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## G = D5×Dic10order 400 = 24·52

### Direct product of D5 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D5×Dic10
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — D5×Dic5 — D5×Dic10
 Lower central C52 — C5×C10 — D5×Dic10
 Upper central C1 — C2 — C4

Generators and relations for D5×Dic10
G = < a,b,c,d | a5=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 468 in 88 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, Q8, D5, C10, C10, C2×Q8, Dic5, Dic5, Dic5, C20, C20, D10, C2×C10, C52, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×C20, C5×Q8, C5×D5, C5×C10, C2×Dic10, Q8×D5, C5×Dic5, C5×Dic5, C526C4, C5×C20, D5×C10, D5×Dic5, C522Q8, C5×Dic10, D5×C20, C524Q8, D5×Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, Dic10, C22×D5, C2×Dic10, Q8×D5, D52, C2×D52, D5×Dic10

Smallest permutation representation of D5×Dic10
On 80 points
Generators in S80
(1 13 5 17 9)(2 14 6 18 10)(3 15 7 19 11)(4 16 8 20 12)(21 29 37 25 33)(22 30 38 26 34)(23 31 39 27 35)(24 32 40 28 36)(41 49 57 45 53)(42 50 58 46 54)(43 51 59 47 55)(44 52 60 48 56)(61 73 65 77 69)(62 74 66 78 70)(63 75 67 79 71)(64 76 68 80 72)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)
(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 40 11 30)(2 39 12 29)(3 38 13 28)(4 37 14 27)(5 36 15 26)(6 35 16 25)(7 34 17 24)(8 33 18 23)(9 32 19 22)(10 31 20 21)(41 79 51 69)(42 78 52 68)(43 77 53 67)(44 76 54 66)(45 75 55 65)(46 74 56 64)(47 73 57 63)(48 72 58 62)(49 71 59 61)(50 70 60 80)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R 20S 20T 20U 20V order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 20 20 20 20 20 20 size 1 1 5 5 2 10 10 10 50 50 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 10 10 10 2 2 2 2 4 ··· 4 10 10 10 10 20 20 20 20

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + + + - - + + - image C1 C2 C2 C2 C2 C2 Q8 D5 D5 D10 D10 D10 Dic10 Q8×D5 D52 C2×D52 D5×Dic10 kernel D5×Dic10 D5×Dic5 C52⋊2Q8 C5×Dic10 D5×C20 C52⋊4Q8 C5×D5 Dic10 C4×D5 Dic5 C20 D10 D5 C5 C4 C2 C1 # reps 1 2 2 1 1 1 2 2 2 6 4 2 8 2 4 4 8

Matrix representation of D5×Dic10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 6 40 0 0 35 35
,
 32 11 0 0 30 27 0 0 0 0 40 0 0 0 0 40
,
 27 28 0 0 12 14 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,6,35,0,0,40,35],[32,30,0,0,11,27,0,0,0,0,40,0,0,0,0,40],[27,12,0,0,28,14,0,0,0,0,40,0,0,0,0,40] >;

D5×Dic10 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_{10}
% in TeX

G:=Group("D5xDic10");
// GroupNames label

G:=SmallGroup(400,163);
// by ID

G=gap.SmallGroup(400,163);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,116,50,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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