Copied to
clipboard

## G = C22×D52order 400 = 24·52

### Direct product of C22, D5 and D5

Aliases: C22×D52, C52⋊C24, C1025C22, C5⋊D5⋊C23, (C5×C10)⋊C23, (C5×D5)⋊C23, (C2×C10)⋊5D10, C51(C23×D5), C101(C22×D5), (D5×C10)⋊10C22, (D5×C2×C10)⋊7C2, (C2×C5⋊D5)⋊8C22, (C22×C5⋊D5)⋊6C2, SmallGroup(400,218)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C22×D52
 Chief series C1 — C5 — C52 — C5×D5 — D52 — C2×D52 — C22×D52
 Lower central C52 — C22×D52
 Upper central C1 — C22

Generators and relations for C22×D52
G = < a,b,c,d,e,f | a2=b2=c5=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1964 in 300 conjugacy classes, 104 normal (6 characteristic)
C1, C2, C2, C22, C22, C5, C5, C23, D5, D5, C10, C10, C24, D10, D10, C2×C10, C2×C10, C52, C22×D5, C22×D5, C22×C10, C5×D5, C5⋊D5, C5×C10, C23×D5, D52, D5×C10, C2×C5⋊D5, C102, C2×D52, D5×C2×C10, C22×C5⋊D5, C22×D52
Quotients: C1, C2, C22, C23, D5, C24, D10, C22×D5, C23×D5, D52, C2×D52, C22×D52

Smallest permutation representation of C22×D52
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 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 38)(2 37)(3 36)(4 40)(5 39)(6 33)(7 32)(8 31)(9 35)(10 34)(11 28)(12 27)(13 26)(14 30)(15 29)(16 23)(17 22)(18 21)(19 25)(20 24)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 20 19 18 17)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)
(1 33)(2 34)(3 35)(4 31)(5 32)(6 38)(7 39)(8 40)(9 36)(10 37)(11 23)(12 24)(13 25)(14 21)(15 22)(16 28)(17 29)(18 30)(19 26)(20 27)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 10Y ··· 10AN order 1 2 2 2 2 ··· 2 2 2 2 2 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 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 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 D5 D10 D10 D52 C2×D52 kernel C22×D52 C2×D52 D5×C2×C10 C22×C5⋊D5 C22×D5 D10 C2×C10 C22 C2 # reps 1 12 2 1 4 24 4 4 12

Matrix representation of C22×D52 in GL6(𝔽11)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 0 0 0 0 0 0 10 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
,
 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 0 10 0 0 0 0 1 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 3 0 0 0 0 0 10
,
 0 10 0 0 0 0 1 7 0 0 0 0 0 0 0 10 0 0 0 0 1 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 4 0 0 0 0 0 1 0 0 0 0 0 0 10 8 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(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,10,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,3,10],[0,1,0,0,0,0,10,7,0,0,0,0,0,0,0,1,0,0,0,0,10,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,4,1,0,0,0,0,0,0,10,0,0,0,0,0,8,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22×D52 in GAP, Magma, Sage, TeX

C_2^2\times D_5^2
% in TeX

G:=Group("C2^2xD5^2");
// GroupNames label

G:=SmallGroup(400,218);
// by ID

G=gap.SmallGroup(400,218);
# by ID

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

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

׿
×
𝔽