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## G = C5×C4.F5order 400 = 24·52

### Direct product of C5 and C4.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C4.F5
 Chief series C1 — C5 — C10 — Dic5 — C5×Dic5 — C5×C5⋊C8 — C5×C4.F5
 Lower central C5 — C10 — C5×C4.F5
 Upper central C1 — C10 — C20

Generators and relations for C5×C4.F5
G = < a,b,c,d | a5=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C5×C4.F5
On 80 points
Generators in S80
(1 21 75 33 57)(2 22 76 34 58)(3 23 77 35 59)(4 24 78 36 60)(5 17 79 37 61)(6 18 80 38 62)(7 19 73 39 63)(8 20 74 40 64)(9 29 45 53 71)(10 30 46 54 72)(11 31 47 55 65)(12 32 48 56 66)(13 25 41 49 67)(14 26 42 50 68)(15 27 43 51 69)(16 28 44 52 70)
(1 69 5 65)(2 66 6 70)(3 71 7 67)(4 68 8 72)(9 19 13 23)(10 24 14 20)(11 21 15 17)(12 18 16 22)(25 77 29 73)(26 74 30 78)(27 79 31 75)(28 76 32 80)(33 43 37 47)(34 48 38 44)(35 45 39 41)(36 42 40 46)(49 59 53 63)(50 64 54 60)(51 61 55 57)(52 58 56 62)
(1 57 33 75 21)(2 76 58 22 34)(3 23 77 35 59)(4 36 24 60 78)(5 61 37 79 17)(6 80 62 18 38)(7 19 73 39 63)(8 40 20 64 74)(9 29 45 53 71)(10 54 30 72 46)(11 65 55 47 31)(12 48 66 32 56)(13 25 41 49 67)(14 50 26 68 42)(15 69 51 43 27)(16 44 70 28 52)
(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)| (1,21,75,33,57)(2,22,76,34,58)(3,23,77,35,59)(4,24,78,36,60)(5,17,79,37,61)(6,18,80,38,62)(7,19,73,39,63)(8,20,74,40,64)(9,29,45,53,71)(10,30,46,54,72)(11,31,47,55,65)(12,32,48,56,66)(13,25,41,49,67)(14,26,42,50,68)(15,27,43,51,69)(16,28,44,52,70), (1,69,5,65)(2,66,6,70)(3,71,7,67)(4,68,8,72)(9,19,13,23)(10,24,14,20)(11,21,15,17)(12,18,16,22)(25,77,29,73)(26,74,30,78)(27,79,31,75)(28,76,32,80)(33,43,37,47)(34,48,38,44)(35,45,39,41)(36,42,40,46)(49,59,53,63)(50,64,54,60)(51,61,55,57)(52,58,56,62), (1,57,33,75,21)(2,76,58,22,34)(3,23,77,35,59)(4,36,24,60,78)(5,61,37,79,17)(6,80,62,18,38)(7,19,73,39,63)(8,40,20,64,74)(9,29,45,53,71)(10,54,30,72,46)(11,65,55,47,31)(12,48,66,32,56)(13,25,41,49,67)(14,50,26,68,42)(15,69,51,43,27)(16,44,70,28,52), (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( (1,21,75,33,57)(2,22,76,34,58)(3,23,77,35,59)(4,24,78,36,60)(5,17,79,37,61)(6,18,80,38,62)(7,19,73,39,63)(8,20,74,40,64)(9,29,45,53,71)(10,30,46,54,72)(11,31,47,55,65)(12,32,48,56,66)(13,25,41,49,67)(14,26,42,50,68)(15,27,43,51,69)(16,28,44,52,70), (1,69,5,65)(2,66,6,70)(3,71,7,67)(4,68,8,72)(9,19,13,23)(10,24,14,20)(11,21,15,17)(12,18,16,22)(25,77,29,73)(26,74,30,78)(27,79,31,75)(28,76,32,80)(33,43,37,47)(34,48,38,44)(35,45,39,41)(36,42,40,46)(49,59,53,63)(50,64,54,60)(51,61,55,57)(52,58,56,62), (1,57,33,75,21)(2,76,58,22,34)(3,23,77,35,59)(4,36,24,60,78)(5,61,37,79,17)(6,80,62,18,38)(7,19,73,39,63)(8,40,20,64,74)(9,29,45,53,71)(10,54,30,72,46)(11,65,55,47,31)(12,48,66,32,56)(13,25,41,49,67)(14,50,26,68,42)(15,69,51,43,27)(16,44,70,28,52), (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([[(1,21,75,33,57),(2,22,76,34,58),(3,23,77,35,59),(4,24,78,36,60),(5,17,79,37,61),(6,18,80,38,62),(7,19,73,39,63),(8,20,74,40,64),(9,29,45,53,71),(10,30,46,54,72),(11,31,47,55,65),(12,32,48,56,66),(13,25,41,49,67),(14,26,42,50,68),(15,27,43,51,69),(16,28,44,52,70)], [(1,69,5,65),(2,66,6,70),(3,71,7,67),(4,68,8,72),(9,19,13,23),(10,24,14,20),(11,21,15,17),(12,18,16,22),(25,77,29,73),(26,74,30,78),(27,79,31,75),(28,76,32,80),(33,43,37,47),(34,48,38,44),(35,45,39,41),(36,42,40,46),(49,59,53,63),(50,64,54,60),(51,61,55,57),(52,58,56,62)], [(1,57,33,75,21),(2,76,58,22,34),(3,23,77,35,59),(4,36,24,60,78),(5,61,37,79,17),(6,80,62,18,38),(7,19,73,39,63),(8,40,20,64,74),(9,29,45,53,71),(10,54,30,72,46),(11,65,55,47,31),(12,48,66,32,56),(13,25,41,49,67),(14,50,26,68,42),(15,69,51,43,27),(16,44,70,28,52)], [(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)]])

