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## G = C5×C13⋊C4order 260 = 22·5·13

### Direct product of C5 and C13⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C5×C13⋊C4
 Chief series C1 — C13 — D13 — C5×D13 — C5×C13⋊C4
 Lower central C13 — C5×C13⋊C4
 Upper central C1 — C5

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

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 4A 4B 5A 5B 5C 5D 10A 10B 10C 10D 13A 13B 13C 20A ··· 20H 65A ··· 65L order 1 2 4 4 5 5 5 5 10 10 10 10 13 13 13 20 ··· 20 65 ··· 65 size 1 13 13 13 1 1 1 1 13 13 13 13 4 4 4 13 ··· 13 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C5 C10 C20 C13⋊C4 C5×C13⋊C4 kernel C5×C13⋊C4 C5×D13 C65 C13⋊C4 D13 C13 C5 C1 # reps 1 1 2 4 4 8 3 12

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

 516 0 0 0 0 516 0 0 0 0 516 0 0 0 0 516
,
 27 368 152 520 182 190 305 495 54 216 304 519 245 1 126 126
,
 115 378 118 38 358 234 285 101 483 36 485 216 134 420 252 208
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

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