metabelian, supersoluble, monomial, A-group
Aliases: C25⋊D5, C5⋊D25, C52.3D5, (C5×C25)⋊3C2, C5.(C5⋊D5), SmallGroup(250,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C25 — C25⋊D5 |
Generators and relations for C25⋊D5
G = < a,b,c | a25=b5=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C25⋊D5
►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 52 43 91 118)(2 53 44 92 119)(3 54 45 93 120)(4 55 46 94 121)(5 56 47 95 122)(6 57 48 96 123)(7 58 49 97 124)(8 59 50 98 125)(9 60 26 99 101)(10 61 27 100 102)(11 62 28 76 103)(12 63 29 77 104)(13 64 30 78 105)(14 65 31 79 106)(15 66 32 80 107)(16 67 33 81 108)(17 68 34 82 109)(18 69 35 83 110)(19 70 36 84 111)(20 71 37 85 112)(21 72 38 86 113)(22 73 39 87 114)(23 74 40 88 115)(24 75 41 89 116)(25 51 42 90 117)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 108)(12 107)(13 106)(14 105)(15 104)(16 103)(17 102)(18 101)(19 125)(20 124)(21 123)(22 122)(23 121)(24 120)(25 119)(26 35)(27 34)(28 33)(29 32)(30 31)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 100)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)
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), (1,52,43,91,118)(2,53,44,92,119)(3,54,45,93,120)(4,55,46,94,121)(5,56,47,95,122)(6,57,48,96,123)(7,58,49,97,124)(8,59,50,98,125)(9,60,26,99,101)(10,61,27,100,102)(11,62,28,76,103)(12,63,29,77,104)(13,64,30,78,105)(14,65,31,79,106)(15,66,32,80,107)(16,67,33,81,108)(17,68,34,82,109)(18,69,35,83,110)(19,70,36,84,111)(20,71,37,85,112)(21,72,38,86,113)(22,73,39,87,114)(23,74,40,88,115)(24,75,41,89,116)(25,51,42,90,117), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,104)(16,103)(17,102)(18,101)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,35)(27,34)(28,33)(29,32)(30,31)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)>;
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), (1,52,43,91,118)(2,53,44,92,119)(3,54,45,93,120)(4,55,46,94,121)(5,56,47,95,122)(6,57,48,96,123)(7,58,49,97,124)(8,59,50,98,125)(9,60,26,99,101)(10,61,27,100,102)(11,62,28,76,103)(12,63,29,77,104)(13,64,30,78,105)(14,65,31,79,106)(15,66,32,80,107)(16,67,33,81,108)(17,68,34,82,109)(18,69,35,83,110)(19,70,36,84,111)(20,71,37,85,112)(21,72,38,86,113)(22,73,39,87,114)(23,74,40,88,115)(24,75,41,89,116)(25,51,42,90,117), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,104)(16,103)(17,102)(18,101)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,35)(27,34)(28,33)(29,32)(30,31)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93) );
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)], [(1,52,43,91,118),(2,53,44,92,119),(3,54,45,93,120),(4,55,46,94,121),(5,56,47,95,122),(6,57,48,96,123),(7,58,49,97,124),(8,59,50,98,125),(9,60,26,99,101),(10,61,27,100,102),(11,62,28,76,103),(12,63,29,77,104),(13,64,30,78,105),(14,65,31,79,106),(15,66,32,80,107),(16,67,33,81,108),(17,68,34,82,109),(18,69,35,83,110),(19,70,36,84,111),(20,71,37,85,112),(21,72,38,86,113),(22,73,39,87,114),(23,74,40,88,115),(24,75,41,89,116),(25,51,42,90,117)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,112),(8,111),(9,110),(10,109),(11,108),(12,107),(13,106),(14,105),(15,104),(16,103),(17,102),(18,101),(19,125),(20,124),(21,123),(22,122),(23,121),(24,120),(25,119),(26,35),(27,34),(28,33),(29,32),(30,31),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(51,92),(52,91),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,100),(69,99),(70,98),(71,97),(72,96),(73,95),(74,94),(75,93)]])
C25⋊D5 is a maximal subgroup of
C25⋊F5 C25⋊2F5 D5×D25
C25⋊D5 is a maximal quotient of C50.D5
64 conjugacy classes
| class | 1 | 2 | 5A | ··· | 5L | 25A | ··· | 25AX |
| order | 1 | 2 | 5 | ··· | 5 | 25 | ··· | 25 |
| size | 1 | 125 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D5 | D5 | D25 |
| kernel | C25⋊D5 | C5×C25 | C25 | C52 | C5 |
| # reps | 1 | 1 | 10 | 2 | 50 |
Matrix representation of C25⋊D5 ►in GL4(𝔽101) generated by
| 32 | 11 | 0 | 0 |
| 90 | 72 | 0 | 0 |
| 0 | 0 | 43 | 93 |
| 0 | 0 | 8 | 69 |
| 22 | 100 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 100 | 22 |
| 22 | 100 | 0 | 0 |
| 79 | 79 | 0 | 0 |
| 0 | 0 | 0 | 100 |
| 0 | 0 | 100 | 0 |
G:=sub<GL(4,GF(101))| [32,90,0,0,11,72,0,0,0,0,43,8,0,0,93,69],[22,1,0,0,100,0,0,0,0,0,0,100,0,0,1,22],[22,79,0,0,100,79,0,0,0,0,0,100,0,0,100,0] >;
C25⋊D5 in GAP, Magma, Sage, TeX
C_{25}\rtimes D_5 % in TeX
G:=Group("C25:D5"); // GroupNames label
G:=SmallGroup(250,7);
// by ID
G=gap.SmallGroup(250,7);
# by ID
G:=PCGroup([4,-2,-5,-5,-5,465,805,482,3203]);
// Polycyclic
G:=Group<a,b,c|a^25=b^5=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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