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G = C23⋊C11  order 253 = 11·23

The semidirect product of C23 and C11 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 11-hyperelementary

Aliases: C23⋊C11, SmallGroup(253,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C23⋊C11
Chief series
C1 — C23 — C23⋊C11
Lower central
C23 — C23⋊C11
Upper central
C1

Generators and relations for C23⋊C11
 G = < a,b | a23=b11=1, bab-1=a12 >

C1
23C11
C23
C23⋊C11

Character table of C23⋊C11

 class 111A11B11C11D11E11F11G11H11I11J23A23B
 size 1232323232323232323231111
ρ11111111111111    trivial
ρ21ζ117ζ118ζ11ζ115ζ119ζ112ζ116ζ1110ζ113ζ11411    linear of order 11
ρ31ζ112ζ117ζ115ζ113ζ11ζ1110ζ118ζ116ζ114ζ11911    linear of order 11
ρ41ζ118ζ116ζ119ζ11ζ114ζ117ζ1110ζ112ζ115ζ11311    linear of order 11
ρ51ζ115ζ11ζ117ζ112ζ118ζ113ζ119ζ114ζ1110ζ11611    linear of order 11
ρ61ζ113ζ115ζ112ζ1110ζ117ζ114ζ11ζ119ζ116ζ11811    linear of order 11
ρ71ζ116ζ1110ζ114ζ119ζ113ζ118ζ112ζ117ζ11ζ11511    linear of order 11
ρ81ζ114ζ113ζ1110ζ116ζ112ζ119ζ115ζ11ζ118ζ11711    linear of order 11
ρ91ζ11ζ119ζ118ζ117ζ116ζ115ζ114ζ113ζ112ζ111011    linear of order 11
ρ101ζ119ζ114ζ116ζ118ζ1110ζ11ζ113ζ115ζ117ζ11211    linear of order 11
ρ111ζ1110ζ112ζ113ζ114ζ115ζ116ζ117ζ118ζ119ζ1111    linear of order 11
ρ12110000000000-1-√-23/2-1+√-23/2    complex faithful
ρ13110000000000-1+√-23/2-1-√-23/2    complex faithful

Permutation representations of C23⋊C11
►On 23 points: primitive - transitive group 23T3
Generators in S23
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23)

(2 3 5 9 17 10 19 14 4 7 13)(6 11 21 18 12 23 22 20 16 8 15)
 
G:=sub<Sym(23)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (2,3,5,9,17,10,19,14,4,7,13)(6,11,21,18,12,23,22,20,16,8,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (2,3,5,9,17,10,19,14,4,7,13)(6,11,21,18,12,23,22,20,16,8,15) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)], [(2,3,5,9,17,10,19,14,4,7,13),(6,11,21,18,12,23,22,20,16,8,15)]])
 
G:=TransitiveGroup(23,3);
 

Matrix representation of C23⋊C11 ►in GL11(𝔽1013)

10121000000000
10120100000000
10120010000000
10120001000000
10120000100000
10120000010000
10120000001000
10120000000100
10120000000010
10120000000001
783323122922810097827837853230
,
784323122922810097827837853230
10000000000
546145722510063215555586462227
01000000000
4636862227773189856084656871009
00100000000
6894517753198783349467920453780
00010000000
685100331794884797468923915221552
00001000000
450315877100338702923133679545556

G:=sub<GL(11,GF(1013))| [1012,1012,1012,1012,1012,1012,1012,1012,1012,1012,783,1,0,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,231,0,0,1,0,0,0,0,0,0,0,229,0,0,0,1,0,0,0,0,0,0,228,0,0,0,0,1,0,0,0,0,0,1009,0,0,0,0,0,1,0,0,0,0,782,0,0,0,0,0,0,1,0,0,0,783,0,0,0,0,0,0,0,1,0,0,785,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,1,230],[784,1,5,0,463,0,689,0,685,0,450,3,0,461,1,686,0,451,0,1003,0,315,231,0,457,0,222,1,775,0,317,0,877,229,0,225,0,777,0,319,1,94,0,100,228,0,1006,0,318,0,878,0,884,1,338,1009,0,321,0,98,0,334,0,797,0,702,782,0,555,0,560,0,9,0,468,0,923,783,0,558,0,8,0,467,0,923,0,133,785,0,6,0,465,0,920,0,915,0,679,3,0,462,0,687,0,453,0,221,0,545,230,0,227,0,1009,0,780,0,552,0,556] >;
 

C23⋊C11 in GAP, Magma, Sage, TeX 

C_{23}\rtimes C_{11}
 
% in TeX
 
G:=Group("C23:C11");
 
// GroupNames label
 
G:=SmallGroup(253,1);
 
// by ID
 
G=gap.SmallGroup(253,1);
 
# by ID
 
G:=PCGroup([2,-11,-23,89]);
 
// Polycyclic
 
G:=Group<a,b|a^23=b^11=1,b*a*b^-1=a^12>;
 
// generators/relations
 

Export

Subgroup lattice of C23⋊C11 in TeX
Character table of C23⋊C11 in TeX

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