direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C31, C22⋊C93, (C2×C62)⋊1C3, SmallGroup(372,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C31 |
Generators and relations for A4×C31
G = < a,b,c,d | a31=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C31
►On 124 points
Generators in S124
(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 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 94)(31 95)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)
(32 63 109)(33 64 110)(34 65 111)(35 66 112)(36 67 113)(37 68 114)(38 69 115)(39 70 116)(40 71 117)(41 72 118)(42 73 119)(43 74 120)(44 75 121)(45 76 122)(46 77 123)(47 78 124)(48 79 94)(49 80 95)(50 81 96)(51 82 97)(52 83 98)(53 84 99)(54 85 100)(55 86 101)(56 87 102)(57 88 103)(58 89 104)(59 90 105)(60 91 106)(61 92 107)(62 93 108)
G:=sub<Sym(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), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,94)(31,95)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108), (32,63,109)(33,64,110)(34,65,111)(35,66,112)(36,67,113)(37,68,114)(38,69,115)(39,70,116)(40,71,117)(41,72,118)(42,73,119)(43,74,120)(44,75,121)(45,76,122)(46,77,123)(47,78,124)(48,79,94)(49,80,95)(50,81,96)(51,82,97)(52,83,98)(53,84,99)(54,85,100)(55,86,101)(56,87,102)(57,88,103)(58,89,104)(59,90,105)(60,91,106)(61,92,107)(62,93,108)>;
G:=Group( (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), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,94)(31,95)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108), (32,63,109)(33,64,110)(34,65,111)(35,66,112)(36,67,113)(37,68,114)(38,69,115)(39,70,116)(40,71,117)(41,72,118)(42,73,119)(43,74,120)(44,75,121)(45,76,122)(46,77,123)(47,78,124)(48,79,94)(49,80,95)(50,81,96)(51,82,97)(52,83,98)(53,84,99)(54,85,100)(55,86,101)(56,87,102)(57,88,103)(58,89,104)(59,90,105)(60,91,106)(61,92,107)(62,93,108) );
G=PermutationGroup([[(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)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,124),(30,94),(31,95),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,121),(76,122),(77,123),(78,124),(79,94),(80,95),(81,96),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108)], [(32,63,109),(33,64,110),(34,65,111),(35,66,112),(36,67,113),(37,68,114),(38,69,115),(39,70,116),(40,71,117),(41,72,118),(42,73,119),(43,74,120),(44,75,121),(45,76,122),(46,77,123),(47,78,124),(48,79,94),(49,80,95),(50,81,96),(51,82,97),(52,83,98),(53,84,99),(54,85,100),(55,86,101),(56,87,102),(57,88,103),(58,89,104),(59,90,105),(60,91,106),(61,92,107),(62,93,108)]])
124 conjugacy classes
| class | 1 | 2 | 3A | 3B | 31A | ··· | 31AD | 62A | ··· | 62AD | 93A | ··· | 93BH |
| order | 1 | 2 | 3 | 3 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C31 | C93 | A4 | A4×C31 |
| kernel | A4×C31 | C2×C62 | A4 | C22 | C31 | C1 |
| # reps | 1 | 2 | 30 | 60 | 1 | 30 |
Matrix representation of A4×C31 ►in GL3(𝔽373) generated by
| 154 | 0 | 0 |
| 0 | 154 | 0 |
| 0 | 0 | 154 |
| 372 | 0 | 0 |
| 372 | 0 | 1 |
| 372 | 1 | 0 |
| 0 | 1 | 372 |
| 1 | 0 | 372 |
| 0 | 0 | 372 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(373))| [154,0,0,0,154,0,0,0,154],[372,372,372,0,0,1,0,1,0],[0,1,0,1,0,0,372,372,372],[0,0,1,1,0,0,0,1,0] >;
A4×C31 in GAP, Magma, Sage, TeX
A_4\times C_{31} % in TeX
G:=Group("A4xC31"); // GroupNames label
G:=SmallGroup(372,10);
// by ID
G=gap.SmallGroup(372,10);
# by ID
G:=PCGroup([4,-3,-31,-2,2,2234,4467]);
// Polycyclic
G:=Group<a,b,c,d|a^31=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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