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G = C8⋊F5order 160 = 25·5

3rd semidirect product of C8 and F5 acting via F5/D5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83F5, C404C4, D5.M4(2), C10.2C42, C5⋊C81C4, C52C88C4, (C2×F5).C4, C51(C8⋊C4), C2.3(C4×F5), D5⋊C8.2C2, (C4×F5).2C2, C4.17(C2×F5), C20.16(C2×C4), (C8×D5).10C2, D10.6(C2×C4), Dic5.8(C2×C4), (C4×D5).32C22, SmallGroup(160,67)

Series: Derived Chief Lower central Upper central

C1C10 — C8⋊F5
C1C5C10D10C4×D5C4×F5 — C8⋊F5
C5C10 — C8⋊F5
C1C4C8

Generators and relations for C8⋊F5
 G = < a,b,c | a8=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >

5C2
5C2
5C4
5C22
10C4
10C4
5C8
5C2×C4
5C2×C4
5C8
5C2×C4
5C8
2F5
2F5
5C42
5C2×C8
5C2×C8
5C8⋊C4

Character table of C8⋊F5

 class 12A2B2C4A4B4C4D4E4F4G4H58A8B8C8D8E8F8G8H1020A20B40A40B40C40D
 size 11551155101010104221010101010104444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-11-1-11-1111-1111-1-1-1-1    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ411111111-1-1-1-1111-11-1-1-111111111    linear of order 2
ρ511-1-1-1-111-i-iii1-ii-1-i-111i1-1-1-iii-i    linear of order 4
ρ61111-1-1-1-11-11-11-ii-iii-ii-i1-1-1-iii-i    linear of order 4
ρ711-1-1-1-111ii-i-i1i-i-1i-111-i1-1-1i-i-ii    linear of order 4
ρ811-1-1-1-111-i-iii1i-i1i1-1-1-i1-1-1i-i-ii    linear of order 4
ρ911-1-1-1-111ii-i-i1-ii1-i1-1-1i1-1-1-iii-i    linear of order 4
ρ101111-1-1-1-1-11-111i-i-i-ii-iii1-1-1i-i-ii    linear of order 4
ρ1111-1-111-1-1-iii-i111-i-1ii-i-11111111    linear of order 4
ρ1211-1-111-1-1i-i-ii1-1-1-i1ii-i1111-1-1-1-1    linear of order 4
ρ131111-1-1-1-1-11-111-iiii-ii-i-i1-1-1-iii-i    linear of order 4
ρ141111-1-1-1-11-11-11i-ii-i-ii-ii1-1-1i-i-ii    linear of order 4
ρ1511-1-111-1-1-iii-i1-1-1i1-i-ii1111-1-1-1-1    linear of order 4
ρ1611-1-111-1-1i-i-ii111i-1-i-ii-11111111    linear of order 4
ρ172-2-22-2i2i-2i2i0000200000000-22i-2i0000    complex lifted from M4(2)
ρ182-22-2-2i2i2i-2i0000200000000-22i-2i0000    complex lifted from M4(2)
ρ192-2-222i-2i2i-2i0000200000000-2-2i2i0000    complex lifted from M4(2)
ρ202-22-22i-2i-2i2i0000200000000-2-2i2i0000    complex lifted from M4(2)
ρ21440044000000-1-4-4000000-1-1-11111    orthogonal lifted from C2×F5
ρ22440044000000-144000000-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ234400-4-4000000-1-4i4i000000-111i-i-ii    complex lifted from C4×F5
ρ244400-4-4000000-14i-4i000000-111-iii-i    complex lifted from C4×F5
ρ254-4004i-4i000000-1000000001i-i8ζ54+2ζ8ζ5883ζ53+2ζ83ζ528383ζ54+2ζ83ζ58385ζ54+2ζ85ζ585    complex faithful
ρ264-400-4i4i000000-1000000001-ii83ζ53+2ζ83ζ52838ζ54+2ζ8ζ5885ζ54+2ζ85ζ58583ζ54+2ζ83ζ583    complex faithful
ρ274-400-4i4i000000-1000000001-ii83ζ54+2ζ83ζ58385ζ54+2ζ85ζ5858ζ54+2ζ8ζ5883ζ53+2ζ83ζ5283    complex faithful
ρ284-4004i-4i000000-1000000001i-i85ζ54+2ζ85ζ58583ζ54+2ζ83ζ58383ζ53+2ζ83ζ52838ζ54+2ζ8ζ58    complex faithful

Smallest permutation representation of C8⋊F5
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 26 22 9 35)(2 27 23 10 36)(3 28 24 11 37)(4 29 17 12 38)(5 30 18 13 39)(6 31 19 14 40)(7 32 20 15 33)(8 25 21 16 34)
(1 7 5 3)(2 4 6 8)(9 32 18 37)(10 29 19 34)(11 26 20 39)(12 31 21 36)(13 28 22 33)(14 25 23 38)(15 30 24 35)(16 27 17 40)

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

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

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

C8⋊F5 is a maximal subgroup of
C16⋊F5  C164F5  C20.12C42  M4(2)×F5  M4(2)⋊5F5  D40⋊C4  D8⋊F5  SD16⋊F5  SD162F5  Dic20⋊C4  Q16⋊F5  C30.3C42  C30.4C42  C24⋊F5
C8⋊F5 is a maximal quotient of
C16⋊F5  C164F5  C40⋊C8  C20.31M4(2)  D10.3M4(2)  C30.3C42  C30.4C42  C24⋊F5

Matrix representation of C8⋊F5 in GL6(𝔽41)

090000
4000000
001000
000100
000010
000001
,
100000
010000
0000040
0010040
0001040
0000140
,
100000
0400000
000010
001000
000001
000100

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

C8⋊F5 in GAP, Magma, Sage, TeX

C_8\rtimes F_5
% in TeX

G:=Group("C8:F5");
// GroupNames label

G:=SmallGroup(160,67);
// by ID

G=gap.SmallGroup(160,67);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,69,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^8=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C8⋊F5 in TeX
Character table of C8⋊F5 in TeX

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