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## G = D5.Dic10order 400 = 24·52

### The non-split extension by D5 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D5.Dic10
 Chief series C1 — C5 — C52 — C5×D5 — D5×C10 — C10×F5 — D5.Dic10
 Lower central C52 — C5×C10 — D5.Dic10
 Upper central C1 — C2

Generators and relations for D5.Dic10
G = < a,b,c,d | a5=b2=c20=1, d2=a-1bc10, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1bc-1 >

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 10F 10G 10H 10I 20A ··· 20H 20I 20J order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 size 1 1 5 5 10 10 10 50 50 50 2 2 4 8 8 2 2 4 8 8 10 10 10 10 10 ··· 10 20 20

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 8 8 type + + + + + - + + - + + + - image C1 C2 C2 C2 C4 C4 D4 Q8 D5 D10 Dic10 C5⋊D4 C4×D5 F5 C2×F5 C4⋊F5 D5×F5 D5.Dic10 kernel D5.Dic10 D5×Dic5 C10×F5 C2×D5.D5 C5×Dic5 C52⋊6C4 C5×D5 C5×D5 C2×F5 D10 D5 D5 C10 Dic5 C10 C5 C2 C1 # reps 1 1 1 1 2 2 1 1 2 2 4 4 4 1 1 2 2 2

Matrix representation of D5.Dic10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 40 40 40 40 0 0 0 0 1 0 0 0 1 0 0 0
,
 25 39 0 0 0 0 2 13 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 1 0 0 0 0 0 0 1 0
,
 20 23 0 0 0 0 20 21 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,1,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,1,0,0,0,40,0,0,0,0,0,40,1,0,0,0,1,40,0,0],[25,2,0,0,0,0,39,13,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0],[20,20,0,0,0,0,23,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D5.Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("D5.Dic10");
// GroupNames label

G:=SmallGroup(400,119);
// by ID

G=gap.SmallGroup(400,119);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,970,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^20=1,d^2=a^-1*b*c^10,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

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