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G = C22.F5order 80 = 24·5

The non-split extension by C22 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.F5, C52M4(2), Dic5.3C4, Dic5.7C22, C5⋊C82C2, C2.6(C2×F5), C10.6(C2×C4), (C2×C10).2C4, (C2×Dic5).5C2, SmallGroup(80,33)

Series: Derived Chief Lower central Upper central

C1C10 — C22.F5
C1C5C10Dic5C5⋊C8 — C22.F5
C5C10 — C22.F5
C1C2C22

Generators and relations for C22.F5
 G = < a,b,c,d | a2=b2=c5=1, d4=b, dad-1=ab=ba, ac=ca, bc=cb, bd=db, dcd-1=c3 >

2C2
5C4
5C4
2C10
5C8
5C2×C4
5C8
5M4(2)

Character table of C22.F5

 class 12A2B4A4B4C58A8B8C8D10A10B10C
 size 1125510410101010444
ρ111111111111111    trivial
ρ211-111-11-11-11-11-1    linear of order 2
ρ311-111-111-11-1-11-1    linear of order 2
ρ41111111-1-1-1-1111    linear of order 2
ρ511-1-1-111-iii-i-11-1    linear of order 4
ρ6111-1-1-11ii-i-i111    linear of order 4
ρ7111-1-1-11-i-iii111    linear of order 4
ρ811-1-1-111i-i-ii-11-1    linear of order 4
ρ92-20-2i2i0200000-20    complex lifted from M4(2)
ρ102-202i-2i0200000-20    complex lifted from M4(2)
ρ1144-4000-100001-11    orthogonal lifted from C2×F5
ρ12444000-10000-1-1-1    orthogonal lifted from F5
ρ134-40000-10000-515    symplectic faithful, Schur index 2
ρ144-40000-1000051-5    symplectic faithful, Schur index 2

Smallest permutation representation of C22.F5
On 40 points
Generators in S40
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(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 11 20 27 35)(2 28 12 36 21)(3 37 29 22 13)(4 23 38 14 30)(5 15 24 31 39)(6 32 16 40 17)(7 33 25 18 9)(8 19 34 10 26)
(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)| (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(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), (1,11,20,27,35)(2,28,12,36,21)(3,37,29,22,13)(4,23,38,14,30)(5,15,24,31,39)(6,32,16,40,17)(7,33,25,18,9)(8,19,34,10,26), (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( (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(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), (1,11,20,27,35)(2,28,12,36,21)(3,37,29,22,13)(4,23,38,14,30)(5,15,24,31,39)(6,32,16,40,17)(7,33,25,18,9)(8,19,34,10,26), (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([(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(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)], [(1,11,20,27,35),(2,28,12,36,21),(3,37,29,22,13),(4,23,38,14,30),(5,15,24,31,39),(6,32,16,40,17),(7,33,25,18,9),(8,19,34,10,26)], [(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)])

C22.F5 is a maximal subgroup of
Dic5.D4  C23.F5  D5⋊M4(2)  D4.F5  D6.F5  C158M4(2)  C25⋊M4(2)  D10.F5  C524M4(2)  C102.C4  C5213M4(2)  C5214M4(2)  C22.S5
C22.F5 is a maximal quotient of
C10.C42  Dic5⋊C8  C23.2F5  D6.F5  C158M4(2)  C25⋊M4(2)  D10.F5  C524M4(2)  C102.C4  C5213M4(2)  C5214M4(2)

Matrix representation of C22.F5 in GL4(𝔽41) generated by

1000
0100
00400
00040
,
40000
04000
00400
00040
,
64000
1000
003535
00640
,
0010
0001
23600
211800
G:=sub<GL(4,GF(41))| [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],[6,1,0,0,40,0,0,0,0,0,35,6,0,0,35,40],[0,0,23,21,0,0,6,18,1,0,0,0,0,1,0,0] >;

C22.F5 in GAP, Magma, Sage, TeX

C_2^2.F_5
% in TeX

G:=Group("C2^2.F5");
// GroupNames label

G:=SmallGroup(80,33);
// by ID

G=gap.SmallGroup(80,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,42,804,414]);
// Polycyclic

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

Export

Subgroup lattice of C22.F5 in TeX
Character table of C22.F5 in TeX

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