metabelian, supersoluble, monomial, A-group
Aliases: C53⋊3C2, C52⋊4D5, C5⋊(C5⋊D5), SmallGroup(250,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C53⋊C2 |
Generators and relations for C53⋊C2
G = < a,b,c,d | a5=b5=c5=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1120 in 128 conjugacy classes, 65 normal (3 characteristic)
C1, C2, C5, D5, C52, C5⋊D5, C53, C53⋊C2
Quotients: C1, C2, D5, C5⋊D5, C53⋊C2
►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)
(1 108 89 14 64)(2 109 90 15 65)(3 110 86 11 61)(4 106 87 12 62)(5 107 88 13 63)(6 56 31 39 81)(7 57 32 40 82)(8 58 33 36 83)(9 59 34 37 84)(10 60 35 38 85)(16 66 41 115 91)(17 67 42 111 92)(18 68 43 112 93)(19 69 44 113 94)(20 70 45 114 95)(21 71 46 120 96)(22 72 47 116 97)(23 73 48 117 98)(24 74 49 118 99)(25 75 50 119 100)(26 76 51 125 101)(27 77 52 121 102)(28 78 53 122 103)(29 79 54 123 104)(30 80 55 124 105)
(1 124 49 34 44)(2 125 50 35 45)(3 121 46 31 41)(4 122 47 32 42)(5 123 48 33 43)(6 16 11 77 21)(7 17 12 78 22)(8 18 13 79 23)(9 19 14 80 24)(10 20 15 76 25)(26 100 85 95 90)(27 96 81 91 86)(28 97 82 92 87)(29 98 83 93 88)(30 99 84 94 89)(36 112 107 104 117)(37 113 108 105 118)(38 114 109 101 119)(39 115 110 102 120)(40 111 106 103 116)(51 75 60 70 65)(52 71 56 66 61)(53 72 57 67 62)(54 73 58 68 63)(55 74 59 69 64)
(2 5)(3 4)(6 97)(7 96)(8 100)(9 99)(10 98)(11 87)(12 86)(13 90)(14 89)(15 88)(16 28)(17 27)(18 26)(19 30)(20 29)(21 82)(22 81)(23 85)(24 84)(25 83)(31 47)(32 46)(33 50)(34 49)(35 48)(36 75)(37 74)(38 73)(39 72)(40 71)(41 122)(42 121)(43 125)(44 124)(45 123)(51 112)(52 111)(53 115)(54 114)(55 113)(56 116)(57 120)(58 119)(59 118)(60 117)(61 106)(62 110)(63 109)(64 108)(65 107)(66 103)(67 102)(68 101)(69 105)(70 104)(76 93)(77 92)(78 91)(79 95)(80 94)
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), (1,108,89,14,64)(2,109,90,15,65)(3,110,86,11,61)(4,106,87,12,62)(5,107,88,13,63)(6,56,31,39,81)(7,57,32,40,82)(8,58,33,36,83)(9,59,34,37,84)(10,60,35,38,85)(16,66,41,115,91)(17,67,42,111,92)(18,68,43,112,93)(19,69,44,113,94)(20,70,45,114,95)(21,71,46,120,96)(22,72,47,116,97)(23,73,48,117,98)(24,74,49,118,99)(25,75,50,119,100)(26,76,51,125,101)(27,77,52,121,102)(28,78,53,122,103)(29,79,54,123,104)(30,80,55,124,105), (1,124,49,34,44)(2,125,50,35,45)(3,121,46,31,41)(4,122,47,32,42)(5,123,48,33,43)(6,16,11,77,21)(7,17,12,78,22)(8,18,13,79,23)(9,19,14,80,24)(10,20,15,76,25)(26,100,85,95,90)(27,96,81,91,86)(28,97,82,92,87)(29,98,83,93,88)(30,99,84,94,89)(36,112,107,104,117)(37,113,108,105,118)(38,114,109,101,119)(39,115,110,102,120)(40,111,106,103,116)(51,75,60,70,65)(52,71,56,66,61)(53,72,57,67,62)(54,73,58,68,63)(55,74,59,69,64), (2,5)(3,4)(6,97)(7,96)(8,100)(9,99)(10,98)(11,87)(12,86)(13,90)(14,89)(15,88)(16,28)(17,27)(18,26)(19,30)(20,29)(21,82)(22,81)(23,85)(24,84)(25,83)(31,47)(32,46)(33,50)(34,49)(35,48)(36,75)(37,74)(38,73)(39,72)(40,71)(41,122)(42,121)(43,125)(44,124)(45,123)(51,112)(52,111)(53,115)(54,114)(55,113)(56,116)(57,120)(58,119)(59,118)(60,117)(61,106)(62,110)(63,109)(64,108)(65,107)(66,103)(67,102)(68,101)(69,105)(70,104)(76,93)(77,92)(78,91)(79,95)(80,94)>;
