Copied to
clipboard

G = Q8×C11order 88 = 23·11

Direct product of C11 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C11, C4.C22, C44.3C2, C22.7C22, C2.2(C2×C22), SmallGroup(88,10)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C11
C1C2C22C44 — Q8×C11
C1C2 — Q8×C11
C1C22 — Q8×C11

Generators and relations for Q8×C11
 G = < a,b,c | a11=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C11
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 45 23 38)(2 46 24 39)(3 47 25 40)(4 48 26 41)(5 49 27 42)(6 50 28 43)(7 51 29 44)(8 52 30 34)(9 53 31 35)(10 54 32 36)(11 55 33 37)(12 63 85 70)(13 64 86 71)(14 65 87 72)(15 66 88 73)(16 56 78 74)(17 57 79 75)(18 58 80 76)(19 59 81 77)(20 60 82 67)(21 61 83 68)(22 62 84 69)
(1 67 23 60)(2 68 24 61)(3 69 25 62)(4 70 26 63)(5 71 27 64)(6 72 28 65)(7 73 29 66)(8 74 30 56)(9 75 31 57)(10 76 32 58)(11 77 33 59)(12 48 85 41)(13 49 86 42)(14 50 87 43)(15 51 88 44)(16 52 78 34)(17 53 79 35)(18 54 80 36)(19 55 81 37)(20 45 82 38)(21 46 83 39)(22 47 84 40)

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

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

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

Q8×C11 is a maximal subgroup of   Q8⋊D11  C11⋊Q16  D44⋊C2

55 conjugacy classes

class 1  2 4A4B4C11A···11J22A···22J44A···44AD
order1244411···1122···2244···44
size112221···11···12···2

55 irreducible representations

dim111122
type++-
imageC1C2C11C22Q8Q8×C11
kernelQ8×C11C44Q8C4C11C1
# reps131030110

Matrix representation of Q8×C11 in GL2(𝔽23) generated by

20
02
,
220
1721
,
09
50
G:=sub<GL(2,GF(23))| [2,0,0,2],[2,17,20,21],[0,5,9,0] >;

Q8×C11 in GAP, Magma, Sage, TeX

Q_8\times C_{11}
% in TeX

G:=Group("Q8xC11");
// GroupNames label

G:=SmallGroup(88,10);
// by ID

G=gap.SmallGroup(88,10);
# by ID

G:=PCGroup([4,-2,-2,-11,-2,176,369,181]);
// Polycyclic

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

Export

Subgroup lattice of Q8×C11 in TeX

׿
×
𝔽