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G = C22⋊F11  order 440 = 23·5·11

The semidirect product of C22 and F11 acting via F11/C11⋊C5=C2

metabelian, supersoluble, monomial

Aliases: C22⋊F11, D22⋊2C10, Dic11⋊C10, C11⋊C20⋊C2, C11⋊C5⋊2D4, C11⋊D4⋊C5, C11⋊2(C5×D4), (C2×C22)⋊1C10, (C2×F11)⋊2C2, C2.5(C2×F11), C22.5(C2×C10), (C22×C11⋊C5)⋊1C2, (C2×C11⋊C5).5C22, SmallGroup(440,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C22⋊F11
Chief series
C1 — C11 — C22 — C2×C11⋊C5 — C2×F11 — C22⋊F11
Lower central
C11 — C22 — C22⋊F11
Upper central
C1 — C2 — C22

Generators and relations for C22⋊F11
 G = < a,b,c,d | a2=b2=c11=d10=1, dad-1=ab=ba, ac=ca, bc=cb, bd=db, dcd-1=c6 >

C1
C2
2C2
22C2
11C5
C11
C22
11C22
11C4
11C10
22C10
22C10
C22
2D11
2C22
C11⋊C5
11D4
11C2×C10
11C2×C10
11C20
D22
C2×C22
Dic11
C2×C11⋊C5
2C2×C11⋊C5
2F11
11C5×D4
C11⋊D4
C2×F11
C11⋊C20
C22×C11⋊C5
C22⋊F11

Character table of C22⋊F11

 class 12A2B2C45A5B5C5D10A10B10C10D10E10F10G10H10I10J10K10L1120A20B20C20D22A22B22C
 size 1122222111111111111111122222222222222221022222222101010
ρ111111111111111111111111111111    trivial
ρ2111-1-111111111111-1-1-1-111-1-1-1-1111    linear of order 2
ρ311-11-111111111-1-1-11111-11-1-1-1-1-11-1    linear of order 2
ρ411-1-1111111111-1-1-1-1-1-1-1-111111-11-1    linear of order 2
ρ511-1-11ζ54ζ5ζ53ζ52ζ53ζ5ζ54ζ52-ζ53-ζ52-ζ5-ζ54-ζ53-ζ52-ζ5-ζ541ζ5ζ53ζ52ζ54-11-1    linear of order 10
ρ611111ζ53ζ52ζ5ζ54ζ5ζ52ζ53ζ54ζ5ζ54ζ52ζ53ζ5ζ54ζ52ζ531ζ52ζ5ζ54ζ53111    linear of order 5
ρ711-1-11ζ5ζ54ζ52ζ53ζ52ζ54ζ5ζ53-ζ52-ζ53-ζ54-ζ5-ζ52-ζ53-ζ54-ζ51ζ54ζ52ζ53ζ5-11-1    linear of order 10
ρ811111ζ54ζ5ζ53ζ52ζ53ζ5ζ54ζ52ζ53ζ52ζ5ζ54ζ53ζ52ζ5ζ541ζ5ζ53ζ52ζ54111    linear of order 5
ρ911-11-1ζ54ζ5ζ53ζ52ζ53ζ5ζ54ζ52-ζ53-ζ52-ζ5ζ54ζ53ζ52ζ5-ζ541-ζ5-ζ53-ζ52-ζ54-11-1    linear of order 10
ρ1011111ζ52ζ53ζ54ζ5ζ54ζ53ζ52ζ5ζ54ζ5ζ53ζ52ζ54ζ5ζ53ζ521ζ53ζ54ζ5ζ52111    linear of order 5
ρ1111-1-11ζ52ζ53ζ54ζ5ζ54ζ53ζ52ζ5-ζ54-ζ5-ζ53-ζ52-ζ54-ζ5-ζ53-ζ521ζ53ζ54ζ5ζ52-11-1    linear of order 10
ρ12111-1-1ζ54ζ5ζ53ζ52ζ53ζ5ζ54ζ52ζ53ζ52ζ5-ζ54-ζ53-ζ52-ζ5ζ541-ζ5-ζ53-ζ52-ζ54111    linear of order 10
ρ1311-11-1ζ5ζ54ζ52ζ53ζ52ζ54ζ5ζ53-ζ52-ζ53-ζ54ζ5ζ52ζ53ζ54-ζ51-ζ54-ζ52-ζ53-ζ5-11-1    linear of order 10
ρ1411-1-11ζ53ζ52ζ5ζ54ζ5ζ52ζ53ζ54-ζ5-ζ54-ζ52-ζ53-ζ5-ζ54-ζ52-ζ531ζ52ζ5ζ54ζ53-11-1    linear of order 10
ρ15111-1-1ζ5ζ54ζ52ζ53ζ52ζ54ζ5ζ53ζ52ζ53ζ54-ζ5-ζ52-ζ53-ζ54ζ51-ζ54-ζ52-ζ53-ζ5111    linear of order 10
ρ16111-1-1ζ53ζ52ζ5ζ54ζ5ζ52ζ53ζ54ζ5ζ54ζ52-ζ53-ζ5-ζ54-ζ52ζ531-ζ52-ζ5-ζ54-ζ53111    linear of order 10
ρ1711-11-1ζ53ζ52ζ5ζ54ζ5ζ52ζ53ζ54-ζ5-ζ54-ζ52ζ53ζ5ζ54ζ52-ζ531-ζ52-ζ5-ζ54-ζ53-11-1    linear of order 10
ρ18111-1-1ζ52ζ53ζ54ζ5ζ54ζ53ζ52ζ5ζ54ζ5ζ53-ζ52-ζ54-ζ5-ζ53ζ521-ζ53-ζ54-ζ5-ζ52111    linear of order 10
ρ1911111ζ5ζ54ζ52ζ53ζ52ζ54ζ5ζ53ζ52ζ53ζ54ζ5ζ52ζ53ζ54ζ51ζ54ζ52ζ53ζ5111    linear of order 5
ρ2011-11-1ζ52ζ53ζ54ζ5ζ54ζ53ζ52ζ5-ζ54-ζ5-ζ53ζ52ζ54ζ5ζ53-ζ521-ζ53-ζ54-ζ5-ζ52-11-1    linear of order 10
ρ212-20002222-2-2-2-200000000200000-20    orthogonal lifted from D4
ρ222-20002ζ522ζ532ζ542ζ5-2ζ54-2ζ53-2ζ52-2ζ500000000200000-20    complex lifted from C5×D4
ρ232-20002ζ532ζ522ζ52ζ54-2ζ5-2ζ52-2ζ53-2ζ5400000000200000-20    complex lifted from C5×D4
ρ242-20002ζ542ζ52ζ532ζ52-2ζ53-2ζ5-2ζ54-2ζ5200000000200000-20    complex lifted from C5×D4
ρ252-20002ζ52ζ542ζ522ζ53-2ζ52-2ζ54-2ζ5-2ζ5300000000200000-20    complex lifted from C5×D4
ρ26101010000000000000000000-10000-1-1-1    orthogonal lifted from F11
ρ271010-10000000000000000000-100001-11    orthogonal lifted from C2×F11
ρ2810-100000000000000000000-10000√-111-√-11    complex faithful
ρ2910-100000000000000000000-10000-√-111√-11    complex faithful

