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G = C2×C22⋊F5order 160 = 25·5

Direct product of C2 and C22⋊F5

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

Aliases: C2×C22⋊F5, C232F5, D10.20D4, D10.14C23, (C2×F5)⋊C22, C10⋊(C22⋊C4), D108(C2×C4), D5⋊(C22⋊C4), D5.3(C2×D4), C222(C2×F5), (C22×C10)⋊4C4, (C22×D5)⋊5C4, (C22×F5)⋊2C2, (C23×D5).3C2, C2.13(C22×F5), C10.13(C22×C4), (C22×D5).39C22, C5⋊(C2×C22⋊C4), (C2×C10)⋊2(C2×C4), SmallGroup(160,212)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C22⋊F5
C1C5D5D10C2×F5C22×F5 — C2×C22⋊F5
C5C10 — C2×C22⋊F5
C1C22C23

Generators and relations for C2×C22⋊F5
 G = < a,b,c,d,e | a2=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 500 in 132 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C22, C22 [×2], C22 [×20], C5, C2×C4 [×8], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C22⋊C4 [×4], C22×C4 [×2], C24, F5 [×4], D10 [×2], D10 [×6], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4, C2×F5 [×4], C2×F5 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C22⋊F5 [×4], C22×F5 [×2], C23×D5, C2×C22⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], C22⋊F5 [×2], C22×F5, C2×C22⋊F5

Character table of C2×C22⋊F5

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H510A10B10C10D10E10F10G
 size 11112255551010101010101010101044444444
ρ11111111111111111111111111111    trivial
ρ21-1-11-11-11-111-1-1-1-11111-11-1-1-111-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ41-1-11-11-11-111-1111-1-1-1-111-1-1-111-11    linear of order 2
ρ51111-1-11111-1-11-1-111-1-1111-11-1-1-11    linear of order 2
ρ61-1-111-1-11-11-11-11111-1-1-11-11-1-1-111    linear of order 2
ρ71111-1-11111-1-1-111-1-111-111-11-1-1-11    linear of order 2
ρ81-1-111-1-11-11-111-1-1-1-11111-11-1-1-111    linear of order 2
ρ91-1-111-11-11-11-1ii-ii-i-ii-i1-11-1-1-111    linear of order 4
ρ10111111-1-1-1-1-1-1-ii-ii-ii-ii11111111    linear of order 4
ρ111-1-11-111-11-1-11-ii-i-ii-iii1-1-1-111-11    linear of order 4
ρ121111-1-1-1-1-1-111ii-i-iii-i-i11-11-1-1-11    linear of order 4
ρ131111-1-1-1-1-1-111-i-iii-i-iii11-11-1-1-11    linear of order 4
ρ141-1-11-111-11-1-11i-iii-ii-i-i1-1-1-111-11    linear of order 4
ρ15111111-1-1-1-1-1-1i-ii-ii-ii-i11111111    linear of order 4
ρ161-1-111-11-11-11-1-i-ii-iii-ii1-11-1-1-111    linear of order 4
ρ1722-2-2002-2-2200000000002-202000-2    orthogonal lifted from D4
ρ182-22-200-2-2220000000000220-2000-2    orthogonal lifted from D4
ρ1922-2-200-222-200000000002-202000-2    orthogonal lifted from D4
ρ202-22-20022-2-20000000000220-2000-2    orthogonal lifted from D4
ρ214-4-444-400000000000000-11-1111-1-1    orthogonal lifted from C2×F5
ρ2244444400000000000000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ234-4-44-4400000000000000-1111-1-11-1    orthogonal lifted from C2×F5
ρ244444-4-400000000000000-1-11-1111-1    orthogonal lifted from C2×F5
ρ2544-4-40000000000000000-115-1-55-51    orthogonal lifted from C22⋊F5
ρ264-44-40000000000000000-1-1515-5-51    orthogonal lifted from C22⋊F5
ρ274-44-40000000000000000-1-1-51-5551    orthogonal lifted from C22⋊F5
ρ2844-4-40000000000000000-11-5-15-551    orthogonal lifted from C22⋊F5

Smallest permutation representation of C2×C22⋊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 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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)
(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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (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)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(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×C22⋊F5 is a maximal subgroup of
(C22×F5)⋊C4  C22⋊F5⋊C4  C22⋊C4×F5  D10⋊(C4⋊C4)  C10.(C4×D4)  (C22×C4)⋊7F5  D106(C4⋊C4)  (C2×D4)⋊7F5  (C2×F5)⋊D4  C2.(D4×F5)  C244F5  C2×D4×F5  D10.C24
C2×C22⋊F5 is a maximal quotient of
C23⋊F55C2  D10.11M4(2)  D109M4(2)  D1010M4(2)  (C4×D5).D4  (C22×C4)⋊7F5  D106(C4⋊C4)  (D4×C10)⋊C4  (C2×D4)⋊6F5  (C2×D4)⋊7F5  (C2×D4)⋊8F5  (C2×D4).7F5  (C2×D4).8F5  (C2×D4).9F5  D5⋊(C4.D4)  (C2×F5)⋊D4  C2.(D4×F5)  (C2×Q8)⋊4F5  (C2×Q8)⋊6F5  (C2×Q8)⋊7F5  (C2×Q8).5F5  (C2×Q8).7F5  (C2×F5)⋊Q8  D5⋊C4≀C2  C4○D4⋊F5  C4○D20⋊C4  D4⋊F5⋊C2  C24.4F5  C244F5

Matrix representation of C2×C22⋊F5 in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
40390000
010000
00223803
00019383
00338190
00303822
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040100
0040010
0040001
0040000
,
900000
32320000
0000400
0040000
0000040
0004000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,0,40,0] >;

C2×C22⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes F_5
% in TeX

G:=Group("C2xC2^2:F5");
// GroupNames label

G:=SmallGroup(160,212);
// by ID

G=gap.SmallGroup(160,212);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,2309,599]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=e^4=1,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

Character table of C2×C22⋊F5 in TeX

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