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G = C53order 125 = 53

Elementary abelian group of type [5,5,5]

direct product, p-group, elementary abelian, monomial

Aliases: C53, SmallGroup(125,5)

Series: Derived Chief Lower central Upper central Jennings

C1 — C53
C1C5C52 — C53
C1 — C53
C1 — C53
C1 — C53

Generators and relations for C53
 G = < a,b,c | a5=b5=c5=1, ab=ba, ac=ca, bc=cb >

Subgroups: 64, all normal (2 characteristic)
C1, C5 [×31], C52 [×31], C53
Quotients: C1, C5 [×31], C52 [×31], C53

Smallest permutation representation of C53
Regular action 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 7 84 59 34)(2 8 85 60 35)(3 9 81 56 31)(4 10 82 57 32)(5 6 83 58 33)(11 110 86 61 36)(12 106 87 62 37)(13 107 88 63 38)(14 108 89 64 39)(15 109 90 65 40)(16 115 91 66 41)(17 111 92 67 42)(18 112 93 68 43)(19 113 94 69 44)(20 114 95 70 45)(21 120 96 71 46)(22 116 97 72 47)(23 117 98 73 48)(24 118 99 74 49)(25 119 100 75 50)(26 125 101 76 51)(27 121 102 77 52)(28 122 103 78 53)(29 123 104 79 54)(30 124 105 80 55)
(1 124 24 19 14)(2 125 25 20 15)(3 121 21 16 11)(4 122 22 17 12)(5 123 23 18 13)(6 104 117 112 107)(7 105 118 113 108)(8 101 119 114 109)(9 102 120 115 110)(10 103 116 111 106)(26 50 45 40 35)(27 46 41 36 31)(28 47 42 37 32)(29 48 43 38 33)(30 49 44 39 34)(51 75 70 65 60)(52 71 66 61 56)(53 72 67 62 57)(54 73 68 63 58)(55 74 69 64 59)(76 100 95 90 85)(77 96 91 86 81)(78 97 92 87 82)(79 98 93 88 83)(80 99 94 89 84)

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), (1,7,84,59,34)(2,8,85,60,35)(3,9,81,56,31)(4,10,82,57,32)(5,6,83,58,33)(11,110,86,61,36)(12,106,87,62,37)(13,107,88,63,38)(14,108,89,64,39)(15,109,90,65,40)(16,115,91,66,41)(17,111,92,67,42)(18,112,93,68,43)(19,113,94,69,44)(20,114,95,70,45)(21,120,96,71,46)(22,116,97,72,47)(23,117,98,73,48)(24,118,99,74,49)(25,119,100,75,50)(26,125,101,76,51)(27,121,102,77,52)(28,122,103,78,53)(29,123,104,79,54)(30,124,105,80,55), (1,124,24,19,14)(2,125,25,20,15)(3,121,21,16,11)(4,122,22,17,12)(5,123,23,18,13)(6,104,117,112,107)(7,105,118,113,108)(8,101,119,114,109)(9,102,120,115,110)(10,103,116,111,106)(26,50,45,40,35)(27,46,41,36,31)(28,47,42,37,32)(29,48,43,38,33)(30,49,44,39,34)(51,75,70,65,60)(52,71,66,61,56)(53,72,67,62,57)(54,73,68,63,58)(55,74,69,64,59)(76,100,95,90,85)(77,96,91,86,81)(78,97,92,87,82)(79,98,93,88,83)(80,99,94,89,84)>;

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), (1,7,84,59,34)(2,8,85,60,35)(3,9,81,56,31)(4,10,82,57,32)(5,6,83,58,33)(11,110,86,61,36)(12,106,87,62,37)(13,107,88,63,38)(14,108,89,64,39)(15,109,90,65,40)(16,115,91,66,41)(17,111,92,67,42)(18,112,93,68,43)(19,113,94,69,44)(20,114,95,70,45)(21,120,96,71,46)(22,116,97,72,47)(23,117,98,73,48)(24,118,99,74,49)(25,119,100,75,50)(26,125,101,76,51)(27,121,102,77,52)(28,122,103,78,53)(29,123,104,79,54)(30,124,105,80,55), (1,124,24,19,14)(2,125,25,20,15)(3,121,21,16,11)(4,122,22,17,12)(5,123,23,18,13)(6,104,117,112,107)(7,105,118,113,108)(8,101,119,114,109)(9,102,120,115,110)(10,103,116,111,106)(26,50,45,40,35)(27,46,41,36,31)(28,47,42,37,32)(29,48,43,38,33)(30,49,44,39,34)(51,75,70,65,60)(52,71,66,61,56)(53,72,67,62,57)(54,73,68,63,58)(55,74,69,64,59)(76,100,95,90,85)(77,96,91,86,81)(78,97,92,87,82)(79,98,93,88,83)(80,99,94,89,84) );

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)], [(1,7,84,59,34),(2,8,85,60,35),(3,9,81,56,31),(4,10,82,57,32),(5,6,83,58,33),(11,110,86,61,36),(12,106,87,62,37),(13,107,88,63,38),(14,108,89,64,39),(15,109,90,65,40),(16,115,91,66,41),(17,111,92,67,42),(18,112,93,68,43),(19,113,94,69,44),(20,114,95,70,45),(21,120,96,71,46),(22,116,97,72,47),(23,117,98,73,48),(24,118,99,74,49),(25,119,100,75,50),(26,125,101,76,51),(27,121,102,77,52),(28,122,103,78,53),(29,123,104,79,54),(30,124,105,80,55)], [(1,124,24,19,14),(2,125,25,20,15),(3,121,21,16,11),(4,122,22,17,12),(5,123,23,18,13),(6,104,117,112,107),(7,105,118,113,108),(8,101,119,114,109),(9,102,120,115,110),(10,103,116,111,106),(26,50,45,40,35),(27,46,41,36,31),(28,47,42,37,32),(29,48,43,38,33),(30,49,44,39,34),(51,75,70,65,60),(52,71,66,61,56),(53,72,67,62,57),(54,73,68,63,58),(55,74,69,64,59),(76,100,95,90,85),(77,96,91,86,81),(78,97,92,87,82),(79,98,93,88,83),(80,99,94,89,84)])

C53 is a maximal subgroup of   C53⋊C2

125 conjugacy classes

class 1 5A···5DT
order15···5
size11···1

125 irreducible representations

dim11
type+
imageC1C5
kernelC53C52
# reps1124

Matrix representation of C53 in GL3(𝔽11) generated by

400
050
001
,
400
030
004
,
500
090
001
G:=sub<GL(3,GF(11))| [4,0,0,0,5,0,0,0,1],[4,0,0,0,3,0,0,0,4],[5,0,0,0,9,0,0,0,1] >;

C53 in GAP, Magma, Sage, TeX

C_5^3
% in TeX

G:=Group("C5^3");
// GroupNames label

G:=SmallGroup(125,5);
// by ID

G=gap.SmallGroup(125,5);
# by ID

G:=PCGroup([3,-5,5,5]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^5=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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