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## G = C5×D10⋊C4order 400 = 24·52

### Direct product of C5 and D10⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×D10⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C102 — D5×C2×C10 — C5×D10⋊C4
 Lower central C5 — C10 — C5×D10⋊C4
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C5×D10⋊C4
G = < a,b,c,d | a5=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 244 in 84 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C2×C4, C23, D5, C10, C10, C22⋊C4, Dic5, C20, D10, D10, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C5×D5, C5×C10, D10⋊C4, C5×C22⋊C4, C5×Dic5, C5×C20, D5×C10, D5×C10, C102, C10×Dic5, C10×C20, D5×C2×C10, C5×D10⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, D5, C10, C22⋊C4, C20, D10, C2×C10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D5, D10⋊C4, C5×C22⋊C4, D5×C10, D5×C20, C5×D20, C5×C5⋊D4, C5×D10⋊C4

Smallest permutation representation of C5×D10⋊C4
On 80 points
Generators in S80
(1 5 9 3 7)(2 6 10 4 8)(11 15 19 13 17)(12 16 20 14 18)(21 25 29 23 27)(22 26 30 24 28)(31 35 39 33 37)(32 36 40 34 38)(41 47 43 49 45)(42 48 44 50 46)(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 52)(2 51)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 80)(30 79)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)
(1 37 17 27)(2 38 18 28)(3 39 19 29)(4 40 20 30)(5 31 11 21)(6 32 12 22)(7 33 13 23)(8 34 14 24)(9 35 15 25)(10 36 16 26)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)

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

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

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

130 conjugacy classes

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

130 irreducible representations

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

Matrix representation of C5×D10⋊C4 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 18 0 0 0 0 18
,
 10 0 0 0 0 37 0 0 0 0 25 0 0 0 0 23
,
 0 4 0 0 31 0 0 0 0 0 0 18 0 0 16 0
,
 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 32
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,18,0,0,0,0,18],[10,0,0,0,0,37,0,0,0,0,25,0,0,0,0,23],[0,31,0,0,4,0,0,0,0,0,0,16,0,0,18,0],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,32] >;

C5×D10⋊C4 in GAP, Magma, Sage, TeX

C_5\times D_{10}\rtimes C_4
% in TeX

G:=Group("C5xD10:C4");
// GroupNames label

G:=SmallGroup(400,86);
// by ID

G=gap.SmallGroup(400,86);
# by ID

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

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

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