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## G = C4×C52⋊C4order 400 = 24·52

### Direct product of C4 and C52⋊C4

Aliases: C4×C52⋊C4, C204F5, C527C42, C53(C4×F5), (C5×C20)⋊8C4, C526C48C4, C10.23(C2×F5), C5⋊D5.9(C2×C4), (C4×C5⋊D5).13C2, C2.2(C2×C52⋊C4), (C5×C10).36(C2×C4), (C2×C52⋊C4).7C2, (C2×C5⋊D5).24C22, SmallGroup(400,158)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×C52⋊C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C52⋊C4 — C4×C52⋊C4
 Lower central C52 — C4×C52⋊C4
 Upper central C1 — C4

Generators and relations for C4×C52⋊C4
G = < a,b,c,d | a4=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >

Subgroups: 556 in 76 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D5, C10, C10, C42, Dic5, C20, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4×F5, C526C4, C5×C20, C52⋊C4, C2×C5⋊D5, C4×C5⋊D5, C2×C52⋊C4, C4×C52⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C42, F5, C2×F5, C4×F5, C52⋊C4, C2×C52⋊C4, C4×C52⋊C4

Smallest permutation representation of C4×C52⋊C4
On 40 points
Generators in S40
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(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)
(1 3 5 2 4)(6 8 10 7 9)(11 13 15 12 14)(16 18 20 17 19)(21 24 22 25 23)(26 29 27 30 28)(31 34 32 35 33)(36 39 37 40 38)
(1 33)(2 31 5 35)(3 34 4 32)(6 39 7 37)(8 40 10 36)(9 38)(11 29 12 27)(13 30 15 26)(14 28)(16 24 17 22)(18 25 20 21)(19 23)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4L 5A ··· 5F 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 ··· 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 25 25 1 1 25 ··· 25 4 ··· 4 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C4 F5 C2×F5 C4×F5 C52⋊C4 C2×C52⋊C4 C4×C52⋊C4 kernel C4×C52⋊C4 C4×C5⋊D5 C2×C52⋊C4 C52⋊6C4 C5×C20 C52⋊C4 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 8 2 2 4 4 4 8

Matrix representation of C4×C52⋊C4 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 0 1 0 0 40 6 0 0 0 0 35 35 0 0 6 40
,
 40 6 0 0 35 35 0 0 0 0 6 40 0 0 1 0
,
 0 0 9 0 0 0 0 9 32 0 0 0 28 9 0 0
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[0,40,0,0,1,6,0,0,0,0,35,6,0,0,35,40],[40,35,0,0,6,35,0,0,0,0,6,1,0,0,40,0],[0,0,32,28,0,0,0,9,9,0,0,0,0,9,0,0] >;

C4×C52⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_5^2\rtimes C_4
% in TeX

G:=Group("C4xC5^2:C4");
// GroupNames label

G:=SmallGroup(400,158);
// by ID

G=gap.SmallGroup(400,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,1444,496,5765,2897]);
// Polycyclic

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

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