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## G = C5×C4⋊Dic5order 400 = 24·52

### Direct product of C5 and C4⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C4⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C102 — C10×Dic5 — C5×C4⋊Dic5
 Lower central C5 — C10 — C5×C4⋊Dic5
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C5×C4⋊Dic5
G = < a,b,c,d | a5=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 148 in 68 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C4, C4, C22, C5, C5, C2×C4, C2×C4, C10, C10, C4⋊C4, Dic5, C20, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C2×C20, C5×C10, C4⋊Dic5, C5×C4⋊C4, C5×Dic5, C5×C20, C102, C10×Dic5, C10×C20, C5×C4⋊Dic5
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, D5, C10, C4⋊C4, Dic5, C20, D10, C2×C10, Dic10, D20, C2×Dic5, C2×C20, C5×D4, C5×Q8, C5×D5, C4⋊Dic5, C5×C4⋊C4, C5×Dic5, D5×C10, C5×Dic10, C5×D20, C10×Dic5, C5×C4⋊Dic5

Smallest permutation representation of C5×C4⋊Dic5
On 80 points
Generators in S80
(1 7 3 9 5)(2 8 4 10 6)(11 15 19 13 17)(12 16 20 14 18)(21 27 23 29 25)(22 28 24 30 26)(31 37 33 39 35)(32 38 34 40 36)(41 47 43 49 45)(42 48 44 50 46)(51 55 59 53 57)(52 56 60 54 58)(61 65 69 63 67)(62 66 70 64 68)(71 75 79 73 77)(72 76 80 74 78)
(1 50 27 40)(2 41 28 31)(3 42 29 32)(4 43 30 33)(5 44 21 34)(6 45 22 35)(7 46 23 36)(8 47 24 37)(9 48 25 38)(10 49 26 39)(11 60 80 68)(12 51 71 69)(13 52 72 70)(14 53 73 61)(15 54 74 62)(16 55 75 63)(17 56 76 64)(18 57 77 65)(19 58 78 66)(20 59 79 67)
(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 55 6 60)(2 54 7 59)(3 53 8 58)(4 52 9 57)(5 51 10 56)(11 50 16 45)(12 49 17 44)(13 48 18 43)(14 47 19 42)(15 46 20 41)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)

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

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

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

130 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 20A ··· 20AV 20AW ··· 20BL order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 10 ··· 10

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 C5 C10 C10 C20 D4 Q8 D5 Dic5 D10 Dic10 D20 C5×D4 C5×Q8 C5×D5 C5×Dic5 D5×C10 C5×Dic10 C5×D20 kernel C5×C4⋊Dic5 C10×Dic5 C10×C20 C5×C20 C4⋊Dic5 C2×Dic5 C2×C20 C20 C5×C10 C5×C10 C2×C20 C20 C2×C10 C10 C10 C10 C10 C2×C4 C4 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 1 2 4 2 4 4 4 4 8 16 8 16 16

Matrix representation of C5×C4⋊Dic5 in GL3(𝔽41) generated by

 16 0 0 0 37 0 0 0 37
,
 40 0 0 0 9 0 0 0 32
,
 40 0 0 0 37 0 0 0 10
,
 9 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(41))| [16,0,0,0,37,0,0,0,37],[40,0,0,0,9,0,0,0,32],[40,0,0,0,37,0,0,0,10],[9,0,0,0,0,1,0,1,0] >;

C5×C4⋊Dic5 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C5xC4:Dic5");
// GroupNames label

G:=SmallGroup(400,85);
// by ID

G=gap.SmallGroup(400,85);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,247,11525]);
// Polycyclic

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

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