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G = C5×C52C16order 400 = 24·52

Direct product of C5 and C52C16

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

Aliases: C5×C52C16, C52C80, C526C16, 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(C52C8), C2.(C5×C52C8), SmallGroup(400,49)

Series: Derived Chief Lower central Upper central

C1C5 — C5×C52C16
C1C5C10C20C40C5×C40 — C5×C52C16
C5 — C5×C52C16
C1C40

Generators and relations for C5×C52C16
 G = < a,b,c | a5=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

2C5
2C5
2C10
2C10
2C20
2C20
5C16
2C40
2C40
5C80

Smallest permutation representation of C5×C52C16
On 80 points
Generators in S80
(1 72 25 50 46)(2 73 26 51 47)(3 74 27 52 48)(4 75 28 53 33)(5 76 29 54 34)(6 77 30 55 35)(7 78 31 56 36)(8 79 32 57 37)(9 80 17 58 38)(10 65 18 59 39)(11 66 19 60 40)(12 67 20 61 41)(13 68 21 62 42)(14 69 22 63 43)(15 70 23 64 44)(16 71 24 49 45)
(1 50 72 46 25)(2 26 47 73 51)(3 52 74 48 27)(4 28 33 75 53)(5 54 76 34 29)(6 30 35 77 55)(7 56 78 36 31)(8 32 37 79 57)(9 58 80 38 17)(10 18 39 65 59)(11 60 66 40 19)(12 20 41 67 61)(13 62 68 42 21)(14 22 43 69 63)(15 64 70 44 23)(16 24 45 71 49)
(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)| (1,72,25,50,46)(2,73,26,51,47)(3,74,27,52,48)(4,75,28,53,33)(5,76,29,54,34)(6,77,30,55,35)(7,78,31,56,36)(8,79,32,57,37)(9,80,17,58,38)(10,65,18,59,39)(11,66,19,60,40)(12,67,20,61,41)(13,68,21,62,42)(14,69,22,63,43)(15,70,23,64,44)(16,71,24,49,45), (1,50,72,46,25)(2,26,47,73,51)(3,52,74,48,27)(4,28,33,75,53)(5,54,76,34,29)(6,30,35,77,55)(7,56,78,36,31)(8,32,37,79,57)(9,58,80,38,17)(10,18,39,65,59)(11,60,66,40,19)(12,20,41,67,61)(13,62,68,42,21)(14,22,43,69,63)(15,64,70,44,23)(16,24,45,71,49), (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( (1,72,25,50,46)(2,73,26,51,47)(3,74,27,52,48)(4,75,28,53,33)(5,76,29,54,34)(6,77,30,55,35)(7,78,31,56,36)(8,79,32,57,37)(9,80,17,58,38)(10,65,18,59,39)(11,66,19,60,40)(12,67,20,61,41)(13,68,21,62,42)(14,69,22,63,43)(15,70,23,64,44)(16,71,24,49,45), (1,50,72,46,25)(2,26,47,73,51)(3,52,74,48,27)(4,28,33,75,53)(5,54,76,34,29)(6,30,35,77,55)(7,56,78,36,31)(8,32,37,79,57)(9,58,80,38,17)(10,18,39,65,59)(11,60,66,40,19)(12,20,41,67,61)(13,62,68,42,21)(14,22,43,69,63)(15,64,70,44,23)(16,24,45,71,49), (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([(1,72,25,50,46),(2,73,26,51,47),(3,74,27,52,48),(4,75,28,53,33),(5,76,29,54,34),(6,77,30,55,35),(7,78,31,56,36),(8,79,32,57,37),(9,80,17,58,38),(10,65,18,59,39),(11,66,19,60,40),(12,67,20,61,41),(13,68,21,62,42),(14,69,22,63,43),(15,70,23,64,44),(16,71,24,49,45)], [(1,50,72,46,25),(2,26,47,73,51),(3,52,74,48,27),(4,28,33,75,53),(5,54,76,34,29),(6,30,35,77,55),(7,56,78,36,31),(8,32,37,79,57),(9,58,80,38,17),(10,18,39,65,59),(11,60,66,40,19),(12,20,41,67,61),(13,62,68,42,21),(14,22,43,69,63),(15,64,70,44,23),(16,24,45,71,49)], [(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+++-
imageC1C2C4C5C8C10C16C20C40C80D5Dic5C52C8C5×D5C52C16C5×Dic5C5×C52C8C5×C52C16
kernelC5×C52C16C5×C40C5×C20C52C16C5×C10C40C52C20C10C5C40C20C10C8C5C4C2C1
# reps1124448816322248881632

Matrix representation of C5×C52C16 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×C52C16 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×C52C16 in TeX

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