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## G = C10×C5⋊D4order 400 = 24·52

### Direct product of C10 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10×C5⋊D4
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C2×C10 — C10×C5⋊D4
 Lower central C5 — C10 — C10×C5⋊D4
 Upper central C1 — C2×C10 — C22×C10

Generators and relations for C10×C5⋊D4
G = < a,b,c,d | a10=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 356 in 140 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C5, C2×C4, D4, C23, C23, D5, C10, C10, C10, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C52, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C5×D5, C5×C10, C5×C10, C5×C10, C2×C5⋊D4, D4×C10, C5×Dic5, D5×C10, D5×C10, C102, C102, C102, C10×Dic5, C5×C5⋊D4, D5×C2×C10, C2×C102, C10×C5⋊D4
Quotients: C1, C2, C22, C5, D4, C23, D5, C10, C2×D4, D10, C2×C10, C5⋊D4, C5×D4, C22×D5, C22×C10, C5×D5, C2×C5⋊D4, D4×C10, D5×C10, C5×C5⋊D4, D5×C2×C10, C10×C5⋊D4

Smallest permutation representation of C10×C5⋊D4
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)
(1 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)(21 23 25 27 29)(22 24 26 28 30)(31 39 37 35 33)(32 40 38 36 34)
(1 32 22 16)(2 33 23 17)(3 34 24 18)(4 35 25 19)(5 36 26 20)(6 37 27 11)(7 38 28 12)(8 39 29 13)(9 40 30 14)(10 31 21 15)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

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

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

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

130 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10CL 10CM ··· 10CT 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 10 ··· 10 10 ··· 10

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 D4 D5 D10 C5⋊D4 C5×D4 C5×D5 D5×C10 C5×C5⋊D4 kernel C10×C5⋊D4 C10×Dic5 C5×C5⋊D4 D5×C2×C10 C2×C102 C2×C5⋊D4 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C5×C10 C22×C10 C2×C10 C10 C10 C23 C22 C2 # reps 1 1 4 1 1 4 4 16 4 4 2 2 6 8 8 8 24 32

Matrix representation of C10×C5⋊D4 in GL4(𝔽41) generated by

 23 0 0 0 0 23 0 0 0 0 40 0 0 0 0 40
,
 37 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 40 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [23,0,0,0,0,23,0,0,0,0,40,0,0,0,0,40],[37,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C10×C5⋊D4 in GAP, Magma, Sage, TeX

C_{10}\times C_5\rtimes D_4
% in TeX

G:=Group("C10xC5:D4");
// GroupNames label

G:=SmallGroup(400,190);
// by ID

G=gap.SmallGroup(400,190);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,794,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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