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## G = Dic5⋊Dic5order 400 = 24·52

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5⋊Dic5
G = < a,b,c,d | a10=c10=1, b2=a5, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 316 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C4, C22, C5, C5, C2×C4, C10, C10, C4⋊C4, Dic5, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×Dic5, C2×C20, C5×C10, C10.D4, C4⋊Dic5, C5×Dic5, C5×Dic5, C526C4, C102, C10×Dic5, C2×C526C4, Dic5⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C4⋊C4, Dic5, D10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C10.D4, C4⋊Dic5, D52, D5×Dic5, C5⋊D20, C522Q8, Dic5⋊Dic5

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

58 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + - + - + + - + - image C1 C2 C2 C4 D4 Q8 D5 Dic5 D10 Dic10 C4×D5 D20 C5⋊D4 D52 D5×Dic5 C5⋊D20 C52⋊2Q8 kernel Dic5⋊Dic5 C10×Dic5 C2×C52⋊6C4 C5×Dic5 C5×C10 C5×C10 C2×Dic5 Dic5 C2×C10 C10 C10 C10 C10 C22 C2 C2 C2 # reps 1 2 1 4 1 1 4 4 4 8 4 4 4 4 4 4 4

Matrix representation of Dic5⋊Dic5 in GL6(𝔽41)

 0 1 0 0 0 0 40 35 0 0 0 0 0 0 0 1 0 0 0 0 40 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 10 0 0 0 0 27 37 0 0 0 0 0 0 4 10 0 0 0 0 27 37 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 34
,
 39 13 0 0 0 0 28 2 0 0 0 0 0 0 18 6 0 0 0 0 35 23 0 0 0 0 0 0 0 40 0 0 0 0 40 0

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,27,0,0,0,0,10,37,0,0,0,0,0,0,4,27,0,0,0,0,10,37,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,34],[39,28,0,0,0,0,13,2,0,0,0,0,0,0,18,35,0,0,0,0,6,23,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

Dic5⋊Dic5 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(400,74);
// by ID

G=gap.SmallGroup(400,74);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,121,31,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^10=1,b^2=a^5,d^2=c^5,b*a*b^-1=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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