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## G = C5×C13⋊C3order 195 = 3·5·13

### Direct product of C5 and C13⋊C3

Aliases: C5×C13⋊C3, C65⋊C3, C13⋊C15, SmallGroup(195,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C5×C13⋊C3
 Chief series C1 — C13 — C65 — C5×C13⋊C3
 Lower central C13 — C5×C13⋊C3
 Upper central C1 — C5

Generators and relations for C5×C13⋊C3
G = < a,b,c | a5=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Smallest permutation representation of C5×C13⋊C3
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 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)

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

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

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

C5×C13⋊C3 is a maximal subgroup of   D65⋊C3

35 conjugacy classes

 class 1 3A 3B 5A 5B 5C 5D 13A 13B 13C 13D 15A ··· 15H 65A ··· 65P order 1 3 3 5 5 5 5 13 13 13 13 15 ··· 15 65 ··· 65 size 1 13 13 1 1 1 1 3 3 3 3 13 ··· 13 3 ··· 3

35 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C5 C15 C13⋊C3 C5×C13⋊C3 kernel C5×C13⋊C3 C65 C13⋊C3 C13 C5 C1 # reps 1 2 4 8 4 16

Matrix representation of C5×C13⋊C3 in GL3(𝔽1171) generated by

 216 0 0 0 216 0 0 0 216
,
 1077 852 1 710 525 1151 249 410 832
,
 1169 931 460 838 75 865 821 790 1098
G:=sub<GL(3,GF(1171))| [216,0,0,0,216,0,0,0,216],[1077,710,249,852,525,410,1,1151,832],[1169,838,821,931,75,790,460,865,1098] >;

C5×C13⋊C3 in GAP, Magma, Sage, TeX

C_5\times C_{13}\rtimes C_3
% in TeX

G:=Group("C5xC13:C3");
// GroupNames label

G:=SmallGroup(195,1);
// by ID

G=gap.SmallGroup(195,1);
# by ID

G:=PCGroup([3,-3,-5,-13,407]);
// Polycyclic

G:=Group<a,b,c|a^5=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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