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## G = C20×F5order 400 = 24·52

### Direct product of C20 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C20×F5
 Chief series C1 — C5 — C10 — D10 — D5×C10 — C10×F5 — C20×F5
 Lower central C5 — C20×F5
 Upper central C1 — C20

Generators and relations for C20×F5
G = < a,b,c | a20=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 168 in 63 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D5, C10, C10, C42, Dic5, C20, C20, F5, D10, C2×C10, C52, C4×D5, C2×C20, C2×F5, C5×D5, C5×C10, C4×C20, C4×F5, C5×Dic5, C5×C20, C5×F5, D5×C10, D5×C20, C10×F5, C20×F5
Quotients: C1, C2, C4, C22, C5, C2×C4, C10, C42, C20, F5, C2×C10, C2×C20, C2×F5, C4×C20, C4×F5, C5×F5, C10×F5, C20×F5

Smallest permutation representation of C20×F5
On 80 points
Generators in S80
(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)
(1 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 29 37 25 33)(22 30 38 26 34)(23 31 39 27 35)(24 32 40 28 36)(41 57 53 49 45)(42 58 54 50 46)(43 59 55 51 47)(44 60 56 52 48)(61 73 65 77 69)(62 74 66 78 70)(63 75 67 79 71)(64 76 68 80 72)
(1 61 51 23)(2 62 52 24)(3 63 53 25)(4 64 54 26)(5 65 55 27)(6 66 56 28)(7 67 57 29)(8 68 58 30)(9 69 59 31)(10 70 60 32)(11 71 41 33)(12 72 42 34)(13 73 43 35)(14 74 44 36)(15 75 45 37)(16 76 46 38)(17 77 47 39)(18 78 48 40)(19 79 49 21)(20 80 50 22)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4L 5A 5B 5C 5D 5E ··· 5I 10A 10B 10C 10D 10E ··· 10I 10J ··· 10Q 20A ··· 20H 20I ··· 20R 20S ··· 20BF order 1 2 2 2 4 4 4 ··· 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 5 5 1 1 5 ··· 5 1 1 1 1 4 ··· 4 1 1 1 1 4 ··· 4 5 ··· 5 1 ··· 1 4 ··· 4 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C4 C5 C10 C10 C20 C20 C20 F5 C2×F5 C4×F5 C5×F5 C10×F5 C20×F5 kernel C20×F5 D5×C20 C10×F5 C5×Dic5 C5×C20 C5×F5 C4×F5 C4×D5 C2×F5 Dic5 C20 F5 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 8 4 4 8 8 8 32 1 1 2 4 4 8

Matrix representation of C20×F5 in GL4(𝔽41) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 16 0 0 0 24 18 0 0 8 0 10 0 28 0 0 37
,
 1 0 30 0 0 0 9 9 0 9 40 0 0 0 1 0
G:=sub<GL(4,GF(41))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[16,24,8,28,0,18,0,0,0,0,10,0,0,0,0,37],[1,0,0,0,0,0,9,0,30,9,40,1,0,9,0,0] >;

C20×F5 in GAP, Magma, Sage, TeX

C_{20}\times F_5
% in TeX

G:=Group("C20xF5");
// GroupNames label

G:=SmallGroup(400,137);
// by ID

G=gap.SmallGroup(400,137);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,5765,599]);
// Polycyclic

G:=Group<a,b,c|a^20=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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