direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C4×C31⋊C3, C124⋊C3, C31⋊2C12, C62.2C6, C2.(C2×C31⋊C3), (C2×C31⋊C3).2C2, SmallGroup(372,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C4×C31⋊C3 |
Generators and relations for C4×C31⋊C3
G = < a,b,c | a4=b31=c3=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C4×C31⋊C3
►On 124 points
Generators in S124
(1 94 32 63)(2 95 33 64)(3 96 34 65)(4 97 35 66)(5 98 36 67)(6 99 37 68)(7 100 38 69)(8 101 39 70)(9 102 40 71)(10 103 41 72)(11 104 42 73)(12 105 43 74)(13 106 44 75)(14 107 45 76)(15 108 46 77)(16 109 47 78)(17 110 48 79)(18 111 49 80)(19 112 50 81)(20 113 51 82)(21 114 52 83)(22 115 53 84)(23 116 54 85)(24 117 55 86)(25 118 56 87)(26 119 57 88)(27 120 58 89)(28 121 59 90)(29 122 60 91)(30 123 61 92)(31 124 62 93)
(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)| (1,94,32,63)(2,95,33,64)(3,96,34,65)(4,97,35,66)(5,98,36,67)(6,99,37,68)(7,100,38,69)(8,101,39,70)(9,102,40,71)(10,103,41,72)(11,104,42,73)(12,105,43,74)(13,106,44,75)(14,107,45,76)(15,108,46,77)(16,109,47,78)(17,110,48,79)(18,111,49,80)(19,112,50,81)(20,113,51,82)(21,114,52,83)(22,115,53,84)(23,116,54,85)(24,117,55,86)(25,118,56,87)(26,119,57,88)(27,120,58,89)(28,121,59,90)(29,122,60,91)(30,123,61,92)(31,124,62,93), (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( (1,94,32,63)(2,95,33,64)(3,96,34,65)(4,97,35,66)(5,98,36,67)(6,99,37,68)(7,100,38,69)(8,101,39,70)(9,102,40,71)(10,103,41,72)(11,104,42,73)(12,105,43,74)(13,106,44,75)(14,107,45,76)(15,108,46,77)(16,109,47,78)(17,110,48,79)(18,111,49,80)(19,112,50,81)(20,113,51,82)(21,114,52,83)(22,115,53,84)(23,116,54,85)(24,117,55,86)(25,118,56,87)(26,119,57,88)(27,120,58,89)(28,121,59,90)(29,122,60,91)(30,123,61,92)(31,124,62,93), (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([[(1,94,32,63),(2,95,33,64),(3,96,34,65),(4,97,35,66),(5,98,36,67),(6,99,37,68),(7,100,38,69),(8,101,39,70),(9,102,40,71),(10,103,41,72),(11,104,42,73),(12,105,43,74),(13,106,44,75),(14,107,45,76),(15,108,46,77),(16,109,47,78),(17,110,48,79),(18,111,49,80),(19,112,50,81),(20,113,51,82),(21,114,52,83),(22,115,53,84),(23,116,54,85),(24,117,55,86),(25,118,56,87),(26,119,57,88),(27,120,58,89),(28,121,59,90),(29,122,60,91),(30,123,61,92),(31,124,62,93)], [(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 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 31A | ··· | 31J | 62A | ··· | 62J | 124A | ··· | 124T |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 31 | 31 | 1 | 1 | 31 | 31 | 31 | 31 | 31 | 31 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C31⋊C3 | C2×C31⋊C3 | C4×C31⋊C3 |
| kernel | C4×C31⋊C3 | C2×C31⋊C3 | C124 | C31⋊C3 | C62 | C31 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 10 | 10 | 20 |
Matrix representation of C4×C31⋊C3 ►in GL3(𝔽5) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 3 | 0 | 2 |
| 2 | 3 | 3 |
| 4 | 2 | 0 |
| 0 | 3 | 0 |
| 3 | 4 | 0 |
| 0 | 4 | 1 |
G:=sub<GL(3,GF(5))| [2,0,0,0,2,0,0,0,2],[3,2,4,0,3,2,2,3,0],[0,3,0,3,4,4,0,0,1] >;
C4×C31⋊C3 in GAP, Magma, Sage, TeX
C_4\times C_{31}\rtimes C_3 % in TeX
G:=Group("C4xC31:C3"); // GroupNames label
G:=SmallGroup(372,2);
// by ID
G=gap.SmallGroup(372,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-31,24,2407]);
// Polycyclic
G:=Group<a,b,c|a^4=b^31=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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