direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: F5×C23, C5⋊C92, C115⋊2C4, D5.C46, (D5×C23).2C2, SmallGroup(460,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C23 |
Generators and relations for F5×C23
G = < a,b,c | a23=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of F5×C23
►On 115 points
Generators in S115
(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)
(1 82 94 54 42)(2 83 95 55 43)(3 84 96 56 44)(4 85 97 57 45)(5 86 98 58 46)(6 87 99 59 24)(7 88 100 60 25)(8 89 101 61 26)(9 90 102 62 27)(10 91 103 63 28)(11 92 104 64 29)(12 70 105 65 30)(13 71 106 66 31)(14 72 107 67 32)(15 73 108 68 33)(16 74 109 69 34)(17 75 110 47 35)(18 76 111 48 36)(19 77 112 49 37)(20 78 113 50 38)(21 79 114 51 39)(22 80 115 52 40)(23 81 93 53 41)
(24 59 87 99)(25 60 88 100)(26 61 89 101)(27 62 90 102)(28 63 91 103)(29 64 92 104)(30 65 70 105)(31 66 71 106)(32 67 72 107)(33 68 73 108)(34 69 74 109)(35 47 75 110)(36 48 76 111)(37 49 77 112)(38 50 78 113)(39 51 79 114)(40 52 80 115)(41 53 81 93)(42 54 82 94)(43 55 83 95)(44 56 84 96)(45 57 85 97)(46 58 86 98)
G:=sub<Sym(115)| (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), (1,82,94,54,42)(2,83,95,55,43)(3,84,96,56,44)(4,85,97,57,45)(5,86,98,58,46)(6,87,99,59,24)(7,88,100,60,25)(8,89,101,61,26)(9,90,102,62,27)(10,91,103,63,28)(11,92,104,64,29)(12,70,105,65,30)(13,71,106,66,31)(14,72,107,67,32)(15,73,108,68,33)(16,74,109,69,34)(17,75,110,47,35)(18,76,111,48,36)(19,77,112,49,37)(20,78,113,50,38)(21,79,114,51,39)(22,80,115,52,40)(23,81,93,53,41), (24,59,87,99)(25,60,88,100)(26,61,89,101)(27,62,90,102)(28,63,91,103)(29,64,92,104)(30,65,70,105)(31,66,71,106)(32,67,72,107)(33,68,73,108)(34,69,74,109)(35,47,75,110)(36,48,76,111)(37,49,77,112)(38,50,78,113)(39,51,79,114)(40,52,80,115)(41,53,81,93)(42,54,82,94)(43,55,83,95)(44,56,84,96)(45,57,85,97)(46,58,86,98)>;
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), (1,82,94,54,42)(2,83,95,55,43)(3,84,96,56,44)(4,85,97,57,45)(5,86,98,58,46)(6,87,99,59,24)(7,88,100,60,25)(8,89,101,61,26)(9,90,102,62,27)(10,91,103,63,28)(11,92,104,64,29)(12,70,105,65,30)(13,71,106,66,31)(14,72,107,67,32)(15,73,108,68,33)(16,74,109,69,34)(17,75,110,47,35)(18,76,111,48,36)(19,77,112,49,37)(20,78,113,50,38)(21,79,114,51,39)(22,80,115,52,40)(23,81,93,53,41), (24,59,87,99)(25,60,88,100)(26,61,89,101)(27,62,90,102)(28,63,91,103)(29,64,92,104)(30,65,70,105)(31,66,71,106)(32,67,72,107)(33,68,73,108)(34,69,74,109)(35,47,75,110)(36,48,76,111)(37,49,77,112)(38,50,78,113)(39,51,79,114)(40,52,80,115)(41,53,81,93)(42,54,82,94)(43,55,83,95)(44,56,84,96)(45,57,85,97)(46,58,86,98) );
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)], [(1,82,94,54,42),(2,83,95,55,43),(3,84,96,56,44),(4,85,97,57,45),(5,86,98,58,46),(6,87,99,59,24),(7,88,100,60,25),(8,89,101,61,26),(9,90,102,62,27),(10,91,103,63,28),(11,92,104,64,29),(12,70,105,65,30),(13,71,106,66,31),(14,72,107,67,32),(15,73,108,68,33),(16,74,109,69,34),(17,75,110,47,35),(18,76,111,48,36),(19,77,112,49,37),(20,78,113,50,38),(21,79,114,51,39),(22,80,115,52,40),(23,81,93,53,41)], [(24,59,87,99),(25,60,88,100),(26,61,89,101),(27,62,90,102),(28,63,91,103),(29,64,92,104),(30,65,70,105),(31,66,71,106),(32,67,72,107),(33,68,73,108),(34,69,74,109),(35,47,75,110),(36,48,76,111),(37,49,77,112),(38,50,78,113),(39,51,79,114),(40,52,80,115),(41,53,81,93),(42,54,82,94),(43,55,83,95),(44,56,84,96),(45,57,85,97),(46,58,86,98)]])
115 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 23A | ··· | 23V | 46A | ··· | 46V | 92A | ··· | 92AR | 115A | ··· | 115V |
| order | 1 | 2 | 4 | 4 | 5 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 115 | ··· | 115 |
| size | 1 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C23 | C46 | C92 | F5 | F5×C23 |
| kernel | F5×C23 | D5×C23 | C115 | F5 | D5 | C5 | C23 | C1 |
| # reps | 1 | 1 | 2 | 22 | 22 | 44 | 1 | 22 |
Matrix representation of F5×C23 ►in GL4(𝔽461) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 460 | 460 | 460 | 460 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(461))| [23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[460,1,0,0,460,0,1,0,460,0,0,1,460,0,0,0],[0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0] >;
F5×C23 in GAP, Magma, Sage, TeX
F_5\times C_{23} % in TeX
G:=Group("F5xC23"); // GroupNames label
G:=SmallGroup(460,5);
// by ID
G=gap.SmallGroup(460,5);
# by ID
G:=PCGroup([4,-2,-23,-2,-5,184,2947,139]);
// Polycyclic
G:=Group<a,b,c|a^23=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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