metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C31⋊2D4, C22⋊D31, D62⋊2C2, Dic31⋊C2, C2.5D62, C62.5C22, (C2×C62)⋊2C2, SmallGroup(248,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C31⋊D4
G = < a,b,c | a31=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C31⋊D4
►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 87 42 111)(2 86 43 110)(3 85 44 109)(4 84 45 108)(5 83 46 107)(6 82 47 106)(7 81 48 105)(8 80 49 104)(9 79 50 103)(10 78 51 102)(11 77 52 101)(12 76 53 100)(13 75 54 99)(14 74 55 98)(15 73 56 97)(16 72 57 96)(17 71 58 95)(18 70 59 94)(19 69 60 124)(20 68 61 123)(21 67 62 122)(22 66 32 121)(23 65 33 120)(24 64 34 119)(25 63 35 118)(26 93 36 117)(27 92 37 116)(28 91 38 115)(29 90 39 114)(30 89 40 113)(31 88 41 112)
(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(53 62)(54 61)(55 60)(56 59)(57 58)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 124)(75 123)(76 122)(77 121)(78 120)(79 119)(80 118)(81 117)(82 116)(83 115)(84 114)(85 113)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)
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), (1,87,42,111)(2,86,43,110)(3,85,44,109)(4,84,45,108)(5,83,46,107)(6,82,47,106)(7,81,48,105)(8,80,49,104)(9,79,50,103)(10,78,51,102)(11,77,52,101)(12,76,53,100)(13,75,54,99)(14,74,55,98)(15,73,56,97)(16,72,57,96)(17,71,58,95)(18,70,59,94)(19,69,60,124)(20,68,61,123)(21,67,62,122)(22,66,32,121)(23,65,33,120)(24,64,34,119)(25,63,35,118)(26,93,36,117)(27,92,37,116)(28,91,38,115)(29,90,39,114)(30,89,40,113)(31,88,41,112), (2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(53,62)(54,61)(55,60)(56,59)(57,58)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,124)(75,123)(76,122)(77,121)(78,120)(79,119)(80,118)(81,117)(82,116)(83,115)(84,114)(85,113)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)>;
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), (1,87,42,111)(2,86,43,110)(3,85,44,109)(4,84,45,108)(5,83,46,107)(6,82,47,106)(7,81,48,105)(8,80,49,104)(9,79,50,103)(10,78,51,102)(11,77,52,101)(12,76,53,100)(13,75,54,99)(14,74,55,98)(15,73,56,97)(16,72,57,96)(17,71,58,95)(18,70,59,94)(19,69,60,124)(20,68,61,123)(21,67,62,122)(22,66,32,121)(23,65,33,120)(24,64,34,119)(25,63,35,118)(26,93,36,117)(27,92,37,116)(28,91,38,115)(29,90,39,114)(30,89,40,113)(31,88,41,112), (2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(53,62)(54,61)(55,60)(56,59)(57,58)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,124)(75,123)(76,122)(77,121)(78,120)(79,119)(80,118)(81,117)(82,116)(83,115)(84,114)(85,113)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105) );
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)], [(1,87,42,111),(2,86,43,110),(3,85,44,109),(4,84,45,108),(5,83,46,107),(6,82,47,106),(7,81,48,105),(8,80,49,104),(9,79,50,103),(10,78,51,102),(11,77,52,101),(12,76,53,100),(13,75,54,99),(14,74,55,98),(15,73,56,97),(16,72,57,96),(17,71,58,95),(18,70,59,94),(19,69,60,124),(20,68,61,123),(21,67,62,122),(22,66,32,121),(23,65,33,120),(24,64,34,119),(25,63,35,118),(26,93,36,117),(27,92,37,116),(28,91,38,115),(29,90,39,114),(30,89,40,113),(31,88,41,112)], [(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(53,62),(54,61),(55,60),(56,59),(57,58),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,124),(75,123),(76,122),(77,121),(78,120),(79,119),(80,118),(81,117),(82,116),(83,115),(84,114),(85,113),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105)]])
C31⋊D4 is a maximal subgroup of
D124⋊5C2 D4×D31 D4⋊2D31
C31⋊D4 is a maximal quotient of Dic31⋊C4 D62⋊C4 D4⋊D31 D4.D31 Q8⋊D31 C31⋊Q16 C23.D31
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 31A | ··· | 31O | 62A | ··· | 62AS |
| order | 1 | 2 | 2 | 2 | 4 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | 2 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D31 | D62 | C31⋊D4 |
| kernel | C31⋊D4 | Dic31 | D62 | C2×C62 | C31 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 15 | 15 | 30 |
Matrix representation of C31⋊D4 ►in GL2(𝔽373) generated by
| 30 | 1 |
| 275 | 233 |
| 100 | 177 |
| 346 | 273 |
| 13 | 289 |
| 2 | 360 |
G:=sub<GL(2,GF(373))| [30,275,1,233],[100,346,177,273],[13,2,289,360] >;
C31⋊D4 in GAP, Magma, Sage, TeX
C_{31}\rtimes D_4 % in TeX
G:=Group("C31:D4"); // GroupNames label
G:=SmallGroup(248,7);
// by ID
G=gap.SmallGroup(248,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-31,49,3843]);
// Polycyclic
G:=Group<a,b,c|a^31=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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