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G = C5×C15order 75 = 3·52

Abelian group of type [5,15]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C15, SmallGroup(75,3)

Series: Derived Chief Lower central Upper central

C1 — C5×C15
C1C5C52 — C5×C15
C1 — C5×C15
C1 — C5×C15

Generators and relations for C5×C15
 G = < a,b | a5=b15=1, ab=ba >


Smallest permutation representation of C5×C15
Regular action on 75 points
Generators in S75
(1 48 75 23 43)(2 49 61 24 44)(3 50 62 25 45)(4 51 63 26 31)(5 52 64 27 32)(6 53 65 28 33)(7 54 66 29 34)(8 55 67 30 35)(9 56 68 16 36)(10 57 69 17 37)(11 58 70 18 38)(12 59 71 19 39)(13 60 72 20 40)(14 46 73 21 41)(15 47 74 22 42)
(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)

G:=sub<Sym(75)| (1,48,75,23,43)(2,49,61,24,44)(3,50,62,25,45)(4,51,63,26,31)(5,52,64,27,32)(6,53,65,28,33)(7,54,66,29,34)(8,55,67,30,35)(9,56,68,16,36)(10,57,69,17,37)(11,58,70,18,38)(12,59,71,19,39)(13,60,72,20,40)(14,46,73,21,41)(15,47,74,22,42), (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)>;

G:=Group( (1,48,75,23,43)(2,49,61,24,44)(3,50,62,25,45)(4,51,63,26,31)(5,52,64,27,32)(6,53,65,28,33)(7,54,66,29,34)(8,55,67,30,35)(9,56,68,16,36)(10,57,69,17,37)(11,58,70,18,38)(12,59,71,19,39)(13,60,72,20,40)(14,46,73,21,41)(15,47,74,22,42), (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) );

G=PermutationGroup([(1,48,75,23,43),(2,49,61,24,44),(3,50,62,25,45),(4,51,63,26,31),(5,52,64,27,32),(6,53,65,28,33),(7,54,66,29,34),(8,55,67,30,35),(9,56,68,16,36),(10,57,69,17,37),(11,58,70,18,38),(12,59,71,19,39),(13,60,72,20,40),(14,46,73,21,41),(15,47,74,22,42)], [(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)])

C5×C15 is a maximal subgroup of   C5⋊D15  C52⋊C9

75 conjugacy classes

class 1 3A3B5A···5X15A···15AV
order1335···515···15
size1111···11···1

75 irreducible representations

dim1111
type+
imageC1C3C5C15
kernelC5×C15C52C15C5
# reps122448

Matrix representation of C5×C15 in GL2(𝔽31) generated by

160
04
,
20
020
G:=sub<GL(2,GF(31))| [16,0,0,4],[2,0,0,20] >;

C5×C15 in GAP, Magma, Sage, TeX

C_5\times C_{15}
% in TeX

G:=Group("C5xC15");
// GroupNames label

G:=SmallGroup(75,3);
// by ID

G=gap.SmallGroup(75,3);
# by ID

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

G:=Group<a,b|a^5=b^15=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C15 in TeX

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