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## G = C22×C11⋊C5order 220 = 22·5·11

### Direct product of C22 and C11⋊C5

Aliases: C22×C11⋊C5, C222C10, (C2×C22)⋊C5, C112(C2×C10), SmallGroup(220,8)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C11⋊C5
G = < a,b,c,d | a2=b2=c11=d5=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Character table of C22×C11⋊C5

 class 1 2A 2B 2C 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 11A 11B 22A 22B 22C 22D 22E 22F size 1 1 1 1 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 5 5 5 5 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 linear of order 2 ρ4 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ5 1 1 1 1 1 1 1 1 linear of order 5 ρ6 1 1 -1 -1 ζ52 ζ53 ζ54 ζ5 -ζ54 -ζ53 -ζ54 ζ5 ζ52 ζ53 ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ52 1 1 -1 -1 1 -1 -1 1 linear of order 10 ρ7 1 1 -1 -1 ζ54 ζ5 ζ53 ζ52 -ζ53 -ζ5 -ζ53 ζ52 ζ54 ζ5 ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ54 1 1 -1 -1 1 -1 -1 1 linear of order 10 ρ8 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ52 1 1 1 1 1 1 1 1 linear of order 5 ρ9 1 1 -1 -1 ζ5 ζ54 ζ52 ζ53 -ζ52 -ζ54 -ζ52 ζ53 ζ5 ζ54 ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ5 1 1 -1 -1 1 -1 -1 1 linear of order 10 ρ10 1 -1 -1 1 ζ54 ζ5 ζ53 ζ52 ζ53 -ζ5 -ζ53 -ζ52 -ζ54 -ζ5 -ζ53 -ζ52 ζ52 ζ54 ζ5 -ζ54 1 1 1 1 -1 -1 -1 -1 linear of order 10 ρ11 1 -1 1 -1 ζ54 ζ5 ζ53 ζ52 -ζ53 ζ5 ζ53 -ζ52 -ζ54 -ζ5 -ζ53 ζ52 -ζ52 -ζ54 -ζ5 ζ54 1 1 -1 -1 -1 1 1 -1 linear of order 10 ρ12 1 -1 -1 1 ζ53 ζ52 ζ5 ζ54 ζ5 -ζ52 -ζ5 -ζ54 -ζ53 -ζ52 -ζ5 -ζ54 ζ54 ζ53 ζ52 -ζ53 1 1 1 1 -1 -1 -1 -1 linear of order 10 ρ13 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ53 1 1 1 1 1 1 1 1 linear of order 5 ρ14 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ54 1 1 1 1 1 1 1 1 linear of order 5 ρ15 1 -1 1 -1 ζ53 ζ52 ζ5 ζ54 -ζ5 ζ52 ζ5 -ζ54 -ζ53 -ζ52 -ζ5 ζ54 -ζ54 -ζ53 -ζ52 ζ53 1 1 -1 -1 -1 1 1 -1 linear of order 10 ρ16 1 -1 1 -1 ζ52 ζ53 ζ54 ζ5 -ζ54 ζ53 ζ54 -ζ5 -ζ52 -ζ53 -ζ54 ζ5 -ζ5 -ζ52 -ζ53 ζ52 1 1 -1 -1 -1 1 1 -1 linear of order 10 ρ17 1 1 -1 -1 ζ53 ζ52 ζ5 ζ54 -ζ5 -ζ52 -ζ5 ζ54 ζ53 ζ52 ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ53 1 1 -1 -1 1 -1 -1 1 linear of order 10 ρ18 1 -1 -1 1 ζ52 ζ53 ζ54 ζ5 ζ54 -ζ53 -ζ54 -ζ5 -ζ52 -ζ53 -ζ54 -ζ5 ζ5 ζ52 ζ53 -ζ52 1 1 1 1 -1 -1 -1 -1 linear of order 10 ρ19 1 -1 -1 1 ζ5 ζ54 ζ52 ζ53 ζ52 -ζ54 -ζ52 -ζ53 -ζ5 -ζ54 -ζ52 -ζ53 ζ53 ζ5 ζ54 -ζ5 1 1 1 1 -1 -1 -1 -1 linear of order 10 ρ20 1 -1 1 -1 ζ5 ζ54 ζ52 ζ53 -ζ52 ζ54 ζ52 -ζ53 -ζ5 -ζ54 -ζ52 ζ53 -ζ53 -ζ5 -ζ54 ζ5 1 1 -1 -1 -1 1 1 -1 linear of order 10 ρ21 5 -5 -5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 -1-√-11/2 -1+√-11/2 1-√-11/2 1+√-11/2 1-√-11/2 1+√-11/2 complex lifted from C2×C11⋊C5 ρ22 5 5 5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 -1-√-11/2 -1+√-11/2 -1+√-11/2 -1-√-11/2 -1+√-11/2 -1-√-11/2 complex lifted from C11⋊C5 ρ23 5 -5 -5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 -1+√-11/2 -1-√-11/2 1+√-11/2 1-√-11/2 1+√-11/2 1-√-11/2 complex lifted from C2×C11⋊C5 ρ24 5 5 -5 -5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 1+√-11/2 1-√-11/2 -1+√-11/2 1+√-11/2 1-√-11/2 -1-√-11/2 complex lifted from C2×C11⋊C5 ρ25 5 -5 5 -5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-11/2 -1-√-11/2 1+√-11/2 1-√-11/2 1-√-11/2 -1-√-11/2 -1+√-11/2 1+√-11/2 complex lifted from C2×C11⋊C5 ρ26 5 -5 5 -5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 1-√-11/2 1+√-11/2 1+√-11/2 -1+√-11/2 -1-√-11/2 1-√-11/2 complex lifted from C2×C11⋊C5 ρ27 5 5 -5 -5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 1-√-11/2 1+√-11/2 -1-√-11/2 1-√-11/2 1+√-11/2 -1+√-11/2 complex lifted from C2×C11⋊C5 ρ28 5 5 5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-11/2 -1+√-11/2 -1+√-11/2 -1-√-11/2 -1-√-11/2 -1+√-11/2 -1-√-11/2 -1+√-11/2 complex lifted from C11⋊C5