Smallest permutation representation of C22⋊F11
►On 44 points
Generators in S44
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)

(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 41 42 43 44)

(2 3 5 9 6 11 10 8 4 7)(13 14 16 20 17 22 21 19 15 18)(23 34)(24 36 27 42 28 44 32 41 26 40)(25 38 31 39 33 43 30 37 29 35)
 
G:=sub<Sym(44)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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)(34,35,36,37,38,39,40,41,42,43,44), (2,3,5,9,6,11,10,8,4,7)(13,14,16,20,17,22,21,19,15,18)(23,34)(24,36,27,42,28,44,32,41,26,40)(25,38,31,39,33,43,30,37,29,35)>;
 
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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)(34,35,36,37,38,39,40,41,42,43,44), (2,3,5,9,6,11,10,8,4,7)(13,14,16,20,17,22,21,19,15,18)(23,34)(24,36,27,42,28,44,32,41,26,40)(25,38,31,39,33,43,30,37,29,35) );
 
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44)], [(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),(34,35,36,37,38,39,40,41,42,43,44)], [(2,3,5,9,6,11,10,8,4,7),(13,14,16,20,17,22,21,19,15,18),(23,34),(24,36,27,42,28,44,32,41,26,40),(25,38,31,39,33,43,30,37,29,35)]])
 

Matrix representation of C22⋊F11 ►in GL12(𝔽661)

010000000000
100000000000
00660000000000
00066000000000
00006600000000
00000660000000
00000066000000
00000006600000
00000000660000
00000000066000
00000000006600
00000000000660
,
66000000000000
06600000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
100000000000
010000000000
00000000000660
00100000000660
00010000000660
00001000000660
00000100000660
00000010000660
00000001000660
00000000100660
00000000010660
00000000001660
,
47100000000000
01900000000000
000000010000
001000000000
000000001000
000100000000
000000000100
000010000000
000000000010
000001000000
000000000001
000000100000

G:=sub<GL(12,GF(661))| [0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660],[660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,660,660,660,660,660,660,660,660,660,660],[471,0,0,0,0,0,0,0,0,0,0,0,0,190,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] >;
 

C22⋊F11 in GAP, Magma, Sage, TeX 

C_2^2\rtimes F_{11}
 
% in TeX
 
G:=Group("C2^2:F11");
 
// GroupNames label
 
G:=SmallGroup(440,11);
 
// by ID
 
G=gap.SmallGroup(440,11);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-2,-11,221,10004,2264]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^11=d^10=1,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^6>;
 
// generators/relations
 

Export

Subgroup lattice of C22⋊F11 in TeX
Character table of C22⋊F11 in TeX

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