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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

 Derived series C1 — C2 — Q8×C11
 Chief series C1 — C2 — C22 — C44 — Q8×C11
 Lower central C1 — C2 — Q8×C11
 Upper central C1 — C22 — 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 47 25 38)(2 48 26 39)(3 49 27 40)(4 50 28 41)(5 51 29 42)(6 52 30 43)(7 53 31 44)(8 54 32 34)(9 55 33 35)(10 45 23 36)(11 46 24 37)(12 63 85 72)(13 64 86 73)(14 65 87 74)(15 66 88 75)(16 56 78 76)(17 57 79 77)(18 58 80 67)(19 59 81 68)(20 60 82 69)(21 61 83 70)(22 62 84 71)
(1 69 25 60)(2 70 26 61)(3 71 27 62)(4 72 28 63)(5 73 29 64)(6 74 30 65)(7 75 31 66)(8 76 32 56)(9 77 33 57)(10 67 23 58)(11 68 24 59)(12 50 85 41)(13 51 86 42)(14 52 87 43)(15 53 88 44)(16 54 78 34)(17 55 79 35)(18 45 80 36)(19 46 81 37)(20 47 82 38)(21 48 83 39)(22 49 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), (1,47,25,38)(2,48,26,39)(3,49,27,40)(4,50,28,41)(5,51,29,42)(6,52,30,43)(7,53,31,44)(8,54,32,34)(9,55,33,35)(10,45,23,36)(11,46,24,37)(12,63,85,72)(13,64,86,73)(14,65,87,74)(15,66,88,75)(16,56,78,76)(17,57,79,77)(18,58,80,67)(19,59,81,68)(20,60,82,69)(21,61,83,70)(22,62,84,71), (1,69,25,60)(2,70,26,61)(3,71,27,62)(4,72,28,63)(5,73,29,64)(6,74,30,65)(7,75,31,66)(8,76,32,56)(9,77,33,57)(10,67,23,58)(11,68,24,59)(12,50,85,41)(13,51,86,42)(14,52,87,43)(15,53,88,44)(16,54,78,34)(17,55,79,35)(18,45,80,36)(19,46,81,37)(20,47,82,38)(21,48,83,39)(22,49,84,40)>;

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

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

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

55 conjugacy classes

 class 1 2 4A 4B 4C 11A ··· 11J 22A ··· 22J 44A ··· 44AD order 1 2 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2

55 irreducible representations

 dim 1 1 1 1 2 2 type + + - image C1 C2 C11 C22 Q8 Q8×C11 kernel Q8×C11 C44 Q8 C4 C11 C1 # reps 1 3 10 30 1 10

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

 2 0 0 2
,
 2 20 17 21
,
 0 9 5 0
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

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