Smallest permutation representation of C22×C11⋊C5
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 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)

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

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

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

C22×C11⋊C5 is a maximal subgroup of   C22⋊F11

Matrix representation of C22×C11⋊C5 in GL6(𝔽331)

 330 0 0 0 0 0 0 330 0 0 0 0 0 0 330 0 0 0 0 0 0 330 0 0 0 0 0 0 330 0 0 0 0 0 0 330
,
 330 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 103 2 225 104 1 0 104 2 225 104 1 0 103 3 225 104 1 0 103 2 226 104 1 0 103 2 225 105 1
,
 323 0 0 0 0 0 0 0 0 1 0 0 0 105 228 329 106 227 0 106 227 103 2 226 0 1 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(6,GF(331))| [330,0,0,0,0,0,0,330,0,0,0,0,0,0,330,0,0,0,0,0,0,330,0,0,0,0,0,0,330,0,0,0,0,0,0,330],[330,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,103,104,103,103,103,0,2,2,3,2,2,0,225,225,225,226,225,0,104,104,104,104,105,0,1,1,1,1,1],[323,0,0,0,0,0,0,0,105,106,1,0,0,0,228,227,0,0,0,1,329,103,0,0,0,0,106,2,0,1,0,0,227,226,0,0] >;

C22×C11⋊C5 in GAP, Magma, Sage, TeX

C_2^2\times C_{11}\rtimes C_5
% in TeX

G:=Group("C2^2xC11:C5");
// GroupNames label

G:=SmallGroup(220,8);
// by ID

G=gap.SmallGroup(220,8);
# by ID

G:=PCGroup([4,-2,-2,-5,-11,331]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^11=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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