metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C88⋊2C2, C8⋊2D11, C22.1D4, C2.3D44, C4.8D22, C11⋊1SD16, D44.1C2, Dic22⋊1C2, C44.8C22, SmallGroup(176,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D11
G = < a,b,c | a8=b11=c2=1, ab=ba, cac=a3, cbc=b-1 >
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
D88⋊7C2 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 | 2A | 2B | 4A | 4B | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 44 | 2 | 44 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D11 | D22 | D44 | C8⋊D11 |
| kernel | C8⋊D11 | C88 | Dic22 | D44 | C22 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 10 | 20 |
Matrix representation of C8⋊D11 ►in GL2(𝔽43) generated by
| 31 | 36 |
| 36 | 39 |
| 33 | 24 |
| 24 | 24 |
| 34 | 8 |
| 33 | 9 |
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
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