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G = C5×C13⋊C4order 260 = 22·5·13

Direct product of C5 and C13⋊C4

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×C13⋊C4, C13⋊C20, C655C4, D13.C10, (C5×D13).2C2, SmallGroup(260,5)

Series: Derived Chief Lower central Upper central

C1C13 — C5×C13⋊C4
C1C13D13C5×D13 — C5×C13⋊C4
C13 — C5×C13⋊C4
C1C5

Generators and relations for C5×C13⋊C4
 G = < a,b,c | a5=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

13C2
13C4
13C10
13C20

Smallest permutation representation of C5×C13⋊C4
On 65 points
Generators in S65
(1 53 40 27 14)(2 54 41 28 15)(3 55 42 29 16)(4 56 43 30 17)(5 57 44 31 18)(6 58 45 32 19)(7 59 46 33 20)(8 60 47 34 21)(9 61 48 35 22)(10 62 49 36 23)(11 63 50 37 24)(12 64 51 38 25)(13 65 52 39 26)
(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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)

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

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

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

35 conjugacy classes

class 1  2 4A4B5A5B5C5D10A10B10C10D13A13B13C20A···20H65A···65L
order124455551010101013131320···2065···65
size113131311111313131344413···134···4

35 irreducible representations

dim11111144
type+++
imageC1C2C4C5C10C20C13⋊C4C5×C13⋊C4
kernelC5×C13⋊C4C5×D13C65C13⋊C4D13C13C5C1
# reps112448312

Matrix representation of C5×C13⋊C4 in GL4(𝔽521) generated by

516000
051600
005160
000516
,
27368152520
182190305495
54216304519
2451126126
,
11537811838
358234285101
48336485216
134420252208
G:=sub<GL(4,GF(521))| [516,0,0,0,0,516,0,0,0,0,516,0,0,0,0,516],[27,182,54,245,368,190,216,1,152,305,304,126,520,495,519,126],[115,358,483,134,378,234,36,420,118,285,485,252,38,101,216,208] >;

C5×C13⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_{13}\rtimes C_4
% in TeX

G:=Group("C5xC13:C4");
// GroupNames label

G:=SmallGroup(260,5);
// by ID

G=gap.SmallGroup(260,5);
# by ID

G:=PCGroup([4,-2,-5,-2,-13,40,2563,395]);
// Polycyclic

G:=Group<a,b,c|a^5=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C5×C13⋊C4 in TeX

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