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G = C11⋊C8order 88 = 23·11

The semidirect product of C11 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C11⋊C8, C22.C4, C44.2C2, C4.2D11, C2.Dic11, SmallGroup(88,1)

Series: Derived Chief Lower central Upper central

C1C11 — C11⋊C8
C1C11C22C44 — C11⋊C8
C11 — C11⋊C8
C1C4

Generators and relations for C11⋊C8
 G = < a,b | a11=b8=1, bab-1=a-1 >

11C8

Character table of C11⋊C8

 class 124A4B8A8B8C8D11A11B11C11D11E22A22B22C22D22E44A44B44C44D44E44F44G44H44I44J
 size 11111111111122222222222222222222
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-111111111111111111111    linear of order 2
ρ311-1-1-ii-ii1111111111-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ411-1-1i-ii-i1111111111-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ51-1-iiζ85ζ87ζ8ζ8311111-1-1-1-1-1-iiiiii-i-i-i-i    linear of order 8
ρ61-1i-iζ83ζ8ζ87ζ8511111-1-1-1-1-1i-i-i-i-i-iiiii    linear of order 8
ρ71-1-iiζ8ζ83ζ85ζ8711111-1-1-1-1-1-iiiiii-i-i-i-i    linear of order 8
ρ81-1i-iζ87ζ85ζ83ζ811111-1-1-1-1-1i-i-i-i-i-iiiii    linear of order 8
ρ922220000ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ111011ζ116115ζ118113ζ119112ζ119112ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ116115ζ118113ζ111011    orthogonal lifted from D11
ρ1022220000ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ117114ζ119112ζ111011ζ118113ζ118113ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ119112ζ111011ζ117114    orthogonal lifted from D11
ρ1122220000ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ116115ζ118113ζ117114ζ111011ζ111011ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ118113ζ117114ζ116115    orthogonal lifted from D11
ρ1222220000ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ119112ζ111011ζ116115ζ117114ζ117114ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ111011ζ116115ζ119112    orthogonal lifted from D11
ρ1322220000ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ118113ζ117114ζ119112ζ116115ζ116115ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ117114ζ119112ζ118113    orthogonal lifted from D11
ρ1422-2-20000ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ117114ζ119112ζ111011ζ118113118113118113116115119112111011117114116115119112111011117114    symplectic lifted from Dic11, Schur index 2
ρ1522-2-20000ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ111011ζ116115ζ118113ζ119112119112119112117114116115118113111011117114116115118113111011    symplectic lifted from Dic11, Schur index 2
ρ1622-2-20000ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ118113ζ117114ζ119112ζ116115116115116115111011117114119112118113111011117114119112118113    symplectic lifted from Dic11, Schur index 2
ρ1722-2-20000ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ119112ζ111011ζ116115ζ117114117114117114118113111011116115119112118113111011116115119112    symplectic lifted from Dic11, Schur index 2
ρ1822-2-20000ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ116115ζ118113ζ117114ζ111011111011111011119112118113117114116115119112118113117114116115    symplectic lifted from Dic11, Schur index 2
ρ192-22i-2i0000ζ118113ζ116115ζ119112ζ111011ζ117114116115117114119112111011118113ζ4ζ1184ζ113ζ43ζ11843ζ113ζ43ζ11643ζ115ζ43ζ11943ζ112ζ43ζ111043ζ11ζ43ζ11743ζ114ζ4ζ1164ζ115ζ4ζ1194ζ112ζ4ζ11104ζ11ζ4ζ1174ζ114    complex faithful
ρ202-22i-2i0000ζ117114ζ118113ζ111011ζ116115ζ119112118113119112111011116115117114ζ4ζ1174ζ114ζ43ζ11743ζ114ζ43ζ11843ζ113ζ43ζ111043ζ11ζ43ζ11643ζ115ζ43ζ11943ζ112ζ4ζ1184ζ113ζ4ζ11104ζ11ζ4ζ1164ζ115ζ4ζ1194ζ112    complex faithful
ρ212-2-2i2i0000ζ111011ζ119112ζ118113ζ117114ζ116115119112116115118113117114111011ζ43ζ111043ζ11ζ4ζ11104ζ11ζ4ζ1194ζ112ζ4ζ1184ζ113ζ4ζ1174ζ114ζ4ζ1164ζ115ζ43ζ11943ζ112ζ43ζ11843ζ113ζ43ζ11743ζ114ζ43ζ11643ζ115    complex faithful
ρ222-22i-2i0000ζ116115ζ111011ζ117114ζ119112ζ118113111011118113117114119112116115ζ4ζ1164ζ115ζ43ζ11643ζ115ζ43ζ111043ζ11ζ43ζ11743ζ114ζ43ζ11943ζ112ζ43ζ11843ζ113ζ4ζ11104ζ11ζ4ζ1174ζ114ζ4ζ1194ζ112ζ4ζ1184ζ113    complex faithful
ρ232-2-2i2i0000ζ117114ζ118113ζ111011ζ116115ζ119112118113119112111011116115117114ζ43ζ11743ζ114ζ4ζ1174ζ114ζ4ζ1184ζ113ζ4ζ11104ζ11ζ4ζ1164ζ115ζ4ζ1194ζ112ζ43ζ11843ζ113ζ43ζ111043ζ11ζ43ζ11643ζ115ζ43ζ11943ζ112    complex faithful
ρ242-2-2i2i0000ζ119112ζ117114ζ116115ζ118113ζ111011117114111011116115118113119112ζ43ζ11943ζ112ζ4ζ1194ζ112ζ4ζ1174ζ114ζ4ζ1164ζ115ζ4ζ1184ζ113ζ4ζ11104ζ11ζ43ζ11743ζ114ζ43ζ11643ζ115ζ43ζ11843ζ113ζ43ζ111043ζ11    complex faithful
ρ252-2-2i2i0000ζ116115ζ111011ζ117114ζ119112ζ118113111011118113117114119112116115ζ43ζ11643ζ115ζ4ζ1164ζ115ζ4ζ11104ζ11ζ4ζ1174ζ114ζ4ζ1194ζ112ζ4ζ1184ζ113ζ43ζ111043ζ11ζ43ζ11743ζ114ζ43ζ11943ζ112ζ43ζ11843ζ113    complex faithful
ρ262-22i-2i0000ζ119112ζ117114ζ116115ζ118113ζ111011117114111011116115118113119112ζ4ζ1194ζ112ζ43ζ11943ζ112ζ43ζ11743ζ114ζ43ζ11643ζ115ζ43ζ11843ζ113ζ43ζ111043ζ11ζ4ζ1174ζ114ζ4ζ1164ζ115ζ4ζ1184ζ113ζ4ζ11104ζ11    complex faithful
ρ272-22i-2i0000ζ111011ζ119112ζ118113ζ117114ζ116115119112116115118113117114111011ζ4ζ11104ζ11ζ43ζ111043ζ11ζ43ζ11943ζ112ζ43ζ11843ζ113ζ43ζ11743ζ114ζ43ζ11643ζ115ζ4ζ1194ζ112ζ4ζ1184ζ113ζ4ζ1174ζ114ζ4ζ1164ζ115    complex faithful
ρ282-2-2i2i0000ζ118113ζ116115ζ119112ζ111011ζ117114116115117114119112111011118113ζ43ζ11843ζ113ζ4ζ1184ζ113ζ4ζ1164ζ115ζ4ζ1194ζ112ζ4ζ11104ζ11ζ4ζ1174ζ114ζ43ζ11643ζ115ζ43ζ11943ζ112ζ43ζ111043ζ11ζ43ζ11743ζ114    complex faithful

