direct product, abelian, monomial, 2-elementary
Aliases: C2×C42, SmallGroup(84,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C42 |
| C1 — C2×C42 |
| C1 — C2×C42 |
Generators and relations for C2×C42
G = < a,b | a2=b42=1, ab=ba >
Smallest permutation representation of C2×C42
►Regular action on 84 points
Generators in S84
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)
(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)
G:=sub<Sym(84)| (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77), (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)>;
G:=Group( (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77), (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) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77)], [(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)]])
C2×C42 is a maximal subgroup of
C21⋊7D4 C21.A4
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 7A | ··· | 7F | 14A | ··· | 14R | 21A | ··· | 21L | 42A | ··· | 42AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C2×C42 | C42 | C2×C14 | C14 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 12 | 36 |
Matrix representation of C2×C42 ►in GL2(𝔽43) generated by
| 42 | 0 |
| 0 | 42 |
| 3 | 0 |
| 0 | 40 |
G:=sub<GL(2,GF(43))| [42,0,0,42],[3,0,0,40] >;
C2×C42 in GAP, Magma, Sage, TeX
C_2\times C_{42} % in TeX
G:=Group("C2xC42"); // GroupNames label
G:=SmallGroup(84,15);
// by ID
G=gap.SmallGroup(84,15);
# by ID
G:=PCGroup([4,-2,-2,-3,-7]);
// Polycyclic
G:=Group<a,b|a^2=b^42=1,a*b=b*a>;
// generators/relations
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