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G = C2×F11  order 220 = 22·5·11

Direct product of C2 and F11

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C2×F11, D22⋊C5, C22⋊C10, D11⋊C10, C11⋊(C2×C10), C11⋊C5⋊C22, (C2×C11⋊C5)⋊C2, Aut(D22), Hol(C22), SmallGroup(220,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C11 — C2×F11
Chief series
C1 — C11 — C11⋊C5 — F11 — C2×F11
Lower central
C11 — C2×F11
Upper central
C1 — C2

Generators and relations for C2×F11
 G = < a,b,c | a2=b11=c10=1, ab=ba, ac=ca, cbc-1=b6 >

C1
C2
11C2
11C2
11C5
C11
11C22
11C10
11C10
11C10
D11
D11
C22
C11⋊C5
11C2×C10
D22
F11
F11
C2×C11⋊C5
C2×F11

Character table of C2×F11

 class 12A2B2C5A5B5C5D10A10B10C10D10E10F10G10H10I10J10K10L1122
 size 111111111111111111111111111111111111111010
ρ11111111111111111111111    trivial
ρ21-11-111111-1-1-1-1-1-1-1111-11-1    linear of order 2
ρ31-1-111111-1-1-11111-1-1-1-1-11-1    linear of order 2
ρ411-1-11111-111-1-1-1-11-1-1-1111    linear of order 2
ρ51-11-1ζ5ζ54ζ53ζ52ζ5-ζ5-ζ54-ζ54-ζ53-ζ52-ζ5-ζ53ζ54ζ53ζ52-ζ521-1    linear of order 10
ρ61-11-1ζ54ζ5ζ52ζ53ζ54-ζ54-ζ5-ζ5-ζ52-ζ53-ζ54-ζ52ζ5ζ52ζ53-ζ531-1    linear of order 10
ρ71-1-11ζ53ζ52ζ54ζ5-ζ53-ζ53-ζ52ζ52ζ54ζ5ζ53-ζ54-ζ52-ζ54-ζ5-ζ51-1    linear of order 10
ρ81-11-1ζ53ζ52ζ54ζ5ζ53-ζ53-ζ52-ζ52-ζ54-ζ5-ζ53-ζ54ζ52ζ54ζ5-ζ51-1    linear of order 10
ρ91-1-11ζ54ζ5ζ52ζ53-ζ54-ζ54-ζ5ζ5ζ52ζ53ζ54-ζ52-ζ5-ζ52-ζ53-ζ531-1    linear of order 10
ρ1011-1-1ζ5ζ54ζ53ζ52-ζ5ζ5ζ54-ζ54-ζ53-ζ52-ζ5ζ53-ζ54-ζ53-ζ52ζ5211    linear of order 10
ρ111111ζ5ζ54ζ53ζ52ζ5ζ5ζ54ζ54ζ53ζ52ζ5ζ53ζ54ζ53ζ52ζ5211    linear of order 5
ρ1211-1-1ζ52ζ53ζ5ζ54-ζ52ζ52ζ53-ζ53-ζ5-ζ54-ζ52ζ5-ζ53-ζ5-ζ54ζ5411    linear of order 10
ρ1311-1-1ζ53ζ52ζ54ζ5-ζ53ζ53ζ52-ζ52-ζ54-ζ5-ζ53ζ54-ζ52-ζ54-ζ5ζ511    linear of order 10
ρ141111ζ54ζ5ζ52ζ53ζ54ζ54ζ5ζ5ζ52ζ53ζ54ζ52ζ5ζ52ζ53ζ5311    linear of order 5
ρ151111ζ52ζ53ζ5ζ54ζ52ζ52ζ53ζ53ζ5ζ54ζ52ζ5ζ53ζ5ζ54ζ5411    linear of order 5
ρ161111ζ53ζ52ζ54ζ5ζ53ζ53ζ52ζ52ζ54ζ5ζ53ζ54ζ52ζ54ζ5ζ511    linear of order 5
ρ171-1-11ζ52ζ53ζ5ζ54-ζ52-ζ52-ζ53ζ53ζ5ζ54ζ52-ζ5-ζ53-ζ5-ζ54-ζ541-1    linear of order 10
ρ1811-1-1ζ54ζ5ζ52ζ53-ζ54ζ54ζ5-ζ5-ζ52-ζ53-ζ54ζ52-ζ5-ζ52-ζ53ζ5311    linear of order 10
ρ191-1-11ζ5ζ54ζ53ζ52-ζ5-ζ5-ζ54ζ54ζ53ζ52ζ5-ζ53-ζ54-ζ53-ζ52-ζ521-1    linear of order 10
ρ201-11-1ζ52ζ53ζ5ζ54ζ52-ζ52-ζ53-ζ53-ζ5-ζ54-ζ52-ζ5ζ53ζ5ζ54-ζ541-1    linear of order 10
ρ211010000000000000000000-1-1    orthogonal lifted from F11
ρ2210-10000000000000000000-11    orthogonal faithful

Permutation representations of C2×F11
►On 22 points - transitive group 22T6
Generators in S22
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)

(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)

(1 12)(2 14 5 20 6 22 10 19 4 18)(3 16 9 17 11 21 8 15 7 13)
 
G:=sub<Sym(22)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,12)(2,14,5,20,6,22,10,19,4,18)(3,16,9,17,11,21,8,15,7,13)>;
 
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,12)(2,14,5,20,6,22,10,19,4,18)(3,16,9,17,11,21,8,15,7,13) );
 
G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22)], [(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,12),(2,14,5,20,6,22,10,19,4,18),(3,16,9,17,11,21,8,15,7,13)]])
 
G:=TransitiveGroup(22,6);
 

C2×F11 is a maximal subgroup of   D44⋊C5  C22⋊F11
C2×F11 is a maximal quotient of   C4.F11  D44⋊C5  C22⋊F11

Matrix representation of C2×F11 ►in GL10(ℤ)

-1000000000
0-100000000
00-10000000
000-1000000
0000-100000
00000-10000
000000-1000
0000000-100
00000000-10
000000000-1
,
-1-1-1-1-1-1-1-1-1-1
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
,
1000000000
0000001000
0100000000
0000000100
0010000000
0000000010
0001000000
0000000001
0000100000
-1-1-1-1-1-1-1-1-1-1

G:=sub<GL(10,Integers())| [-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1] >;
 

C2×F11 in GAP, Magma, Sage, TeX 

C_2\times F_{11}
 
% in TeX
 
G:=Group("C2xF11");
 
// GroupNames label
 
G:=SmallGroup(220,7);
 
// by ID
 
G=gap.SmallGroup(220,7);
 
# by ID
 
G:=PCGroup([4,-2,-2,-5,-11,3203,731]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^11=c^10=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C2×F11 in TeX
Character table of C2×F11 in TeX

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