direct product, abelian, monomial
Aliases: C2×C62, SmallGroup(72,50)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C62 |
| C1 — C2×C62 |
| C1 — C2×C62 |
Generators and relations for C2×C62
G = < a,b,c | a2=b6=c6=1, ab=ba, ac=ca, bc=cb >
Subgroups: 96, all normal (4 characteristic)
C1, C2, C3, C22, C6, C23, C32, C2×C6, C3×C6, C22×C6, C62, C2×C62
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C3×C6, C22×C6, C62, C2×C62
►Regular action on 72 points
Generators in S72
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 59)(8 60)(9 55)(10 56)(11 57)(12 58)(13 52)(14 53)(15 54)(16 49)(17 50)(18 51)(19 63)(20 64)(21 65)(22 66)(23 61)(24 62)(25 38)(26 39)(27 40)(28 41)(29 42)(30 37)(31 45)(32 46)(33 47)(34 48)(35 43)(36 44)
(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)
(1 31 58 14 39 61)(2 32 59 15 40 62)(3 33 60 16 41 63)(4 34 55 17 42 64)(5 35 56 18 37 65)(6 36 57 13 38 66)(7 54 27 24 68 46)(8 49 28 19 69 47)(9 50 29 20 70 48)(10 51 30 21 71 43)(11 52 25 22 72 44)(12 53 26 23 67 45)
G:=sub<Sym(72)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44), (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), (1,31,58,14,39,61)(2,32,59,15,40,62)(3,33,60,16,41,63)(4,34,55,17,42,64)(5,35,56,18,37,65)(6,36,57,13,38,66)(7,54,27,24,68,46)(8,49,28,19,69,47)(9,50,29,20,70,48)(10,51,30,21,71,43)(11,52,25,22,72,44)(12,53,26,23,67,45)>;
G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44), (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), (1,31,58,14,39,61)(2,32,59,15,40,62)(3,33,60,16,41,63)(4,34,55,17,42,64)(5,35,56,18,37,65)(6,36,57,13,38,66)(7,54,27,24,68,46)(8,49,28,19,69,47)(9,50,29,20,70,48)(10,51,30,21,71,43)(11,52,25,22,72,44)(12,53,26,23,67,45) );
G=PermutationGroup([[(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,59),(8,60),(9,55),(10,56),(11,57),(12,58),(13,52),(14,53),(15,54),(16,49),(17,50),(18,51),(19,63),(20,64),(21,65),(22,66),(23,61),(24,62),(25,38),(26,39),(27,40),(28,41),(29,42),(30,37),(31,45),(32,46),(33,47),(34,48),(35,43),(36,44)], [(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)], [(1,31,58,14,39,61),(2,32,59,15,40,62),(3,33,60,16,41,63),(4,34,55,17,42,64),(5,35,56,18,37,65),(6,36,57,13,38,66),(7,54,27,24,68,46),(8,49,28,19,69,47),(9,50,29,20,70,48),(10,51,30,21,71,43),(11,52,25,22,72,44),(12,53,26,23,67,45)]])
C2×C62 is a maximal subgroup of
C62⋊5C4
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | ··· | 3H | 6A | ··· | 6BD |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C2×C62 | C62 | C22×C6 | C2×C6 |
| # reps | 1 | 7 | 8 | 56 |
Matrix representation of C2×C62 ►in GL3(𝔽7) generated by
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 6 |
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,1],[1,0,0,0,2,0,0,0,6],[5,0,0,0,5,0,0,0,4] >;
C2×C62 in GAP, Magma, Sage, TeX
C_2\times C_6^2
% in TeX
G:=Group("C2xC6^2"); // GroupNames label
G:=SmallGroup(72,50);
// by ID
G=gap.SmallGroup(72,50);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3]);
// Polycyclic
G:=Group<a,b,c|a^2=b^6=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations