metabelian, supersoluble, monomial, A-group
Aliases: C25⋊1F5, C52.4F5, (C5×C25)⋊5C4, C5⋊1(C25⋊C4), C25⋊D5.1C2, C5.(C5⋊F5), SmallGroup(500,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C25 — C25⋊F5 |
Generators and relations for C25⋊F5
G = < a,b,c | a25=b5=c4=1, ab=ba, cac-1=a18, cbc-1=b3 >
Smallest permutation representation of C25⋊F5
►On 125 points
Generators in S125
(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)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125)
(1 84 57 28 106)(2 85 58 29 107)(3 86 59 30 108)(4 87 60 31 109)(5 88 61 32 110)(6 89 62 33 111)(7 90 63 34 112)(8 91 64 35 113)(9 92 65 36 114)(10 93 66 37 115)(11 94 67 38 116)(12 95 68 39 117)(13 96 69 40 118)(14 97 70 41 119)(15 98 71 42 120)(16 99 72 43 121)(17 100 73 44 122)(18 76 74 45 123)(19 77 75 46 124)(20 78 51 47 125)(21 79 52 48 101)(22 80 53 49 102)(23 81 54 50 103)(24 82 55 26 104)(25 83 56 27 105)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 95 59 120)(27 77 58 113)(28 84 57 106)(29 91 56 124)(30 98 55 117)(31 80 54 110)(32 87 53 103)(33 94 52 121)(34 76 51 114)(35 83 75 107)(36 90 74 125)(37 97 73 118)(38 79 72 111)(39 86 71 104)(40 93 70 122)(41 100 69 115)(42 82 68 108)(43 89 67 101)(44 96 66 119)(45 78 65 112)(46 85 64 105)(47 92 63 123)(48 99 62 116)(49 81 61 109)(50 88 60 102)
G:=sub<Sym(125)| (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125), (1,84,57,28,106)(2,85,58,29,107)(3,86,59,30,108)(4,87,60,31,109)(5,88,61,32,110)(6,89,62,33,111)(7,90,63,34,112)(8,91,64,35,113)(9,92,65,36,114)(10,93,66,37,115)(11,94,67,38,116)(12,95,68,39,117)(13,96,69,40,118)(14,97,70,41,119)(15,98,71,42,120)(16,99,72,43,121)(17,100,73,44,122)(18,76,74,45,123)(19,77,75,46,124)(20,78,51,47,125)(21,79,52,48,101)(22,80,53,49,102)(23,81,54,50,103)(24,82,55,26,104)(25,83,56,27,105), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,95,59,120)(27,77,58,113)(28,84,57,106)(29,91,56,124)(30,98,55,117)(31,80,54,110)(32,87,53,103)(33,94,52,121)(34,76,51,114)(35,83,75,107)(36,90,74,125)(37,97,73,118)(38,79,72,111)(39,86,71,104)(40,93,70,122)(41,100,69,115)(42,82,68,108)(43,89,67,101)(44,96,66,119)(45,78,65,112)(46,85,64,105)(47,92,63,123)(48,99,62,116)(49,81,61,109)(50,88,60,102)>;
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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125), (1,84,57,28,106)(2,85,58,29,107)(3,86,59,30,108)(4,87,60,31,109)(5,88,61,32,110)(6,89,62,33,111)(7,90,63,34,112)(8,91,64,35,113)(9,92,65,36,114)(10,93,66,37,115)(11,94,67,38,116)(12,95,68,39,117)(13,96,69,40,118)(14,97,70,41,119)(15,98,71,42,120)(16,99,72,43,121)(17,100,73,44,122)(18,76,74,45,123)(19,77,75,46,124)(20,78,51,47,125)(21,79,52,48,101)(22,80,53,49,102)(23,81,54,50,103)(24,82,55,26,104)(25,83,56,27,105), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,95,59,120)(27,77,58,113)(28,84,57,106)(29,91,56,124)(30,98,55,117)(31,80,54,110)(32,87,53,103)(33,94,52,121)(34,76,51,114)(35,83,75,107)(36,90,74,125)(37,97,73,118)(38,79,72,111)(39,86,71,104)(40,93,70,122)(41,100,69,115)(42,82,68,108)(43,89,67,101)(44,96,66,119)(45,78,65,112)(46,85,64,105)(47,92,63,123)(48,99,62,116)(49,81,61,109)(50,88,60,102) );
