10Dic3 C42⋊11Dic3 C42⋊7Dic3 (C2×C4)⋊6D12 (C2×C42)⋊3S3 C42.274D6 C42.276D6 C42.277D6
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4X | 6A | ··· | 6N | 12A | ··· | 12AV |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C2×C4×C12 | C4×C12 | C22×C12 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 3 | 2 | 24 | 8 | 6 | 48 |
Matrix representation of C2×C4×C12 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 1 |
| 5 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 5 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,5,0,0,0,1],[5,0,0,0,3,0,0,0,5] >;
C2×C4×C12 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{12} % in TeX
G:=Group("C2xC4xC12"); // GroupNames label
G:=SmallGroup(96,161);
// by ID
G=gap.SmallGroup(96,161);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,295]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations