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G = C5×C5⋊2C16  order 400 = 24·52

Direct product of C5 and C5⋊2C16

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C5×C5⋊2C16, C5⋊2C80, C52⋊6C16, C40.2C10, C20.5C20, C10.2C40, C40.10D5, C20.13Dic5, C8.2(C5×D5), (C5×C40).3C2, (C5×C10).6C8, (C5×C20).15C4, C4.2(C5×Dic5), C10.5(C5⋊2C8), C2.(C5×C5⋊2C8), SmallGroup(400,49)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C5×C5⋊2C16
Chief series
C1 — C5 — C10 — C20 — C40 — C5×C40 — C5×C5⋊2C16
Lower central
C5 — C5×C5⋊2C16
Upper central
C1 — C40

Generators and relations for C5×C5⋊2C16
 G = < a,b,c | a5=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

C1
C2
C5
C5
2C5
2C5
C4
C10
C10
2C10
2C10
C52
C8
C20
C20
2C20
2C20
C5×C10
5C16
C40
C40
2C40
2C40
C5×C20
C5⋊2C16
5C80
C5×C40
C5×C5⋊2C16

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

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

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

160 conjugacy classes

class 1  2 4A4B5A5B5C5D5E···5N8A8B8C8D10A10B10C10D10E···10N16A···16H20A···20H20I···20AB40A···40P40Q···40BD80A···80AF
order124455555···588881010101010···1016···1620···2020···2040···4040···4080···80
size111111112···2111111112···25···51···12···21···12···25···5

160 irreducible representations

dim111111111122222222
type+++-
imageC1C2C4C5C8C10C16C20C40C80D5Dic5C5⋊2C8C5×D5C5⋊2C16C5×Dic5C5×C5⋊2C8C5×C5⋊2C16
kernelC5×C5⋊2C16C5×C40C5×C20C5⋊2C16C5×C10C40C52C20C10C5C40C20C10C8C5C4C2C1
# reps1124448816322248881632

Matrix representation of C5×C5⋊2C16 ►in GL2(𝔽41) generated by

160
016
,
370
010
,
027
10
G:=sub<GL(2,GF(41))| [16,0,0,16],[37,0,0,10],[0,1,27,0] >;
 

C5×C5⋊2C16 in GAP, Magma, Sage, TeX 

C_5\times C_5\rtimes_2C_{16}
 
% in TeX
 
G:=Group("C5xC5:2C16");
 
// GroupNames label
 
G:=SmallGroup(400,49);
 
// by ID
 
G=gap.SmallGroup(400,49);
 
# by ID
 
G:=PCGroup([6,-2,-5,-2,-2,-2,-5,60,50,69,11525]);
 
// 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^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C5×C5⋊2C16 in TeX

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