direct product, abelian, monomial, 2-elementary
Aliases: C22×C30, SmallGroup(120,47)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C30 |
| C1 — C22×C30 |
| C1 — C22×C30 |
Generators and relations for C22×C30
G = < a,b,c | a2=b2=c30=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C2×C10, C22×C6, C30, C22×C10, C2×C30, C22×C30
Quotients: C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C2×C10, C22×C6, C30, C22×C10, C2×C30, C22×C30
►Regular action on 120 points
Generators in S120
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 91)(57 92)(58 93)(59 94)(60 95)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
G:=sub<Sym(120)| (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,91)(57,92)(58,93)(59,94)(60,95), (1,56)(2,57)(3,58)(4,59)(5,60)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)>;
G:=Group( (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,91)(57,92)(58,93)(59,94)(60,95), (1,56)(2,57)(3,58)(4,59)(5,60)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120) );
G=PermutationGroup([[(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,91),(57,92),(58,93),(59,94),(60,95)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100)], [(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),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)]])
C22×C30 is a maximal subgroup of
C30.38D4
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6N | 10A | ··· | 10AB | 15A | ··· | 15H | 30A | ··· | 30BD |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C22×C30 | C2×C30 | C22×C10 | C22×C6 | C2×C10 | C2×C6 | C23 | C22 |
| # reps | 1 | 7 | 2 | 4 | 14 | 28 | 8 | 56 |
Matrix representation of C22×C30 ►in GL3(𝔽31) generated by
| 30 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 30 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 1 |
| 30 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 17 |
G:=sub<GL(3,GF(31))| [30,0,0,0,1,0,0,0,1],[30,0,0,0,30,0,0,0,1],[30,0,0,0,26,0,0,0,17] >;
C22×C30 in GAP, Magma, Sage, TeX
C_2^2\times C_{30} % in TeX
G:=Group("C2^2xC30"); // GroupNames label
G:=SmallGroup(120,47);
// by ID
G=gap.SmallGroup(120,47);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^30=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations