Copied to
clipboard

G = C5xC22.F5order 400 = 24·52

Direct product of C5 and C22.F5

direct product, metabelian, supersoluble, monomial

Aliases: C5xC22.F5, C102.2C4, Dic5.3C20, C52:11M4(2), C5:C8:2C10, C22.(C5xF5), (C2xC10).9F5, C2.6(C10xF5), C10.6(C2xC20), (C2xC10).2C20, C5:2(C5xM4(2)), C10.47(C2xF5), (C5xDic5).11C4, (C2xDic5).5C10, Dic5.8(C2xC10), (C10xDic5).10C2, (C5xDic5).13C22, (C5xC5:C8):6C2, (C5xC10).18(C2xC4), SmallGroup(400,140)

Series: Derived Chief Lower central Upper central

C1C10 — C5xC22.F5
C1C5C10Dic5C5xDic5C5xC5:C8 — C5xC22.F5
C5C10 — C5xC22.F5
C1C10C2xC10

Generators and relations for C5xC22.F5
 G = < a,b,c,d,e | a5=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 112 in 45 conjugacy classes, 24 normal (20 characteristic)
Quotients: C1, C2, C4, C22, C5, C2xC4, C10, M4(2), C20, F5, C2xC10, C2xC20, C2xF5, C5xM4(2), C22.F5, C5xF5, C10xF5, C5xC22.F5
2C2
4C5
5C4
5C4
2C10
2C10
4C10
4C10
4C10
5C2xC4
5C8
5C8
4C2xC10
5C20
5C20
2C5xC10
5M4(2)
5C40
5C40
5C2xC20
5C5xM4(2)

Smallest permutation representation of C5xC22.F5
On 40 points
Generators in S40
(1 22 35 26 9)(2 23 36 27 10)(3 24 37 28 11)(4 17 38 29 12)(5 18 39 30 13)(6 19 40 31 14)(7 20 33 32 15)(8 21 34 25 16)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)
(1 35 9 22 26)(2 23 36 27 10)(3 28 24 11 37)(4 12 29 38 17)(5 39 13 18 30)(6 19 40 31 14)(7 32 20 15 33)(8 16 25 34 21)
(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)

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

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

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

70 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D5E···5I8A8B8C8D10A10B10C10D10E10F10G10H10I···10W20A···20H20I20J20K20L40A···40P
order12244455555···58888101010101010101010···1020···202020202040···40
size112551011114···410101010111122224···45···51010101010···10

70 irreducible representations

dim111111111122444444
type+++++-
imageC1C2C2C4C4C5C10C10C20C20M4(2)C5xM4(2)F5C2xF5C22.F5C5xF5C10xF5C5xC22.F5
kernelC5xC22.F5C5xC5:C8C10xDic5C5xDic5C102C22.F5C5:C8C2xDic5Dic5C2xC10C52C5C2xC10C10C5C22C2C1
# reps121224848828112448

Matrix representation of C5xC22.F5 in GL4(F41) generated by

37000
03700
00370
00037
,
1000
0100
00400
00040
,
40000
04000
00400
00040
,
16000
71800
00100
003337
,
0010
0001
321500
0900
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[16,7,0,0,0,18,0,0,0,0,10,33,0,0,0,37],[0,0,32,0,0,0,15,9,1,0,0,0,0,1,0,0] >;

C5xC22.F5 in GAP, Magma, Sage, TeX

C_5\times C_2^2.F_5
% in TeX

G:=Group("C5xC2^2.F5");
// GroupNames label

G:=SmallGroup(400,140);
// by ID

G=gap.SmallGroup(400,140);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,69,5765,599]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

Export

Subgroup lattice of C5xC22.F5 in TeX

׿
x
:
Z
F
o
wr
Q
<