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), (1,108,89,14,64)(2,109,90,15,65)(3,110,86,11,61)(4,106,87,12,62)(5,107,88,13,63)(6,56,31,39,81)(7,57,32,40,82)(8,58,33,36,83)(9,59,34,37,84)(10,60,35,38,85)(16,66,41,115,91)(17,67,42,111,92)(18,68,43,112,93)(19,69,44,113,94)(20,70,45,114,95)(21,71,46,120,96)(22,72,47,116,97)(23,73,48,117,98)(24,74,49,118,99)(25,75,50,119,100)(26,76,51,125,101)(27,77,52,121,102)(28,78,53,122,103)(29,79,54,123,104)(30,80,55,124,105), (1,124,49,34,44)(2,125,50,35,45)(3,121,46,31,41)(4,122,47,32,42)(5,123,48,33,43)(6,16,11,77,21)(7,17,12,78,22)(8,18,13,79,23)(9,19,14,80,24)(10,20,15,76,25)(26,100,85,95,90)(27,96,81,91,86)(28,97,82,92,87)(29,98,83,93,88)(30,99,84,94,89)(36,112,107,104,117)(37,113,108,105,118)(38,114,109,101,119)(39,115,110,102,120)(40,111,106,103,116)(51,75,60,70,65)(52,71,56,66,61)(53,72,57,67,62)(54,73,58,68,63)(55,74,59,69,64), (2,5)(3,4)(6,97)(7,96)(8,100)(9,99)(10,98)(11,87)(12,86)(13,90)(14,89)(15,88)(16,28)(17,27)(18,26)(19,30)(20,29)(21,82)(22,81)(23,85)(24,84)(25,83)(31,47)(32,46)(33,50)(34,49)(35,48)(36,75)(37,74)(38,73)(39,72)(40,71)(41,122)(42,121)(43,125)(44,124)(45,123)(51,112)(52,111)(53,115)(54,114)(55,113)(56,116)(57,120)(58,119)(59,118)(60,117)(61,106)(62,110)(63,109)(64,108)(65,107)(66,103)(67,102)(68,101)(69,105)(70,104)(76,93)(77,92)(78,91)(79,95)(80,94) );
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)], [(1,108,89,14,64),(2,109,90,15,65),(3,110,86,11,61),(4,106,87,12,62),(5,107,88,13,63),(6,56,31,39,81),(7,57,32,40,82),(8,58,33,36,83),(9,59,34,37,84),(10,60,35,38,85),(16,66,41,115,91),(17,67,42,111,92),(18,68,43,112,93),(19,69,44,113,94),(20,70,45,114,95),(21,71,46,120,96),(22,72,47,116,97),(23,73,48,117,98),(24,74,49,118,99),(25,75,50,119,100),(26,76,51,125,101),(27,77,52,121,102),(28,78,53,122,103),(29,79,54,123,104),(30,80,55,124,105)], [(1,124,49,34,44),(2,125,50,35,45),(3,121,46,31,41),(4,122,47,32,42),(5,123,48,33,43),(6,16,11,77,21),(7,17,12,78,22),(8,18,13,79,23),(9,19,14,80,24),(10,20,15,76,25),(26,100,85,95,90),(27,96,81,91,86),(28,97,82,92,87),(29,98,83,93,88),(30,99,84,94,89),(36,112,107,104,117),(37,113,108,105,118),(38,114,109,101,119),(39,115,110,102,120),(40,111,106,103,116),(51,75,60,70,65),(52,71,56,66,61),(53,72,57,67,62),(54,73,58,68,63),(55,74,59,69,64)], [(2,5),(3,4),(6,97),(7,96),(8,100),(9,99),(10,98),(11,87),(12,86),(13,90),(14,89),(15,88),(16,28),(17,27),(18,26),(19,30),(20,29),(21,82),(22,81),(23,85),(24,84),(25,83),(31,47),(32,46),(33,50),(34,49),(35,48),(36,75),(37,74),(38,73),(39,72),(40,71),(41,122),(42,121),(43,125),(44,124),(45,123),(51,112),(52,111),(53,115),(54,114),(55,113),(56,116),(57,120),(58,119),(59,118),(60,117),(61,106),(62,110),(63,109),(64,108),(65,107),(66,103),(67,102),(68,101),(69,105),(70,104),(76,93),(77,92),(78,91),(79,95),(80,94)]])
C53⋊C2 is a maximal subgroup of
C53⋊8C4 C53⋊9C4 D5×C5⋊D5
C53⋊C2 is a maximal quotient of C53⋊12C4
64 conjugacy classes
| class | 1 | 2 | 5A | ··· | 5BJ |
| order | 1 | 2 | 5 | ··· | 5 |
| size | 1 | 125 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D5 |
| kernel | C53⋊C2 | C53 | C52 |
| # reps | 1 | 1 | 62 |
Matrix representation of C53⋊C2 ►in GL6(𝔽11)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 4 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 10 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 10 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 7 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 1 | 8 |
G:=sub<GL(6,GF(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,0,1,4,0,0,0,0,0,0,8,3,0,0,0,0,8,10],[0,10,0,0,0,0,1,7,0,0,0,0,0,0,0,4,0,0,0,0,8,7,0,0,0,0,0,0,0,10,0,0,0,0,1,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,0,1,4,0,0,0,0,0,0,0,10,0,0,0,0,1,3],[1,7,0,0,0,0,0,10,0,0,0,0,0,0,7,7,0,0,0,0,1,4,0,0,0,0,0,0,3,1,0,0,0,0,3,8] >;
C53⋊C2 in GAP, Magma, Sage, TeX
C_5^3\rtimes C_2
% in TeX
G:=Group("C5^3:C2"); // GroupNames label
G:=SmallGroup(250,14);
// by ID
G=gap.SmallGroup(250,14);
# by ID
G:=PCGroup([4,-2,-5,-5,-5,65,482,3203]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations