Copied to
clipboard

G = C5×C22⋊C4order 80 = 24·5

Direct product of C5 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22⋊C4, C22⋊C20, C23.C10, C10.12D4, (C2×C20)⋊2C2, (C2×C10)⋊3C4, (C2×C4)⋊1C10, C2.1(C5×D4), C2.1(C2×C20), C10.17(C2×C4), (C22×C10).1C2, C22.2(C2×C10), (C2×C10).13C22, SmallGroup(80,21)

Series: Derived Chief Lower central Upper central

C1C2 — C5×C22⋊C4
C1C2C22C2×C10C2×C20 — C5×C22⋊C4
C1C2 — C5×C22⋊C4
C1C2×C10 — C5×C22⋊C4

Generators and relations for C5×C22⋊C4
 G = < a,b,c,d | a5=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

2C2
2C2
2C4
2C22
2C4
2C22
2C10
2C10
2C20
2C20
2C2×C10
2C2×C10

Smallest permutation representation of C5×C22⋊C4
On 40 points
Generators in S40
(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 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)(21 31)(22 32)(23 33)(24 34)(25 35)
(1 6 21 36)(2 7 22 37)(3 8 23 38)(4 9 24 39)(5 10 25 40)(11 26 16 31)(12 27 17 32)(13 28 18 33)(14 29 19 34)(15 30 20 35)

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

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

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

C5×C22⋊C4 is a maximal subgroup of
C23.1D10  C23.11D10  Dic5.14D4  C23.D10  Dic54D4  C22⋊D20  D10.12D4  D10⋊D4  Dic5.5D4  C22.D20  D4×C20

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B5C5D10A···10L10M···10T20A···20P
order1222224444555510···1010···1020···20
size111122222211111···12···22···2

50 irreducible representations

dim1111111122
type++++
imageC1C2C2C4C5C10C10C20D4C5×D4
kernelC5×C22⋊C4C2×C20C22×C10C2×C10C22⋊C4C2×C4C23C22C10C2
# reps12144841628

Matrix representation of C5×C22⋊C4 in GL3(𝔽41) generated by

100
0100
0010
,
100
010
0040
,
100
0400
0040
,
900
001
010
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[1,0,0,0,1,0,0,0,40],[1,0,0,0,40,0,0,0,40],[9,0,0,0,0,1,0,1,0] >;

C5×C22⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes C_4
% in TeX

G:=Group("C5xC2^2:C4");
// GroupNames label

G:=SmallGroup(80,21);
// by ID

G=gap.SmallGroup(80,21);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,200,221]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C5×C22⋊C4 in TeX

׿
×
𝔽