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G = C2×C4⋊F5order 160 = 25·5

Direct product of C2 and C4⋊F5

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

Aliases: C2×C4⋊F5, D10.4Q8, D10.11D4, D10.10C23, C10⋊(C4⋊C4), D5⋊(C4⋊C4), (C2×C4)⋊3F5, C42(C2×F5), C202(C2×C4), (C2×C20)⋊3C4, (C4×D5)⋊4C4, D5.1(C2×D4), D5.1(C2×Q8), Dic57(C2×C4), (C2×Dic5)⋊8C4, C2.6(C22×F5), D10.16(C2×C4), C10.5(C22×C4), (C22×F5).2C2, (C2×F5).1C22, C22.18(C2×F5), (C4×D5).30C22, (C22×D5).37C22, C5⋊(C2×C4⋊C4), (C2×C4×D5).14C2, (C2×C10).17(C2×C4), SmallGroup(160,204)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C4⋊F5
C1C5D5D10C2×F5C22×F5 — C2×C4⋊F5
C5C10 — C2×C4⋊F5
C1C22C2×C4

Generators and relations for C2×C4⋊F5
 G = < a,b,c,d | a2=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 292 in 92 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C2×C4, C2×C4 [×13], C23, D5 [×2], D5 [×2], C10, C10 [×2], C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×C4⋊C4, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C4⋊F5 [×4], C2×C4×D5, C22×F5 [×2], C2×C4⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×F5 [×3], C4⋊F5 [×2], C22×F5, C2×C4⋊F5

Character table of C2×C4⋊F5

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L510A10B10C20A20B20C20D
 size 11115555221010101010101010101044444444
ρ11111111111111111111111111111    trivial
ρ211-1-1-111-11-11-1111-1-1-1-111-11-11-11-1    linear of order 2
ρ311111111-1-11-1-1-1-111-1-111111-1-1-1-1    linear of order 2
ρ411-1-1-111-1-1111-1-1-1-1-11111-11-1-11-11    linear of order 2
ρ511-1-1-111-1-11-11-11111-1-1-11-11-1-11-11    linear of order 2
ρ611111111-1-1-1-1-111-1-111-11111-1-1-1-1    linear of order 2
ρ71111111111-111-1-1-1-1-1-1-111111111    linear of order 2
ρ811-1-1-111-11-1-1-11-1-11111-11-11-11-11-1    linear of order 2
ρ911-1-11-1-11-11i-11i-ii-i-ii-i1-11-1-11-11    linear of order 4
ρ1011-1-11-1-111-1-i1-1i-i-ii-iii1-11-11-11-1    linear of order 4
ρ111111-1-1-1-111-i-1-1i-ii-ii-ii11111111    linear of order 4
ρ121111-1-1-1-1-1-1i11i-i-iii-i-i1111-1-1-1-1    linear of order 4
ρ131111-1-1-1-1-1-1-i11-iii-i-iii1111-1-1-1-1    linear of order 4
ρ141111-1-1-1-111i-1-1-ii-ii-ii-i11111111    linear of order 4
ρ1511-1-11-1-111-1i1-1-iii-ii-i-i1-11-11-11-1    linear of order 4
ρ1611-1-11-1-11-11-i-11-ii-iii-ii1-11-1-11-11    linear of order 4
ρ172-2-22-2-22200000000000022-2-20000    orthogonal lifted from D4
ρ182-22-22-22-20000000000002-2-220000    orthogonal lifted from D4
ρ192-22-2-22-220000000000002-2-220000    symplectic lifted from Q8, Schur index 2
ρ202-2-2222-2-200000000000022-2-20000    symplectic lifted from Q8, Schur index 2
ρ2144-4-400004-40000000000-11-11-11-11    orthogonal lifted from C2×F5
ρ2244440000440000000000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2344440000-4-40000000000-1-1-1-11111    orthogonal lifted from C2×F5
ρ2444-4-40000-440000000000-11-111-11-1    orthogonal lifted from C2×F5
ρ254-4-440000000000000000-1-111-5-5--5--5    complex lifted from C4⋊F5
ρ264-4-440000000000000000-1-111--5--5-5-5    complex lifted from C4⋊F5
ρ274-44-40000000000000000-111-1-5--5--5-5    complex lifted from C4⋊F5
ρ284-44-40000000000000000-111-1--5-5-5--5    complex lifted from C4⋊F5

Smallest permutation representation of C2×C4⋊F5
On 40 points
Generators in S40
(1 24)(2 25)(3 21)(4 22)(5 23)(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 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 21 16 26)(12 22 17 27)(13 23 18 28)(14 24 19 29)(15 25 20 30)
(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)
(2 3 5 4)(6 8 7 10)(11 18 12 20)(13 17 15 16)(14 19)(21 23 22 25)(26 28 27 30)(31 38 32 40)(33 37 35 36)(34 39)

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

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

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

C2×C4⋊F5 is a maximal subgroup of
D10.18D8  D10.10D8  M4(2)⋊F5  C428F5  C429F5  D10⋊(C4⋊C4)  C10.(C4×D4)  C4⋊C4×F5  C4⋊C45F5  C20⋊(C4⋊C4)  M4(2)⋊1F5  D106(C4⋊C4)  C2.(D4×F5)  (C2×F5)⋊Q8  C4○D4⋊F5  C2×D4×F5  C2×Q8×F5  D5.2+ 1+4
C2×C4⋊F5 is a maximal quotient of
C42.11F5  C203M4(2)  C42.14F5  C428F5  C429F5  C20⋊C8⋊C2  D10⋊(C4⋊C4)  C4⋊C4.9F5  C20⋊(C4⋊C4)  (C2×C8)⋊6F5  (C8×D5).C4  M4(2)⋊1F5  M4(2).1F5  C208M4(2)  D106(C4⋊C4)

Matrix representation of C2×C4⋊F5 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
3550000
960000
003402727
00147140
00014714
002727034
,
100000
010000
000100
000010
000001
0040404040
,
900000
38320000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,9,0,0,0,0,5,6,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[9,38,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C2×C4⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes F_5
% in TeX

G:=Group("C2xC4:F5");
// GroupNames label

G:=SmallGroup(160,204);
// by ID

G=gap.SmallGroup(160,204);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,2309,599]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of C2×C4⋊F5 in TeX

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