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G = C2xC4xF5order 160 = 25·5

Direct product of C2xC4 and F5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2xC4xF5, C10:C42, D5:C42, D10.9C23, C5:(C2xC42), C20:3(C2xC4), (C2xC20):5C4, (C4xD5):5C4, D5.(C22xC4), (C2xDic5):7C4, Dic5:6(C2xC4), C2.2(C22xF5), D10.15(C2xC4), C10.4(C22xC4), (C2xF5).5C22, (C22xF5).3C2, C22.17(C2xF5), (C4xD5).35C22, (C22xD5).36C22, (C2xC4xD5).18C2, (C2xC10).16(C2xC4), SmallGroup(160,203)

Series: Derived Chief Lower central Upper central

C1C5 — C2xC4xF5
C1C5D5D10C2xF5C22xF5 — C2xC4xF5
C5 — C2xC4xF5
C1C2xC4

Generators and relations for C2xC4xF5
 G = < a,b,c,d | a2=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 292 in 108 conjugacy classes, 62 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C23, D5, C10, C10, C42, C22xC4, Dic5, C20, F5, D10, D10, C2xC10, C2xC42, C4xD5, C2xDic5, C2xC20, C2xF5, C22xD5, C4xF5, C2xC4xD5, C22xF5, C2xC4xF5
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, C22xC4, F5, C2xC42, C2xF5, C4xF5, C22xF5, C2xC4xF5

Smallest permutation representation of C2xC4xF5
On 40 points
Generators in S40
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(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 31 6 36)(2 33 10 39)(3 35 9 37)(4 32 8 40)(5 34 7 38)(11 26 16 21)(12 28 20 24)(13 30 19 22)(14 27 18 25)(15 29 17 23)

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

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

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

C2xC4xF5 is a maximal subgroup of
C20.C42  D10.3M4(2)  M4(2):3F5  C42:4F5  C10.(C4xD4)  C4:C4:5F5  C20:(C4:C4)  C2.(D4xF5)  (C2xF5):Q8  D5:C4wrC2
C2xC4xF5 is a maximal quotient of
C42.5F5  C42:4F5  Dic5.C42  D10.C42  C20.12C42  M4(2):5F5

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4X 5 10A10B10C20A20B20C20D
order1222222244444···4510101020202020
size1111555511115···544444444

40 irreducible representations

dim111111114444
type+++++++
imageC1C2C2C2C4C4C4C4F5C2xF5C2xF5C4xF5
kernelC2xC4xF5C4xF5C2xC4xD5C22xF5C4xD5C2xDic5C2xC20C2xF5C2xC4C4C22C2
# reps1412422161214

Matrix representation of C2xC4xF5 in GL5(F41)

400000
01000
00100
00010
00001
,
400000
032000
003200
000320
000032
,
10000
000040
010040
001040
000140
,
90000
00090
09000
00009
00900

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,9,0,0,0,0,0,0,9,0] >;

C2xC4xF5 in GAP, Magma, Sage, TeX

C_2\times C_4\times F_5
% in TeX

G:=Group("C2xC4xF5");
// GroupNames label

G:=SmallGroup(160,203);
// by ID

G=gap.SmallGroup(160,203);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,2309,599]);
// Polycyclic

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

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