Copied to
clipboard

## G = C2×Dic5⋊2D5order 400 = 24·52

### Direct product of C2 and Dic5⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×Dic5⋊2D5
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — Dic5⋊2D5 — C2×Dic5⋊2D5
 Lower central C52 — C2×Dic5⋊2D5
 Upper central C1 — C22

Generators and relations for C2×Dic52D5
G = < a,b,c,d,e | a2=b10=d5=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 908 in 140 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22×C4, Dic5, C20, D10, C2×C10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C22×D5, C5⋊D5, C5×C10, C5×C10, C2×C4×D5, C5×Dic5, C2×C5⋊D5, C102, Dic52D5, C10×Dic5, C22×C5⋊D5, C2×Dic52D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, C2×C4×D5, D52, Dic52D5, C2×D52, C2×Dic52D5

Smallest permutation representation of C2×Dic52D5
On 40 points
Generators in S40
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(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)
(1 24 6 29)(2 23 7 28)(3 22 8 27)(4 21 9 26)(5 30 10 25)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)
(1 5 9 3 7)(2 6 10 4 8)(11 15 19 13 17)(12 16 20 14 18)(21 27 23 29 25)(22 28 24 30 26)(31 37 33 39 35)(32 38 34 40 36)
(1 7)(2 6)(3 5)(8 10)(11 17)(12 16)(13 15)(18 20)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)

G:=sub<Sym(40)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,24,6,29)(2,23,7,28)(3,22,8,27)(4,21,9,26)(5,30,10,25)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35), (1,5,9,3,7)(2,6,10,4,8)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,37,33,39,35)(32,38,34,40,36), (1,7)(2,6)(3,5)(8,10)(11,17)(12,16)(13,15)(18,20)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)>;

G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,24,6,29)(2,23,7,28)(3,22,8,27)(4,21,9,26)(5,30,10,25)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35), (1,5,9,3,7)(2,6,10,4,8)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,37,33,39,35)(32,38,34,40,36), (1,7)(2,6)(3,5)(8,10)(11,17)(12,16)(13,15)(18,20)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37) );

G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)], [(1,24,6,29),(2,23,7,28),(3,22,8,27),(4,21,9,26),(5,30,10,25),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35)], [(1,5,9,3,7),(2,6,10,4,8),(11,15,19,13,17),(12,16,20,14,18),(21,27,23,29,25),(22,28,24,30,26),(31,37,33,39,35),(32,38,34,40,36)], [(1,7),(2,6),(3,5),(8,10),(11,17),(12,16),(13,15),(18,20),(22,30),(23,29),(24,28),(25,27),(32,40),(33,39),(34,38),(35,37)]])

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 20A ··· 20P order 1 2 2 2 2 2 2 2 4 ··· 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 25 25 25 25 5 ··· 5 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 C2 C4 D5 D10 D10 C4×D5 D52 Dic5⋊2D5 C2×D52 kernel C2×Dic5⋊2D5 Dic5⋊2D5 C10×Dic5 C22×C5⋊D5 C2×C5⋊D5 C2×Dic5 Dic5 C2×C10 C10 C22 C2 C2 # reps 1 4 2 1 8 4 8 4 16 4 8 4

Matrix representation of C2×Dic52D5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 1 0 0 0 0 40 0 0 0 0 0 0 0 6 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 0 0 0 0 0 19 32 0 0 0 0 0 0 1 0 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 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 6 40 0 0 0 0 1 0
,
 1 0 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 6 40 0 0 0 0 0 0 6 40 0 0 0 0 35 35

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,40,0,0,0,0,1,0,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,19,0,0,0,0,0,32,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,40,35] >;

C2×Dic52D5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5\rtimes_2D_5
% in TeX

G:=Group("C2xDic5:2D5");
// GroupNames label

G:=SmallGroup(400,175);
// by ID

G=gap.SmallGroup(400,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,55,970,11525]);
// Polycyclic

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

׿
×
𝔽