Smallest permutation representation of C11⋊C8
Regular action on 88 points
Generators in S88
(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)(45 46 47 48 49 50 51 52 53 54 55)(56 57 58 59 60 61 62 63 64 65 66)(67 68 69 70 71 72 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)
(1 78 43 56 21 67 32 45)(2 88 44 66 22 77 33 55)(3 87 34 65 12 76 23 54)(4 86 35 64 13 75 24 53)(5 85 36 63 14 74 25 52)(6 84 37 62 15 73 26 51)(7 83 38 61 16 72 27 50)(8 82 39 60 17 71 28 49)(9 81 40 59 18 70 29 48)(10 80 41 58 19 69 30 47)(11 79 42 57 20 68 31 46)

G:=sub<Sym(88)| (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (1,78,43,56,21,67,32,45)(2,88,44,66,22,77,33,55)(3,87,34,65,12,76,23,54)(4,86,35,64,13,75,24,53)(5,85,36,63,14,74,25,52)(6,84,37,62,15,73,26,51)(7,83,38,61,16,72,27,50)(8,82,39,60,17,71,28,49)(9,81,40,59,18,70,29,48)(10,80,41,58,19,69,30,47)(11,79,42,57,20,68,31,46)>;

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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (1,78,43,56,21,67,32,45)(2,88,44,66,22,77,33,55)(3,87,34,65,12,76,23,54)(4,86,35,64,13,75,24,53)(5,85,36,63,14,74,25,52)(6,84,37,62,15,73,26,51)(7,83,38,61,16,72,27,50)(8,82,39,60,17,71,28,49)(9,81,40,59,18,70,29,48)(10,80,41,58,19,69,30,47)(11,79,42,57,20,68,31,46) );

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),(45,46,47,48,49,50,51,52,53,54,55),(56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88)], [(1,78,43,56,21,67,32,45),(2,88,44,66,22,77,33,55),(3,87,34,65,12,76,23,54),(4,86,35,64,13,75,24,53),(5,85,36,63,14,74,25,52),(6,84,37,62,15,73,26,51),(7,83,38,61,16,72,27,50),(8,82,39,60,17,71,28,49),(9,81,40,59,18,70,29,48),(10,80,41,58,19,69,30,47),(11,79,42,57,20,68,31,46)])

C11⋊C8 is a maximal subgroup of
C8×D11  C88⋊C2  C44.C4  D4⋊D11  D4.D11  Q8⋊D11  C11⋊Q16  C33⋊C8  C11⋊C40  C553C8  C55⋊C8
C11⋊C8 is a maximal quotient of
C11⋊C16  C33⋊C8  C553C8  C55⋊C8

Matrix representation of C11⋊C8 in GL2(𝔽89) generated by

881
187
,
1913
5770
G:=sub<GL(2,GF(89))| [88,1,1,87],[19,57,13,70] >;

C11⋊C8 in GAP, Magma, Sage, TeX

C_{11}\rtimes C_8
% in TeX

G:=Group("C11:C8");
// GroupNames label

G:=SmallGroup(88,1);
// by ID

G=gap.SmallGroup(88,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,8,21,1283]);
// Polycyclic

G:=Group<a,b|a^11=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C11⋊C8 in TeX
Character table of C11⋊C8 in TeX

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