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## G = C5×C22⋊F5order 400 = 24·52

### Direct product of C5 and C22⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C22⋊F5
 Chief series C1 — C5 — C10 — D10 — D5×C10 — C10×F5 — C5×C22⋊F5
 Lower central C5 — C10 — C5×C22⋊F5
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×C22⋊F5
G = < a,b,c,d,e | a5=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 256 in 73 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22⋊C4, C20, F5, D10, D10, C2×C10, C2×C10, C52, C2×C20, C2×F5, C22×D5, C22×C10, C5×D5, C5×D5, C5×C10, C5×C10, C5×C22⋊C4, C22⋊F5, C5×F5, D5×C10, D5×C10, C102, C10×F5, D5×C2×C10, C5×C22⋊F5
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, F5, C2×C10, C2×C20, C5×D4, C2×F5, C5×C22⋊C4, C22⋊F5, C5×F5, C10×F5, C5×C22⋊F5

Smallest permutation representation of C5×C22⋊F5
On 40 points
Generators in S40
(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)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)(21 24 22 25 23)(26 29 27 30 28)(31 33 35 32 34)(36 38 40 37 39)
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)

G:=sub<Sym(40)| (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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,24,22,25,23)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)>;

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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,24,22,25,23)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30) );

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)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20),(21,24,22,25,23),(26,29,27,30,28),(31,33,35,32,34),(36,38,40,37,39)], [(1,31,11,21),(2,32,12,22),(3,33,13,23),(4,34,14,24),(5,35,15,25),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30)]])

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5I 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10W 10X ··· 10AE 10AF 10AG 10AH 10AI 20A ··· 20P order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 10 10 10 10 ··· 10 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 2 5 5 10 10 10 10 10 1 1 1 1 4 ··· 4 1 1 1 1 2 2 2 2 4 ··· 4 5 ··· 5 10 10 10 10 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 D4 C5×D4 F5 C2×F5 C22⋊F5 C5×F5 C10×F5 C5×C22⋊F5 kernel C5×C22⋊F5 C10×F5 D5×C2×C10 D5×C10 C102 C22⋊F5 C2×F5 C22×D5 D10 C2×C10 C5×D5 D5 C2×C10 C10 C5 C22 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×C22⋊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
,
 1 0 0 0 0 0 0 40 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
,
 40 0 0 0 0 0 0 40 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
,
 0 1 0 0 0 0 1 0 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],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×C22⋊F5 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes F_5
% in TeX

G:=Group("C5xC2^2:F5");
// GroupNames label

G:=SmallGroup(400,141);
// by ID

G=gap.SmallGroup(400,141);
# by ID

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

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

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