direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C22×C31⋊C3, C62⋊2C6, (C2×C62)⋊3C3, C31⋊2(C2×C6), SmallGroup(372,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C22×C31⋊C3 |
Generators and relations for C22×C31⋊C3
G = < a,b,c,d | a2=b2=c31=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Smallest permutation representation of C22×C31⋊C3
►On 124 points
Generators in S124
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 121)(60 122)(61 123)(62 124)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(2 26 6)(3 20 11)(4 14 16)(5 8 21)(7 27 31)(9 15 10)(12 28 25)(13 22 30)(17 29 19)(18 23 24)(33 57 37)(34 51 42)(35 45 47)(36 39 52)(38 58 62)(40 46 41)(43 59 56)(44 53 61)(48 60 50)(49 54 55)(64 88 68)(65 82 73)(66 76 78)(67 70 83)(69 89 93)(71 77 72)(74 90 87)(75 84 92)(79 91 81)(80 85 86)(95 119 99)(96 113 104)(97 107 109)(98 101 114)(100 120 124)(102 108 103)(105 121 118)(106 115 123)(110 122 112)(111 116 117)
G:=sub<Sym(124)| (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124), (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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86)(95,119,99)(96,113,104)(97,107,109)(98,101,114)(100,120,124)(102,108,103)(105,121,118)(106,115,123)(110,122,112)(111,116,117)>;
G:=Group( (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,121)(60,122)(61,123)(62,124), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124), (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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86)(95,119,99)(96,113,104)(97,107,109)(98,101,114)(100,120,124)(102,108,103)(105,121,118)(106,115,123)(110,122,112)(111,116,117) );
G=PermutationGroup([[(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,121),(60,122),(61,123),(62,124)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124)], [(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),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(2,26,6),(3,20,11),(4,14,16),(5,8,21),(7,27,31),(9,15,10),(12,28,25),(13,22,30),(17,29,19),(18,23,24),(33,57,37),(34,51,42),(35,45,47),(36,39,52),(38,58,62),(40,46,41),(43,59,56),(44,53,61),(48,60,50),(49,54,55),(64,88,68),(65,82,73),(66,76,78),(67,70,83),(69,89,93),(71,77,72),(74,90,87),(75,84,92),(79,91,81),(80,85,86),(95,119,99),(96,113,104),(97,107,109),(98,101,114),(100,120,124),(102,108,103),(105,121,118),(106,115,123),(110,122,112),(111,116,117)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 31A | ··· | 31J | 62A | ··· | 62AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | 1 | 1 | 31 | 31 | 31 | ··· | 31 | 3 | ··· | 3 | 3 | ··· | 3 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C31⋊C3 | C2×C31⋊C3 |
| kernel | C22×C31⋊C3 | C2×C31⋊C3 | C2×C62 | C62 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 10 | 30 |
Matrix representation of C22×C31⋊C3 ►in GL4(𝔽373) generated by
| 372 | 0 | 0 | 0 |
| 0 | 372 | 0 | 0 |
| 0 | 0 | 372 | 0 |
| 0 | 0 | 0 | 372 |
| 372 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 179 | 157 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 284 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 225 | 225 | 239 |
| 0 | 65 | 360 | 147 |
G:=sub<GL(4,GF(373))| [372,0,0,0,0,372,0,0,0,0,372,0,0,0,0,372],[372,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,179,1,0,0,157,0,1,0,1,0,0],[284,0,0,0,0,1,225,65,0,0,225,360,0,0,239,147] >;
C22×C31⋊C3 in GAP, Magma, Sage, TeX
C_2^2\times C_{31}\rtimes C_3 % in TeX
G:=Group("C2^2xC31:C3"); // GroupNames label
G:=SmallGroup(372,9);
// by ID
G=gap.SmallGroup(372,9);
# by ID
G:=PCGroup([4,-2,-2,-3,-31,1211]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^31=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations
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