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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 — D10 — D5×C10 — C10×F5 — 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=d4=1, 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 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 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)(21 25 24 23 22)(26 30 29 28 27)(31 32 33 34 35)(36 37 38 39 40)(41 44 42 45 43)(46 49 47 50 48)(51 53 55 52 54)(56 58 60 57 59)(61 64 62 65 63)(66 69 67 70 68)(71 73 75 72 74)(76 78 80 77 79)
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 4A 4B ··· 4F 5A 5B 5C 5D 5E ··· 5I 10A 10B 10C 10D 10E ··· 10I 10J ··· 10Q 20A 20B 20C 20D 20E ··· 20N 20O ··· 20AH order 1 2 2 2 4 4 ··· 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 20 20 20 20 ··· 20 20 ··· 20 size 1 1 5 5 2 10 ··· 10 1 1 1 1 4 ··· 4 1 1 1 1 4 ··· 4 5 ··· 5 2 2 2 2 4 ··· 4 10 ··· 10

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 type + + + + - + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 D4 Q8 C5×D4 C5×Q8 F5 C2×F5 C4⋊F5 C5×F5 C10×F5 C5×C4⋊F5 kernel C5×C4⋊F5 D5×C20 C10×F5 C5×Dic5 C5×C20 C4⋊F5 C4×D5 C2×F5 Dic5 C20 C5×D5 C5×D5 D5 D5 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 8 8 1 1 4 4 1 1 2 4 4 8

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 9 24 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 0 0 0 0 0 0 10 0 0 0 0 0 0 16 0 0 0 0 0 0 18
,
 19 29 0 0 0 0 37 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[9,0,0,0,0,0,24,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,18],[19,37,0,0,0,0,29,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×C4⋊F5 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes F_5
% in TeX

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

G:=SmallGroup(400,138);
// by ID

G=gap.SmallGroup(400,138);
# by ID

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

G:=Group<a,b,c,d|a^5=b^4=c^5=d^4=1,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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