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## G = C5×C10.D4order 400 = 24·52

### Direct product of C5 and C10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C102 — C10×Dic5 — C5×C10.D4
 Lower central C5 — C10 — C5×C10.D4
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C5×C10.D4
G = < a,b,c,d | a5=b10=c4=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 148 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C4, C22, C5, C5, C2×C4, C2×C4, C10, C10, C4⋊C4, Dic5, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C2×C20, C5×C10, C10.D4, C5×C4⋊C4, C5×Dic5, C5×Dic5, C5×C20, C102, C10×Dic5, C10×C20, C5×C10.D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, D5, C10, C4⋊C4, C20, D10, C2×C10, Dic10, C4×D5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C5×D5, C10.D4, C5×C4⋊C4, D5×C10, C5×Dic10, D5×C20, C5×C5⋊D4, C5×C10.D4

Smallest permutation representation of C5×C10.D4
On 80 points
Generators in S80
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)(21 25 29 23 27)(22 26 30 24 28)(31 35 39 33 37)(32 36 40 34 38)(41 45 49 43 47)(42 46 50 44 48)(51 57 53 59 55)(52 58 54 60 56)(61 67 63 69 65)(62 68 64 70 66)(71 77 73 79 75)(72 78 74 80 76)
(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 70 25 58)(2 69 26 57)(3 68 27 56)(4 67 28 55)(5 66 29 54)(6 65 30 53)(7 64 21 52)(8 63 22 51)(9 62 23 60)(10 61 24 59)(11 50 72 38)(12 49 73 37)(13 48 74 36)(14 47 75 35)(15 46 76 34)(16 45 77 33)(17 44 78 32)(18 43 79 31)(19 42 80 40)(20 41 71 39)
(1 73 6 78)(2 72 7 77)(3 71 8 76)(4 80 9 75)(5 79 10 74)(11 21 16 26)(12 30 17 25)(13 29 18 24)(14 28 19 23)(15 27 20 22)(31 59 36 54)(32 58 37 53)(33 57 38 52)(34 56 39 51)(35 55 40 60)(41 63 46 68)(42 62 47 67)(43 61 48 66)(44 70 49 65)(45 69 50 64)

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

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

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

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 D10 Dic10 C4×D5 C5⋊D4 C5×D4 C5×Q8 C5×D5 D5×C10 C5×Dic10 D5×C20 C5×C5⋊D4 kernel C5×C10.D4 C10×Dic5 C10×C20 C5×Dic5 C10.D4 C2×Dic5 C2×C20 Dic5 C5×C10 C5×C10 C2×C20 C2×C10 C10 C10 C10 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 2 1 4 4 8 4 16 1 1 2 2 4 4 4 4 4 8 8 16 16 16

Matrix representation of C5×C10.D4 in GL3(𝔽41) generated by

 18 0 0 0 10 0 0 0 10
,
 40 0 0 0 4 0 0 0 31
,
 40 0 0 0 0 1 0 40 0
,
 32 0 0 0 0 9 0 9 0
G:=sub<GL(3,GF(41))| [18,0,0,0,10,0,0,0,10],[40,0,0,0,4,0,0,0,31],[40,0,0,0,0,40,0,1,0],[32,0,0,0,0,9,0,9,0] >;

C5×C10.D4 in GAP, Magma, Sage, TeX

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

G:=Group("C5xC10.D4");
// GroupNames label

G:=SmallGroup(400,84);
// by ID

G=gap.SmallGroup(400,84);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,505,127,11525]);
// Polycyclic

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

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