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G = Q8⋊D11order 176 = 24·11

The semidirect product of Q8 and D11 acting via D11/C11=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊D11, C4.3D22, C22.9D4, C113SD16, D44.2C2, C44.3C22, C11⋊C83C2, (Q8×C11)⋊1C2, C2.6(C11⋊D4), SmallGroup(176,16)

Series: Derived Chief Lower central Upper central

C1C44 — Q8⋊D11
C1C11C22C44D44 — Q8⋊D11
C11C22C44 — Q8⋊D11
C1C2C4Q8

Generators and relations for Q8⋊D11
 G = < a,b,c,d | a4=c11=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

44C2
2C4
22C22
4D11
11C8
11D4
2D22
2C44
11SD16

Smallest permutation representation of Q8⋊D11
On 88 points
Generators in S88
(1 43 21 32)(2 44 22 33)(3 34 12 23)(4 35 13 24)(5 36 14 25)(6 37 15 26)(7 38 16 27)(8 39 17 28)(9 40 18 29)(10 41 19 30)(11 42 20 31)(45 67 56 78)(46 68 57 79)(47 69 58 80)(48 70 59 81)(49 71 60 82)(50 72 61 83)(51 73 62 84)(52 74 63 85)(53 75 64 86)(54 76 65 87)(55 77 66 88)
(1 65 21 54)(2 66 22 55)(3 56 12 45)(4 57 13 46)(5 58 14 47)(6 59 15 48)(7 60 16 49)(8 61 17 50)(9 62 18 51)(10 63 19 52)(11 64 20 53)(23 78 34 67)(24 79 35 68)(25 80 36 69)(26 81 37 70)(27 82 38 71)(28 83 39 72)(29 84 40 73)(30 85 41 74)(31 86 42 75)(32 87 43 76)(33 88 44 77)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 77)(53 76)(54 75)(55 74)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 88)(64 87)(65 86)(66 85)

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

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

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

Q8⋊D11 is a maximal subgroup of   SD16×D11  D88⋊C2  Q16⋊D11  D885C2  C44.C23  Q8⋊D22  D4.8D22
Q8⋊D11 is a maximal quotient of   C4.Dic22  C22.D8  Q8⋊Dic11

32 conjugacy classes

class 1 2A2B4A4B8A8B11A···11E22A···22E44A···44O
order122448811···1122···2244···44
size11442422222···22···24···4

32 irreducible representations

dim1111222224
type++++++++
imageC1C2C2C2D4SD16D11D22C11⋊D4Q8⋊D11
kernelQ8⋊D11C11⋊C8D44Q8×C11C22C11Q8C4C2C1
# reps11111255105

Matrix representation of Q8⋊D11 in GL4(𝔽89) generated by

88000
08800
004227
005047
,
323100
565700
007013
006819
,
3100
424400
0010
0001
,
448800
664500
0019
00088
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,42,50,0,0,27,47],[32,56,0,0,31,57,0,0,0,0,70,68,0,0,13,19],[3,42,0,0,1,44,0,0,0,0,1,0,0,0,0,1],[44,66,0,0,88,45,0,0,0,0,1,0,0,0,9,88] >;

Q8⋊D11 in GAP, Magma, Sage, TeX

Q_8\rtimes D_{11}
% in TeX

G:=Group("Q8:D11");
// GroupNames label

G:=SmallGroup(176,16);
// by ID

G=gap.SmallGroup(176,16);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,61,46,182,97,42,4004]);
// Polycyclic

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

Export

Subgroup lattice of Q8⋊D11 in TeX

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