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## G = C5×C5⋊C16order 400 = 24·52

### Direct product of C5 and C5⋊C16

Aliases: C5×C5⋊C16, C5⋊C80, C10.C40, C522C16, C20.2C20, C20.16F5, C4.2(C5×F5), C10.6(C5⋊C8), (C5×C10).2C8, (C5×C20).5C4, C52C8.2C10, C2.(C5×C5⋊C8), (C5×C52C8).1C2, SmallGroup(400,56)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5×C5⋊C16
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5×C5⋊2C8 — C5×C5⋊C16
 Lower central C5 — C5×C5⋊C16
 Upper central C1 — C20

Generators and relations for C5×C5⋊C16
G = < a,b,c | a5=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C5×C5⋊C16
On 80 points
Generators in S80
(1 39 69 28 58)(2 40 70 29 59)(3 41 71 30 60)(4 42 72 31 61)(5 43 73 32 62)(6 44 74 17 63)(7 45 75 18 64)(8 46 76 19 49)(9 47 77 20 50)(10 48 78 21 51)(11 33 79 22 52)(12 34 80 23 53)(13 35 65 24 54)(14 36 66 25 55)(15 37 67 26 56)(16 38 68 27 57)
(1 39 69 28 58)(2 29 40 59 70)(3 60 30 71 41)(4 72 61 42 31)(5 43 73 32 62)(6 17 44 63 74)(7 64 18 75 45)(8 76 49 46 19)(9 47 77 20 50)(10 21 48 51 78)(11 52 22 79 33)(12 80 53 34 23)(13 35 65 24 54)(14 25 36 55 66)(15 56 26 67 37)(16 68 57 38 27)
(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)

G:=sub<Sym(80)| (1,39,69,28,58)(2,40,70,29,59)(3,41,71,30,60)(4,42,72,31,61)(5,43,73,32,62)(6,44,74,17,63)(7,45,75,18,64)(8,46,76,19,49)(9,47,77,20,50)(10,48,78,21,51)(11,33,79,22,52)(12,34,80,23,53)(13,35,65,24,54)(14,36,66,25,55)(15,37,67,26,56)(16,38,68,27,57), (1,39,69,28,58)(2,29,40,59,70)(3,60,30,71,41)(4,72,61,42,31)(5,43,73,32,62)(6,17,44,63,74)(7,64,18,75,45)(8,76,49,46,19)(9,47,77,20,50)(10,21,48,51,78)(11,52,22,79,33)(12,80,53,34,23)(13,35,65,24,54)(14,25,36,55,66)(15,56,26,67,37)(16,68,57,38,27), (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)>;

G:=Group( (1,39,69,28,58)(2,40,70,29,59)(3,41,71,30,60)(4,42,72,31,61)(5,43,73,32,62)(6,44,74,17,63)(7,45,75,18,64)(8,46,76,19,49)(9,47,77,20,50)(10,48,78,21,51)(11,33,79,22,52)(12,34,80,23,53)(13,35,65,24,54)(14,36,66,25,55)(15,37,67,26,56)(16,38,68,27,57), (1,39,69,28,58)(2,29,40,59,70)(3,60,30,71,41)(4,72,61,42,31)(5,43,73,32,62)(6,17,44,63,74)(7,64,18,75,45)(8,76,49,46,19)(9,47,77,20,50)(10,21,48,51,78)(11,52,22,79,33)(12,80,53,34,23)(13,35,65,24,54)(14,25,36,55,66)(15,56,26,67,37)(16,68,57,38,27), (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) );

G=PermutationGroup([[(1,39,69,28,58),(2,40,70,29,59),(3,41,71,30,60),(4,42,72,31,61),(5,43,73,32,62),(6,44,74,17,63),(7,45,75,18,64),(8,46,76,19,49),(9,47,77,20,50),(10,48,78,21,51),(11,33,79,22,52),(12,34,80,23,53),(13,35,65,24,54),(14,36,66,25,55),(15,37,67,26,56),(16,38,68,27,57)], [(1,39,69,28,58),(2,29,40,59,70),(3,60,30,71,41),(4,72,61,42,31),(5,43,73,32,62),(6,17,44,63,74),(7,64,18,75,45),(8,76,49,46,19),(9,47,77,20,50),(10,21,48,51,78),(11,52,22,79,33),(12,80,53,34,23),(13,35,65,24,54),(14,25,36,55,66),(15,56,26,67,37),(16,68,57,38,27)], [(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)]])

100 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5I 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10I 16A ··· 16H 20A ··· 20H 20I ··· 20R 40A ··· 40P 80A ··· 80AF order 1 2 4 4 5 5 5 5 5 ··· 5 8 8 8 8 10 10 10 10 10 ··· 10 16 ··· 16 20 ··· 20 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 1 1 1 1 4 ··· 4 5 5 5 5 1 1 1 1 4 ··· 4 5 ··· 5 1 ··· 1 4 ··· 4 5 ··· 5 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + - image C1 C2 C4 C5 C8 C10 C16 C20 C40 C80 F5 C5⋊C8 C5⋊C16 C5×F5 C5×C5⋊C8 C5×C5⋊C16 kernel C5×C5⋊C16 C5×C5⋊2C8 C5×C20 C5⋊C16 C5×C10 C5⋊2C8 C52 C20 C10 C5 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 4 8 8 16 32 1 1 2 4 4 8

Matrix representation of C5×C5⋊C16 in GL5(𝔽241)

 87 0 0 0 0 0 87 0 0 0 0 0 87 0 0 0 0 0 87 0 0 0 0 0 87
,
 1 0 0 0 0 0 87 0 0 0 0 88 205 0 0 0 98 0 98 0 0 189 0 0 91
,
 76 0 0 0 0 0 240 0 204 0 0 0 0 240 1 0 0 1 1 0 0 0 0 1 0

G:=sub<GL(5,GF(241))| [87,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,87],[1,0,0,0,0,0,87,88,98,189,0,0,205,0,0,0,0,0,98,0,0,0,0,0,91],[76,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,204,240,1,1,0,0,1,0,0] >;

C5×C5⋊C16 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes C_{16}
% in TeX

G:=Group("C5xC5:C16");
// GroupNames label

G:=SmallGroup(400,56);
// by ID

G=gap.SmallGroup(400,56);
# by ID

G:=PCGroup([6,-2,-5,-2,-2,-2,-5,60,50,69,5765,1169]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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