direct product, abelian, monomial, 2-elementary
Aliases: C2×C54, SmallGroup(108,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C54 |
| C1 — C2×C54 |
| C1 — C2×C54 |
Generators and relations for C2×C54
G = < a,b | a2=b54=1, ab=ba >
Smallest permutation representation of C2×C54
►Regular action on 108 points
Generators in S108
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)
(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 97 98 99 100 101 102 103 104 105 106 107 108)
G:=sub<Sym(108)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108) );
G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74)], [(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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)]])
C2×C54 is a maximal subgroup of
C27⋊D4 C27.A4 C27⋊A4
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 9A | ··· | 9F | 18A | ··· | 18R | 27A | ··· | 27R | 54A | ··· | 54BB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 |
| kernel | C2×C54 | C54 | C2×C18 | C18 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 18 | 54 |
Matrix representation of C2×C54 ►in GL2(𝔽109) generated by
| 108 | 0 |
| 0 | 1 |
| 80 | 0 |
| 0 | 29 |
G:=sub<GL(2,GF(109))| [108,0,0,1],[80,0,0,29] >;
C2×C54 in GAP, Magma, Sage, TeX
C_2\times C_{54} % in TeX
G:=Group("C2xC54"); // GroupNames label
G:=SmallGroup(108,5);
// by ID
G=gap.SmallGroup(108,5);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,57,78]);
// Polycyclic
G:=Group<a,b|a^2=b^54=1,a*b=b*a>;
// generators/relations
Export