70 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E ··· 5I 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10I 10J 10K 10L 10M 20A 20B 20C 20D 20E ··· 20N 20O ··· 20V 40A ··· 40P order 1 2 2 4 4 4 5 5 5 5 5 ··· 5 8 8 8 8 10 10 10 10 10 ··· 10 10 10 10 10 20 20 20 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 10 2 5 5 1 1 1 1 4 ··· 4 10 10 10 10 1 1 1 1 4 ··· 4 10 10 10 10 2 2 2 2 4 ··· 4 5 ··· 5 10 ··· 10

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 M4(2) C5×M4(2) F5 C2×F5 C4.F5 C5×F5 C10×F5 C5×C4.F5 kernel C5×C4.F5 C5×C5⋊C8 D5×C20 C5×C20 D5×C10 C4.F5 C5⋊C8 C4×D5 C20 D10 C52 C5 C20 C10 C5 C4 C2 C1 # reps 1 2 1 2 2 4 8 4 8 8 2 8 1 1 2 4 4 8

Matrix representation of C5×C4.F5 in GL6(𝔽41)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 32 12 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 24 16 0 0 0 0 23 0 37 0 0 0 27 0 0 10
,
 14 36 0 0 0 0 21 27 0 0 0 0 0 0 40 0 17 0 0 0 0 0 40 1 0 0 0 1 1 0 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[32,0,0,0,0,0,12,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,24,23,27,0,0,0,16,0,0,0,0,0,0,37,0,0,0,0,0,0,10],[14,21,0,0,0,0,36,27,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,17,40,1,1,0,0,0,1,0,0] >;

C5×C4.F5 in GAP, Magma, Sage, TeX

C_5\times C_4.F_5
% in TeX

G:=Group("C5xC4.F5");
// GroupNames label

G:=SmallGroup(400,136);
// by ID

G=gap.SmallGroup(400,136);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,247,69,5765,599]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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