direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D31, D62.C2, C124⋊2C2, C2.1D62, Dic31⋊2C2, C62.2C22, C31⋊1(C2×C4), SmallGroup(248,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C4×D31 |
Generators and relations for C4×D31
G = < a,b,c | a4=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D31
►On 124 points
Generators in S124
(1 95 49 63)(2 96 50 64)(3 97 51 65)(4 98 52 66)(5 99 53 67)(6 100 54 68)(7 101 55 69)(8 102 56 70)(9 103 57 71)(10 104 58 72)(11 105 59 73)(12 106 60 74)(13 107 61 75)(14 108 62 76)(15 109 32 77)(16 110 33 78)(17 111 34 79)(18 112 35 80)(19 113 36 81)(20 114 37 82)(21 115 38 83)(22 116 39 84)(23 117 40 85)(24 118 41 86)(25 119 42 87)(26 120 43 88)(27 121 44 89)(28 122 45 90)(29 123 46 91)(30 124 47 92)(31 94 48 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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 34)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(94 95)(96 124)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)
G:=sub<Sym(124)| (1,95,49,63)(2,96,50,64)(3,97,51,65)(4,98,52,66)(5,99,53,67)(6,100,54,68)(7,101,55,69)(8,102,56,70)(9,103,57,71)(10,104,58,72)(11,105,59,73)(12,106,60,74)(13,107,61,75)(14,108,62,76)(15,109,32,77)(16,110,33,78)(17,111,34,79)(18,112,35,80)(19,113,36,81)(20,114,37,82)(21,115,38,83)(22,116,39,84)(23,117,40,85)(24,118,41,86)(25,119,42,87)(26,120,43,88)(27,121,44,89)(28,122,45,90)(29,123,46,91)(30,124,47,92)(31,94,48,93), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,34)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(94,95)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)>;
G:=Group( (1,95,49,63)(2,96,50,64)(3,97,51,65)(4,98,52,66)(5,99,53,67)(6,100,54,68)(7,101,55,69)(8,102,56,70)(9,103,57,71)(10,104,58,72)(11,105,59,73)(12,106,60,74)(13,107,61,75)(14,108,62,76)(15,109,32,77)(16,110,33,78)(17,111,34,79)(18,112,35,80)(19,113,36,81)(20,114,37,82)(21,115,38,83)(22,116,39,84)(23,117,40,85)(24,118,41,86)(25,119,42,87)(26,120,43,88)(27,121,44,89)(28,122,45,90)(29,123,46,91)(30,124,47,92)(31,94,48,93), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,34)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(94,95)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111) );
G=PermutationGroup([[(1,95,49,63),(2,96,50,64),(3,97,51,65),(4,98,52,66),(5,99,53,67),(6,100,54,68),(7,101,55,69),(8,102,56,70),(9,103,57,71),(10,104,58,72),(11,105,59,73),(12,106,60,74),(13,107,61,75),(14,108,62,76),(15,109,32,77),(16,110,33,78),(17,111,34,79),(18,112,35,80),(19,113,36,81),(20,114,37,82),(21,115,38,83),(22,116,39,84),(23,117,40,85),(24,118,41,86),(25,119,42,87),(26,120,43,88),(27,121,44,89),(28,122,45,90),(29,123,46,91),(30,124,47,92),(31,94,48,93)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,34),(35,62),(36,61),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(94,95),(96,124),(97,123),(98,122),(99,121),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111)]])
C4×D31 is a maximal subgroup of
C8⋊D31 D124⋊5C2 D4⋊2D31 Q8⋊2D31
C4×D31 is a maximal quotient of C8⋊D31 Dic31⋊C4 D62⋊C4
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 31 | 31 | 1 | 1 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D31 | D62 | C4×D31 |
| kernel | C4×D31 | Dic31 | C124 | D62 | D31 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 15 | 15 | 30 |
Matrix representation of C4×D31 ►in GL3(𝔽373) generated by
| 104 | 0 | 0 |
| 0 | 372 | 0 |
| 0 | 0 | 372 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 372 | 194 |
| 372 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(373))| [104,0,0,0,372,0,0,0,372],[1,0,0,0,0,372,0,1,194],[372,0,0,0,0,1,0,1,0] >;
C4×D31 in GAP, Magma, Sage, TeX
C_4\times D_{31} % in TeX
G:=Group("C4xD31"); // GroupNames label
G:=SmallGroup(248,4);
// by ID
G=gap.SmallGroup(248,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-31,21,3843]);
// Polycyclic
G:=Group<a,b,c|a^4=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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