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),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)], [(1,84,57,28,106),(2,85,58,29,107),(3,86,59,30,108),(4,87,60,31,109),(5,88,61,32,110),(6,89,62,33,111),(7,90,63,34,112),(8,91,64,35,113),(9,92,65,36,114),(10,93,66,37,115),(11,94,67,38,116),(12,95,68,39,117),(13,96,69,40,118),(14,97,70,41,119),(15,98,71,42,120),(16,99,72,43,121),(17,100,73,44,122),(18,76,74,45,123),(19,77,75,46,124),(20,78,51,47,125),(21,79,52,48,101),(22,80,53,49,102),(23,81,54,50,103),(24,82,55,26,104),(25,83,56,27,105)], [(2,8,25,19),(3,15,24,12),(4,22,23,5),(6,11,21,16),(7,18,20,9),(10,14,17,13),(26,95,59,120),(27,77,58,113),(28,84,57,106),(29,91,56,124),(30,98,55,117),(31,80,54,110),(32,87,53,103),(33,94,52,121),(34,76,51,114),(35,83,75,107),(36,90,74,125),(37,97,73,118),(38,79,72,111),(39,86,71,104),(40,93,70,122),(41,100,69,115),(42,82,68,108),(43,89,67,101),(44,96,66,119),(45,78,65,112),(46,85,64,105),(47,92,63,123),(48,99,62,116),(49,81,61,109),(50,88,60,102)]])
35 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5F | 25A | ··· | 25Y |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 25 | ··· | 25 |
| size | 1 | 125 | 125 | 125 | 4 | ··· | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | |
| image | C1 | C2 | C4 | F5 | F5 | C25⋊C4 |
| kernel | C25⋊F5 | C25⋊D5 | C5×C25 | C25 | C52 | C5 |
| # reps | 1 | 1 | 2 | 5 | 1 | 25 |
Matrix representation of C25⋊F5 ►in GL8(𝔽101)
| 100 | 100 | 100 | 100 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 78 | 4 | 70 | 32 |
| 0 | 0 | 0 | 0 | 69 | 46 | 73 | 38 |
| 0 | 0 | 0 | 0 | 63 | 31 | 8 | 35 |
| 0 | 0 | 0 | 0 | 66 | 28 | 97 | 74 |
| 100 | 100 | 100 | 100 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 100 | 100 | 0 | 0 | 0 | 0 |
| 100 | 100 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 100 | 100 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 100 | 100 | 100 |
G:=sub<GL(8,GF(101))| [100,1,0,0,0,0,0,0,100,0,1,0,0,0,0,0,100,0,0,1,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,78,69,63,66,0,0,0,0,4,46,31,28,0,0,0,0,70,73,8,97,0,0,0,0,32,38,35,74],[100,1,0,0,0,0,0,0,100,0,1,0,0,0,0,0,100,0,0,1,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,100,1,0,0,0,0,0,0,100,1,100,0,0,0,0,100,0,1,100,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,1,0,0,100,0,0,0,0,0,0,1,100,0,0,0,0,0,0,0,100,0,0,0,0,0,1,0,100] >;
C25⋊F5 in GAP, Magma, Sage, TeX
C_{25}\rtimes F_5 % in TeX
G:=Group("C25:F5"); // GroupNames label
G:=SmallGroup(500,22);
// by ID
G=gap.SmallGroup(500,22);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,1622,3127,387,803,808,5004,5009]);
// Polycyclic
G:=Group<a,b,c|a^25=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^18,c*b*c^-1=b^3>;
// generators/relations
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