metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C23⋊F5, C5⋊Dic23, C115⋊1C4, D5.D23, (D5×C23).1C2, SmallGroup(460,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C115 — C23⋊F5 |
Generators and relations for C23⋊F5
G = < a,b,c | a23=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Smallest permutation representation of C23⋊F5
►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 44 66 76 94)(2 45 67 77 95)(3 46 68 78 96)(4 24 69 79 97)(5 25 47 80 98)(6 26 48 81 99)(7 27 49 82 100)(8 28 50 83 101)(9 29 51 84 102)(10 30 52 85 103)(11 31 53 86 104)(12 32 54 87 105)(13 33 55 88 106)(14 34 56 89 107)(15 35 57 90 108)(16 36 58 91 109)(17 37 59 92 110)(18 38 60 70 111)(19 39 61 71 112)(20 40 62 72 113)(21 41 63 73 114)(22 42 64 74 115)(23 43 65 75 93)
(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(24 63 97 73)(25 62 98 72)(26 61 99 71)(27 60 100 70)(28 59 101 92)(29 58 102 91)(30 57 103 90)(31 56 104 89)(32 55 105 88)(33 54 106 87)(34 53 107 86)(35 52 108 85)(36 51 109 84)(37 50 110 83)(38 49 111 82)(39 48 112 81)(40 47 113 80)(41 69 114 79)(42 68 115 78)(43 67 93 77)(44 66 94 76)(45 65 95 75)(46 64 96 74)
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), (1,44,66,76,94)(2,45,67,77,95)(3,46,68,78,96)(4,24,69,79,97)(5,25,47,80,98)(6,26,48,81,99)(7,27,49,82,100)(8,28,50,83,101)(9,29,51,84,102)(10,30,52,85,103)(11,31,53,86,104)(12,32,54,87,105)(13,33,55,88,106)(14,34,56,89,107)(15,35,57,90,108)(16,36,58,91,109)(17,37,59,92,110)(18,38,60,70,111)(19,39,61,71,112)(20,40,62,72,113)(21,41,63,73,114)(22,42,64,74,115)(23,43,65,75,93), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(24,63,97,73)(25,62,98,72)(26,61,99,71)(27,60,100,70)(28,59,101,92)(29,58,102,91)(30,57,103,90)(31,56,104,89)(32,55,105,88)(33,54,106,87)(34,53,107,86)(35,52,108,85)(36,51,109,84)(37,50,110,83)(38,49,111,82)(39,48,112,81)(40,47,113,80)(41,69,114,79)(42,68,115,78)(43,67,93,77)(44,66,94,76)(45,65,95,75)(46,64,96,74)>;
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), (1,44,66,76,94)(2,45,67,77,95)(3,46,68,78,96)(4,24,69,79,97)(5,25,47,80,98)(6,26,48,81,99)(7,27,49,82,100)(8,28,50,83,101)(9,29,51,84,102)(10,30,52,85,103)(11,31,53,86,104)(12,32,54,87,105)(13,33,55,88,106)(14,34,56,89,107)(15,35,57,90,108)(16,36,58,91,109)(17,37,59,92,110)(18,38,60,70,111)(19,39,61,71,112)(20,40,62,72,113)(21,41,63,73,114)(22,42,64,74,115)(23,43,65,75,93), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(24,63,97,73)(25,62,98,72)(26,61,99,71)(27,60,100,70)(28,59,101,92)(29,58,102,91)(30,57,103,90)(31,56,104,89)(32,55,105,88)(33,54,106,87)(34,53,107,86)(35,52,108,85)(36,51,109,84)(37,50,110,83)(38,49,111,82)(39,48,112,81)(40,47,113,80)(41,69,114,79)(42,68,115,78)(43,67,93,77)(44,66,94,76)(45,65,95,75)(46,64,96,74) );
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)], [(1,44,66,76,94),(2,45,67,77,95),(3,46,68,78,96),(4,24,69,79,97),(5,25,47,80,98),(6,26,48,81,99),(7,27,49,82,100),(8,28,50,83,101),(9,29,51,84,102),(10,30,52,85,103),(11,31,53,86,104),(12,32,54,87,105),(13,33,55,88,106),(14,34,56,89,107),(15,35,57,90,108),(16,36,58,91,109),(17,37,59,92,110),(18,38,60,70,111),(19,39,61,71,112),(20,40,62,72,113),(21,41,63,73,114),(22,42,64,74,115),(23,43,65,75,93)], [(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(24,63,97,73),(25,62,98,72),(26,61,99,71),(27,60,100,70),(28,59,101,92),(29,58,102,91),(30,57,103,90),(31,56,104,89),(32,55,105,88),(33,54,106,87),(34,53,107,86),(35,52,108,85),(36,51,109,84),(37,50,110,83),(38,49,111,82),(39,48,112,81),(40,47,113,80),(41,69,114,79),(42,68,115,78),(43,67,93,77),(44,66,94,76),(45,65,95,75),(46,64,96,74)]])
49 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 23A | ··· | 23K | 46A | ··· | 46K | 115A | ··· | 115V |
| order | 1 | 2 | 4 | 4 | 5 | 23 | ··· | 23 | 46 | ··· | 46 | 115 | ··· | 115 |
| size | 1 | 5 | 115 | 115 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D23 | Dic23 | F5 | C23⋊F5 |
| kernel | C23⋊F5 | D5×C23 | C115 | D5 | C5 | C23 | C1 |
| # reps | 1 | 1 | 2 | 11 | 11 | 1 | 22 |
Matrix representation of C23⋊F5 ►in GL4(𝔽461) generated by
| 0 | 1 | 0 | 0 |
| 460 | 363 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 460 | 363 |
| 96 | 383 | 460 | 0 |
| 78 | 364 | 0 | 460 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 363 | 460 | 0 | 0 |
| 365 | 78 | 365 | 78 |
| 266 | 96 | 266 | 96 |
G:=sub<GL(4,GF(461))| [0,460,0,0,1,363,0,0,0,0,0,460,0,0,1,363],[96,78,1,0,383,364,0,1,460,0,0,0,0,460,0,0],[1,363,365,266,0,460,78,96,0,0,365,266,0,0,78,96] >;
C23⋊F5 in GAP, Magma, Sage, TeX
C_{23}\rtimes F_5 % in TeX
G:=Group("C23:F5"); // GroupNames label
G:=SmallGroup(460,6);
// by ID
G=gap.SmallGroup(460,6);
# by ID
G:=PCGroup([4,-2,-2,-5,-23,8,146,102,7043]);
// Polycyclic
G:=Group<a,b,c|a^23=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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