Copied to
clipboard

G = C11⋊D4order 88 = 23·11

The semidirect product of C11 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C112D4, C22⋊D11, D222C2, Dic11⋊C2, C2.5D22, C22.5C22, (C2×C22)⋊2C2, SmallGroup(88,7)

Series: Derived Chief Lower central Upper central

C1C22 — C11⋊D4
C1C11C22D22 — C11⋊D4
C11C22 — C11⋊D4
C1C2C22

Generators and relations for C11⋊D4
 G = < a,b,c | a11=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
22C2
11C4
11C22
2D11
2C22
11D4

Character table of C11⋊D4

 class 12A2B2C411A11B11C11D11E22A22B22C22D22E22F22G22H22I22J22K22L22M22N22O
 size 112222222222222222222222222
ρ11111111111111111111111111    trivial
ρ211-11-111111111-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ3111-1-111111111111111111111    linear of order 2
ρ411-1-1111111111-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ52-200022222-2-2-20000000000-2-2    orthogonal lifted from D4
ρ622-200ζ111011ζ117114ζ118113ζ116115ζ119112ζ116115ζ119112ζ117114117114118113111011116115119112119112116115111011118113117114ζ118113ζ111011    orthogonal lifted from D22
ρ722200ζ111011ζ117114ζ118113ζ116115ζ119112ζ116115ζ119112ζ117114ζ117114ζ118113ζ111011ζ116115ζ119112ζ119112ζ116115ζ111011ζ118113ζ117114ζ118113ζ111011    orthogonal lifted from D11
ρ822200ζ119112ζ118113ζ116115ζ111011ζ117114ζ111011ζ117114ζ118113ζ118113ζ116115ζ119112ζ111011ζ117114ζ117114ζ111011ζ119112ζ116115ζ118113ζ116115ζ119112    orthogonal lifted from D11
ρ922-200ζ119112ζ118113ζ116115ζ111011ζ117114ζ111011ζ117114ζ118113118113116115119112111011117114117114111011119112116115118113ζ116115ζ119112    orthogonal lifted from D22
ρ1022-200ζ117114ζ116115ζ111011ζ119112ζ118113ζ119112ζ118113ζ116115116115111011117114119112118113118113119112117114111011116115ζ111011ζ117114    orthogonal lifted from D22
ρ1122200ζ116115ζ119112ζ117114ζ118113ζ111011ζ118113ζ111011ζ119112ζ119112ζ117114ζ116115ζ118113ζ111011ζ111011ζ118113ζ116115ζ117114ζ119112ζ117114ζ116115    orthogonal lifted from D11
ρ1222200ζ117114ζ116115ζ111011ζ119112ζ118113ζ119112ζ118113ζ116115ζ116115ζ111011ζ117114ζ119112ζ118113ζ118113ζ119112ζ117114ζ111011ζ116115ζ111011ζ117114    orthogonal lifted from D11
ρ1322-200ζ116115ζ119112ζ117114ζ118113ζ111011ζ118113ζ111011ζ119112119112117114116115118113111011111011118113116115117114119112ζ117114ζ116115    orthogonal lifted from D22
ρ1422-200ζ118113ζ111011ζ119112ζ117114ζ116115ζ117114ζ116115ζ111011111011119112118113117114116115116115117114118113119112111011ζ119112ζ118113    orthogonal lifted from D22
ρ1522200ζ118113ζ111011ζ119112ζ117114ζ116115ζ117114ζ116115ζ111011ζ111011ζ119112ζ118113ζ117114ζ116115ζ116115ζ117114ζ118113ζ119112ζ111011ζ119112ζ118113    orthogonal lifted from D11
ρ162-2000ζ111011ζ117114ζ118113ζ116115ζ119112116115119112117114117114ζ118113111011116115ζ119112119112ζ116115ζ111011118113ζ117114118113111011    complex faithful
ρ172-2000ζ116115ζ119112ζ117114ζ118113ζ111011118113111011119112ζ119112ζ117114116115118113111011ζ111011ζ118113ζ116115117114119112117114116115    complex faithful
ρ182-2000ζ111011ζ117114ζ118113ζ116115ζ119112116115119112117114ζ117114118113ζ111011ζ116115119112ζ119112116115111011ζ118113117114118113111011    complex faithful
ρ192-2000ζ118113ζ111011ζ119112ζ117114ζ116115117114116115111011111011119112118113117114116115ζ116115ζ117114ζ118113ζ119112ζ111011119112118113    complex faithful
ρ202-2000ζ118113ζ111011ζ119112ζ117114ζ116115117114116115111011ζ111011ζ119112ζ118113ζ117114ζ116115116115117114118113119112111011119112118113    complex faithful
ρ212-2000ζ117114ζ116115ζ111011ζ119112ζ118113119112118113116115ζ116115111011ζ117114119112ζ118113118113ζ119112117114ζ111011116115111011117114    complex faithful
ρ222-2000ζ116115ζ119112ζ117114ζ118113ζ111011118113111011119112119112117114ζ116115ζ118113ζ111011111011118113116115ζ117114ζ119112117114116115    complex faithful
ρ232-2000ζ119112ζ118113ζ116115ζ111011ζ117114111011117114118113118113ζ116115ζ119112111011117114ζ117114ζ111011119112116115ζ118113116115119112    complex faithful
ρ242-2000ζ117114ζ116115ζ111011ζ119112ζ118113119112118113116115116115ζ111011117114ζ119112118113ζ118113119112ζ117114111011ζ116115111011117114    complex faithful
ρ252-2000ζ119112ζ118113ζ116115ζ111011ζ117114111011117114118113ζ118113116115119112ζ111011ζ117114117114111011ζ119112ζ116115118113116115119112    complex faithful

Smallest permutation representation of C11⋊D4
On 44 points
Generators in S44
(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 32 21 43)(2 31 22 42)(3 30 12 41)(4 29 13 40)(5 28 14 39)(6 27 15 38)(7 26 16 37)(8 25 17 36)(9 24 18 35)(10 23 19 34)(11 33 20 44)
(2 11)(3 10)(4 9)(5 8)(6 7)(12 19)(13 18)(14 17)(15 16)(20 22)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 44)(32 43)(33 42)

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

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

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

C11⋊D4 is a maximal subgroup of
D445C2  D4×D11  D42D11  C33⋊D4  C11⋊D12  C337D4  C11⋊S4  C22⋊F11  C55⋊D4  C11⋊D20  C557D4
C11⋊D4 is a maximal quotient of
Dic11⋊C4  D22⋊C4  D4⋊D11  D4.D11  Q8⋊D11  C11⋊Q16  C23.D11  C33⋊D4  C11⋊D12  C337D4  C55⋊D4  C11⋊D20  C557D4

Matrix representation of C11⋊D4 in GL2(𝔽23) generated by

1617
1721
,
01
220
,
1511
118
G:=sub<GL(2,GF(23))| [16,17,17,21],[0,22,1,0],[15,11,11,8] >;

C11⋊D4 in GAP, Magma, Sage, TeX

C_{11}\rtimes D_4
% in TeX

G:=Group("C11:D4");
// GroupNames label

G:=SmallGroup(88,7);
// by ID

G=gap.SmallGroup(88,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,49,1283]);
// Polycyclic

G:=Group<a,b,c|a^11=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C11⋊D4 in TeX
Character table of C11⋊D4 in TeX

׿
×
𝔽