Copied to
clipboard

## G = Dic52order 400 = 24·52

### Direct product of Dic5 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Dic52
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C10×Dic5 — Dic52
 Lower central C52 — Dic52
 Upper central C1 — C22

Generators and relations for Dic52
G = < a,b,c,d | a10=c10=1, b2=a5, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 316 in 76 conjugacy classes, 36 normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C10, C10, C42, Dic5, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×Dic5, C2×C20, C5×C10, C5×C10, C4×Dic5, C5×Dic5, C526C4, C102, C10×Dic5, C2×C526C4, Dic52
Quotients: C1, C2, C4, C22, C2×C4, D5, C42, Dic5, D10, C4×D5, C2×Dic5, C4×Dic5, D52, D5×Dic5, Dic52D5, Dic52

Smallest permutation representation of Dic52
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 46 6 41)(2 45 7 50)(3 44 8 49)(4 43 9 48)(5 42 10 47)(11 54 16 59)(12 53 17 58)(13 52 18 57)(14 51 19 56)(15 60 20 55)(21 64 26 69)(22 63 27 68)(23 62 28 67)(24 61 29 66)(25 70 30 65)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
(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 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4H 4I 4J 4K 4L 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 5 ··· 5 25 25 25 25 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + - + + - + image C1 C2 C2 C4 C4 D5 Dic5 D10 C4×D5 D52 D5×Dic5 Dic5⋊2D5 kernel Dic52 C10×Dic5 C2×C52⋊6C4 C5×Dic5 C52⋊6C4 C2×Dic5 Dic5 C2×C10 C10 C22 C2 C2 # reps 1 2 1 8 4 4 8 4 16 4 8 4

Matrix representation of Dic52 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 0 40 0 0 1 34
,
 32 0 0 0 0 32 0 0 0 0 0 40 0 0 40 0
,
 34 40 0 0 1 0 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 7 1 0 0 0 0 32 0 0 0 0 32
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,34],[32,0,0,0,0,32,0,0,0,0,0,40,0,0,40,0],[34,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[40,7,0,0,0,1,0,0,0,0,32,0,0,0,0,32] >;

Dic52 in GAP, Magma, Sage, TeX

{\rm Dic}_5^2
% in TeX

G:=Group("Dic5^2");
// GroupNames label

G:=SmallGroup(400,71);
// by ID

G=gap.SmallGroup(400,71);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,31,970,11525]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