metabelian, soluble, monomial, A-group
Aliases: C31⋊A4, (C2×C62)⋊2C3, C22⋊(C31⋊C3), SmallGroup(372,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C62 — C31⋊A4 |
Generators and relations for C31⋊A4
G = < a,b,c,d | a31=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a5, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C31⋊A4
►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 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(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)(81 121)(82 122)(83 123)(84 124)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(91 100)(92 101)(93 102)
(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)(32 123 69)(33 117 74)(34 111 79)(35 105 84)(36 99 89)(37 124 63)(38 118 68)(39 112 73)(40 106 78)(41 100 83)(42 94 88)(43 119 93)(44 113 67)(45 107 72)(46 101 77)(47 95 82)(48 120 87)(49 114 92)(50 108 66)(51 102 71)(52 96 76)(53 121 81)(54 115 86)(55 109 91)(56 103 65)(57 97 70)(58 122 75)(59 116 80)(60 110 85)(61 104 90)(62 98 64)
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), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(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)(81,121)(82,122)(83,123)(84,124)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(91,100)(92,101)(93,102), (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)(32,123,69)(33,117,74)(34,111,79)(35,105,84)(36,99,89)(37,124,63)(38,118,68)(39,112,73)(40,106,78)(41,100,83)(42,94,88)(43,119,93)(44,113,67)(45,107,72)(46,101,77)(47,95,82)(48,120,87)(49,114,92)(50,108,66)(51,102,71)(52,96,76)(53,121,81)(54,115,86)(55,109,91)(56,103,65)(57,97,70)(58,122,75)(59,116,80)(60,110,85)(61,104,90)(62,98,64)>;
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), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(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)(81,121)(82,122)(83,123)(84,124)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(91,100)(92,101)(93,102), (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)(32,123,69)(33,117,74)(34,111,79)(35,105,84)(36,99,89)(37,124,63)(38,118,68)(39,112,73)(40,106,78)(41,100,83)(42,94,88)(43,119,93)(44,113,67)(45,107,72)(46,101,77)(47,95,82)(48,120,87)(49,114,92)(50,108,66)(51,102,71)(52,96,76)(53,121,81)(54,115,86)(55,109,91)(56,103,65)(57,97,70)(58,122,75)(59,116,80)(60,110,85)(61,104,90)(62,98,64) );
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)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,121),(54,122),(55,123),(56,124),(57,94),(58,95),(59,96),(60,97),(61,98),(62,99)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(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),(81,121),(82,122),(83,123),(84,124),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99),(91,100),(92,101),(93,102)], [(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),(32,123,69),(33,117,74),(34,111,79),(35,105,84),(36,99,89),(37,124,63),(38,118,68),(39,112,73),(40,106,78),(41,100,83),(42,94,88),(43,119,93),(44,113,67),(45,107,72),(46,101,77),(47,95,82),(48,120,87),(49,114,92),(50,108,66),(51,102,71),(52,96,76),(53,121,81),(54,115,86),(55,109,91),(56,103,65),(57,97,70),(58,122,75),(59,116,80),(60,110,85),(61,104,90),(62,98,64)]])
44 conjugacy classes
| class | 1 | 2 | 3A | 3B | 31A | ··· | 31J | 62A | ··· | 62AD |
| order | 1 | 2 | 3 | 3 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 3 | 124 | 124 | 3 | ··· | 3 | 3 | ··· | 3 |
44 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||
| image | C1 | C3 | A4 | C31⋊C3 | C31⋊A4 |
| kernel | C31⋊A4 | C2×C62 | C31 | C22 | C1 |
| # reps | 1 | 2 | 1 | 10 | 30 |
Matrix representation of C31⋊A4 ►in GL3(𝔽373) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 96 | 46 |
| 56 | 196 | 2 |
| 2 | 248 | 288 |
| 288 | 48 | 68 |
| 197 | 288 | 122 |
| 122 | 346 | 305 |
| 305 | 308 | 202 |
| 1 | 0 | 0 |
| 347 | 161 | 237 |
| 155 | 19 | 211 |
G:=sub<GL(3,GF(373))| [0,0,1,1,0,96,0,1,46],[56,2,288,196,248,48,2,288,68],[197,122,305,288,346,308,122,305,202],[1,347,155,0,161,19,0,237,211] >;
C31⋊A4 in GAP, Magma, Sage, TeX
C_{31}\rtimes A_4 % in TeX
G:=Group("C31:A4"); // GroupNames label
G:=SmallGroup(372,11);
// by ID
G=gap.SmallGroup(372,11);
# by ID
G:=PCGroup([4,-3,-2,2,-31,49,110,4803]);
// Polycyclic
G:=Group<a,b,c,d|a^31=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^5,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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