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G = C2×C11⋊C5  order 110 = 2·5·11

Direct product of C2 and C11⋊C5

direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C2×C11⋊C5, C22⋊C5, C11⋊2C10, SmallGroup(110,2)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×C11⋊C5
 G = < a,b,c | a2=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

C1
C2
11C5
C11
11C10
C22
C11⋊C5
C2×C11⋊C5

Character table of C2×C11⋊C5

 class 125A5B5C5D10A10B10C10D11A11B22A22B
 size 1111111111111111115555
ρ111111111111111    trivial
ρ21-11111-1-1-1-111-1-1    linear of order 2
ρ31-1ζ52ζ53ζ5ζ54-ζ52-ζ53-ζ5-ζ5411-1-1    linear of order 10
ρ411ζ53ζ52ζ54ζ5ζ53ζ52ζ54ζ51111    linear of order 5
ρ51-1ζ54ζ5ζ52ζ53-ζ54-ζ5-ζ52-ζ5311-1-1    linear of order 10
ρ611ζ5ζ54ζ53ζ52ζ5ζ54ζ53ζ521111    linear of order 5
ρ71-1ζ5ζ54ζ53ζ52-ζ5-ζ54-ζ53-ζ5211-1-1    linear of order 10
ρ811ζ52ζ53ζ5ζ54ζ52ζ53ζ5ζ541111    linear of order 5
ρ911ζ54ζ5ζ52ζ53ζ54ζ5ζ52ζ531111    linear of order 5
ρ101-1ζ53ζ52ζ54ζ5-ζ53-ζ52-ζ54-ζ511-1-1    linear of order 10
ρ115500000000-1+√-11/2-1-√-11/2-1-√-11/2-1+√-11/2    complex lifted from C11⋊C5
ρ125-500000000-1-√-11/2-1+√-11/21-√-11/21+√-11/2    complex faithful
ρ135-500000000-1+√-11/2-1-√-11/21+√-11/21-√-11/2    complex faithful
ρ145500000000-1-√-11/2-1+√-11/2-1+√-11/2-1-√-11/2    complex lifted from C11⋊C5

Permutation representations of C2×C11⋊C5
►On 22 points - transitive group 22T5
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)

(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)
 
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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)>;
 
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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18) );
 
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)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18)]])
 
G:=TransitiveGroup(22,5);
 

C2×C11⋊C5 is a maximal subgroup of   C11⋊C20

Matrix representation of C2×C11⋊C5 ►in GL5(𝔽3)

20000
02000
00200
00020
00002
,
10002
00121
10022
10011
22020
,
10012
00112
00001
02011
00011

G:=sub<GL(5,GF(3))| [2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,1,1,2,0,0,0,0,2,0,1,0,0,0,0,2,2,1,2,2,1,2,1,0],[1,0,0,0,0,0,0,0,2,0,0,1,0,0,0,1,1,0,1,1,2,2,1,1,1] >;
 

C2×C11⋊C5 in GAP, Magma, Sage, TeX 

C_2\times C_{11}\rtimes C_5
 
% in TeX
 
G:=Group("C2xC11:C5");
 
// GroupNames label
 
G:=SmallGroup(110,2);
 
// by ID
 
G=gap.SmallGroup(110,2);
 
# by ID
 
G:=PCGroup([3,-2,-5,-11,185]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C11⋊C5 in TeX
Character table of C2×C11⋊C5 in TeX

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