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## G = D10⋊Dic5order 400 = 24·52

### 1st semidirect product of D10 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10⋊Dic5
 Chief series C1 — C5 — C52 — C5×C10 — C102 — D5×C2×C10 — D10⋊Dic5
 Lower central C52 — C5×C10 — D10⋊Dic5
 Upper central C1 — C22

Generators and relations for D10⋊Dic5
G = < a,b,c,d | a10=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 412 in 84 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22⋊C4, Dic5, C20, D10, D10, C2×C10, C2×C10, C52, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×C10, C5×D5, C5×C10, D10⋊C4, C23.D5, C5×Dic5, C526C4, D5×C10, D5×C10, C102, C10×Dic5, C2×C526C4, D5×C2×C10, D10⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C4×D5, D20, C2×Dic5, C5⋊D4, D10⋊C4, C23.D5, D52, D5×Dic5, C522D4, C5⋊D20, D10⋊Dic5

Smallest permutation representation of D10⋊Dic5
On 80 points
Generators in S80
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 60)(8 59)(9 58)(10 57)(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 19 3 11 5 13 7 15 9 17)(2 20 4 12 6 14 8 16 10 18)(21 35 29 33 27 31 25 39 23 37)(22 36 30 34 28 32 26 40 24 38)(41 55 49 53 47 51 45 59 43 57)(42 56 50 54 48 52 46 60 44 58)(61 77 63 79 65 71 67 73 69 75)(62 78 64 80 66 72 68 74 70 76)
(1 33 13 23)(2 34 14 24)(3 35 15 25)(4 36 16 26)(5 37 17 27)(6 38 18 28)(7 39 19 29)(8 40 20 30)(9 31 11 21)(10 32 12 22)(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)| (1,2,3,4,5,6,7,8,9,10)(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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,60)(8,59)(9,58)(10,57)(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), (1,19,3,11,5,13,7,15,9,17)(2,20,4,12,6,14,8,16,10,18)(21,35,29,33,27,31,25,39,23,37)(22,36,30,34,28,32,26,40,24,38)(41,55,49,53,47,51,45,59,43,57)(42,56,50,54,48,52,46,60,44,58)(61,77,63,79,65,71,67,73,69,75)(62,78,64,80,66,72,68,74,70,76), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,36,16,26)(5,37,17,27)(6,38,18,28)(7,39,19,29)(8,40,20,30)(9,31,11,21)(10,32,12,22)(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( (1,2,3,4,5,6,7,8,9,10)(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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,60)(8,59)(9,58)(10,57)(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), (1,19,3,11,5,13,7,15,9,17)(2,20,4,12,6,14,8,16,10,18)(21,35,29,33,27,31,25,39,23,37)(22,36,30,34,28,32,26,40,24,38)(41,55,49,53,47,51,45,59,43,57)(42,56,50,54,48,52,46,60,44,58)(61,77,63,79,65,71,67,73,69,75)(62,78,64,80,66,72,68,74,70,76), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,36,16,26)(5,37,17,27)(6,38,18,28)(7,39,19,29)(8,40,20,30)(9,31,11,21)(10,32,12,22)(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([[(1,2,3,4,5,6,7,8,9,10),(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,60),(8,59),(9,58),(10,57),(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)], [(1,19,3,11,5,13,7,15,9,17),(2,20,4,12,6,14,8,16,10,18),(21,35,29,33,27,31,25,39,23,37),(22,36,30,34,28,32,26,40,24,38),(41,55,49,53,47,51,45,59,43,57),(42,56,50,54,48,52,46,60,44,58),(61,77,63,79,65,71,67,73,69,75),(62,78,64,80,66,72,68,74,70,76)], [(1,33,13,23),(2,34,14,24),(3,35,15,25),(4,36,16,26),(5,37,17,27),(6,38,18,28),(7,39,19,29),(8,40,20,30),(9,31,11,21),(10,32,12,22),(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)]])

58 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 10Y ··· 10AF 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 10 10 50 50 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - - + image C1 C2 C2 C2 C4 D4 D5 D5 Dic5 D10 C4×D5 D20 C5⋊D4 D52 D5×Dic5 C52⋊2D4 C5⋊D20 kernel D10⋊Dic5 C10×Dic5 C2×C52⋊6C4 D5×C2×C10 D5×C10 C5×C10 C2×Dic5 C22×D5 D10 C2×C10 C10 C10 C10 C22 C2 C2 C2 # reps 1 1 1 1 4 2 2 2 4 4 4 4 12 4 4 4 4

Matrix representation of D10⋊Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 6 0 0 35 35
,
 17 40 0 0 1 24 0 0 0 0 1 35 0 0 0 40
,
 7 1 0 0 40 0 0 0 0 0 40 0 0 0 0 40
,
 40 17 0 0 24 1 0 0 0 0 9 0 0 0 0 9
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,35,0,0,6,35],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,35,40],[7,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[40,24,0,0,17,1,0,0,0,0,9,0,0,0,0,9] >;

D10⋊Dic5 in GAP, Magma, Sage, TeX

D_{10}\rtimes {\rm Dic}_5
% in TeX

G:=Group("D10:Dic5");
// GroupNames label

G:=SmallGroup(400,72);
// by ID

G=gap.SmallGroup(400,72);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,970,11525]);
// Polycyclic

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

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