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G = C8⋊D11order 176 = 24·11

2nd semidirect product of C8 and D11 acting via D11/C11=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C882C2, C82D11, C22.1D4, C2.3D44, C4.8D22, C111SD16, D44.1C2, Dic221C2, C44.8C22, SmallGroup(176,5)

Series: Derived Chief Lower central Upper central

C1C44 — C8⋊D11
C1C11C22C44D44 — C8⋊D11
C11C22C44 — C8⋊D11
C1C2C4C8

Generators and relations for C8⋊D11
 G = < a,b,c | a8=b11=c2=1, ab=ba, cac=a3, cbc=b-1 >

44C2
22C22
22C4
4D11
11Q8
11D4
2Dic11
2D22
11SD16

Smallest permutation representation of C8⋊D11
On 88 points
Generators in S88
(1 87 43 65 21 76 32 54)(2 88 44 66 22 77 33 55)(3 78 34 56 12 67 23 45)(4 79 35 57 13 68 24 46)(5 80 36 58 14 69 25 47)(6 81 37 59 15 70 26 48)(7 82 38 60 16 71 27 49)(8 83 39 61 17 72 28 50)(9 84 40 62 18 73 29 51)(10 85 41 63 19 74 30 52)(11 86 42 64 20 75 31 53)
(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)| (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53), (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( (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53), (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([(1,87,43,65,21,76,32,54),(2,88,44,66,22,77,33,55),(3,78,34,56,12,67,23,45),(4,79,35,57,13,68,24,46),(5,80,36,58,14,69,25,47),(6,81,37,59,15,70,26,48),(7,82,38,60,16,71,27,49),(8,83,39,61,17,72,28,50),(9,84,40,62,18,73,29,51),(10,85,41,63,19,74,30,52),(11,86,42,64,20,75,31,53)], [(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)])

C8⋊D11 is a maximal subgroup of   D887C2  C8⋊D22  C8.D22  D4⋊D22  SD16×D11  Q8.D22  Q16⋊D11
C8⋊D11 is a maximal quotient of   C44.44D4  C44.4Q8  C2.D88

47 conjugacy classes

class 1 2A2B4A4B8A8B11A···11E22A···22E44A···44J88A···88T
order122448811···1122···2244···4488···88
size1144244222···22···22···22···2

47 irreducible representations

dim1111222222
type++++++++
imageC1C2C2C2D4SD16D11D22D44C8⋊D11
kernelC8⋊D11C88Dic22D44C22C11C8C4C2C1
# reps111112551020

Matrix representation of C8⋊D11 in GL2(𝔽43) generated by

3136
3639
,
3324
2424
,
348
339
G:=sub<GL(2,GF(43))| [31,36,36,39],[33,24,24,24],[34,33,8,9] >;

C8⋊D11 in GAP, Magma, Sage, TeX

C_8\rtimes D_{11}
% in TeX

G:=Group("C8:D11");
// GroupNames label

G:=SmallGroup(176,5);
// by ID

G=gap.SmallGroup(176,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,61,26,182,42,4004]);
// Polycyclic

G:=Group<a,b,c|a^8=b^11=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D11 in TeX

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