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G = C22×D11order 88 = 23·11

Direct product of C22 and D11

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22×D11, C11⋊C23, C22⋊C22, (C2×C22)⋊3C2, SmallGroup(88,11)

Series: Derived Chief Lower central Upper central

C1C11 — C22×D11
C1C11D11D22 — C22×D11
C11 — C22×D11
C1C22

Generators and relations for C22×D11
 G = < a,b,c,d | a2=b2=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

11C2
11C2
11C2
11C2
11C22
11C22
11C22
11C22
11C22
11C22
11C23

Character table of C22×D11

 class 12A2B2C2D2E2F2G11A11B11C11D11E22A22B22C22D22E22F22G22H22I22J22K22L22M22N22O
 size 11111111111122222222222222222222
ρ11111111111111111111111111111    trivial
ρ21-11-111-1-111111-11111-1-1-1-1-1-1-1-11-1    linear of order 2
ρ311-1-11-11-1111111-1-1-1-11-1-1-1-1-111-11    linear of order 2
ρ41-1-111-1-1111111-1-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ51-1-11-111-111111-1-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ611-1-1-11-11111111-1-1-1-11-1-1-1-1-111-11    linear of order 2
ρ71-11-1-1-11111111-11111-1-1-1-1-1-1-1-11-1    linear of order 2
ρ81111-1-1-1-111111111111111111111    linear of order 2
ρ92-22-20000ζ117114ζ118113ζ111011ζ116115ζ119112118113ζ118113ζ111011ζ116115ζ119112116115117114118113111011116115119112119112111011ζ117114117114    orthogonal lifted from D22
ρ1022-2-20000ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115116115119112111011117114ζ111011118113116115119112111011117114ζ117114ζ119112118113ζ118113    orthogonal lifted from D22
ρ112-22-20000ζ119112ζ117114ζ116115ζ118113ζ111011117114ζ117114ζ116115ζ118113ζ111011118113119112117114116115118113111011111011116115ζ119112119112    orthogonal lifted from D22
ρ122-2-220000ζ119112ζ117114ζ116115ζ118113ζ111011117114117114116115118113111011118113ζ119112ζ117114ζ116115ζ118113ζ111011111011116115119112119112    orthogonal lifted from D22
ρ132-22-20000ζ116115ζ111011ζ117114ζ119112ζ118113111011ζ111011ζ117114ζ119112ζ118113119112116115111011117114119112118113118113117114ζ116115116115    orthogonal lifted from D22
ρ1422220000ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ111011ζ117114ζ119112ζ118113ζ119112ζ116115ζ111011ζ117114ζ119112ζ118113ζ118113ζ117114ζ116115ζ116115    orthogonal lifted from D11
ρ152-2-220000ζ118113ζ116115ζ119112ζ111011ζ117114116115116115119112111011117114111011ζ118113ζ116115ζ119112ζ111011ζ117114117114119112118113118113    orthogonal lifted from D22
ρ1622-2-20000ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113118113111011116115119112ζ116115117114118113111011116115119112ζ119112ζ111011117114ζ117114    orthogonal lifted from D22
ρ172-2-220000ζ111011ζ119112ζ118113ζ117114ζ116115119112119112118113117114116115117114ζ111011ζ119112ζ118113ζ117114ζ116115116115118113111011111011    orthogonal lifted from D22
ρ1822-2-20000ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112119112118113117114116115ζ117114111011119112118113117114116115ζ116115ζ118113111011ζ111011    orthogonal lifted from D22
ρ1922-2-20000ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011111011117114119112118113ζ119112116115111011117114119112118113ζ118113ζ117114116115ζ116115    orthogonal lifted from D22
ρ202-2-220000ζ116115ζ111011ζ117114ζ119112ζ118113111011111011117114119112118113119112ζ116115ζ111011ζ117114ζ119112ζ118113118113117114116115116115    orthogonal lifted from D22
ρ2122220000ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ117114ζ116115ζ118113ζ111011ζ118113ζ119112ζ117114ζ116115ζ118113ζ111011ζ111011ζ116115ζ119112ζ119112    orthogonal lifted from D11
ρ2222-2-20000ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114117114116115118113111011ζ118113119112117114116115118113111011ζ111011ζ116115119112ζ119112    orthogonal lifted from D22
ρ2322220000ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ118113ζ111011ζ116115ζ119112ζ116115ζ117114ζ118113ζ111011ζ116115ζ119112ζ119112ζ111011ζ117114ζ117114    orthogonal lifted from D11
ρ242-22-20000ζ118113ζ116115ζ119112ζ111011ζ117114116115ζ116115ζ119112ζ111011ζ117114111011118113116115119112111011117114117114119112ζ118113118113    orthogonal lifted from D22
ρ252-22-20000ζ111011ζ119112ζ118113ζ117114ζ116115119112ζ119112ζ118113ζ117114ζ116115117114111011119112118113117114116115116115118113ζ111011111011    orthogonal lifted from D22
ρ2622220000ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ116115ζ119112ζ111011ζ117114ζ111011ζ118113ζ116115ζ119112ζ111011ζ117114ζ117114ζ119112ζ118113ζ118113    orthogonal lifted from D11
ρ272-2-220000ζ117114ζ118113ζ111011ζ116115ζ119112118113118113111011116115119112116115ζ117114ζ118113ζ111011ζ116115ζ119112119112111011117114117114    orthogonal lifted from D22
ρ2822220000ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ119112ζ118113ζ117114ζ116115ζ117114ζ111011ζ119112ζ118113ζ117114ζ116115ζ116115ζ118113ζ111011ζ111011    orthogonal lifted from D11

Smallest permutation representation of C22×D11
On 44 points
Generators in S44
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(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)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 33)(20 32)(21 31)(22 30)

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

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

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

C22×D11 is a maximal subgroup of   D22⋊C4
C22×D11 is a maximal quotient of   D445C2  D42D11  D44⋊C2

Matrix representation of C22×D11 in GL3(𝔽23) generated by

100
0220
0022
,
2200
010
001
,
100
001
0224
,
2200
0022
0220
G:=sub<GL(3,GF(23))| [1,0,0,0,22,0,0,0,22],[22,0,0,0,1,0,0,0,1],[1,0,0,0,0,22,0,1,4],[22,0,0,0,0,22,0,22,0] >;

C22×D11 in GAP, Magma, Sage, TeX

C_2^2\times D_{11}
% in TeX

G:=Group("C2^2xD11");
// GroupNames label

G:=SmallGroup(88,11);
// by ID

G=gap.SmallGroup(88,11);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,1283]);
// Polycyclic

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

Export

Subgroup lattice of C22×D11 in TeX
Character table of C22×D11 in TeX

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