metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D11, C11⋊2D8, D44⋊2C2, C4.1D22, C22.7D4, C44.1C22, C11⋊C8⋊1C2, (D4×C11)⋊1C2, C2.4(C11⋊D4), SmallGroup(176,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D11
G = < a,b,c,d | a4=b2=c11=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊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 87)(2 88)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)
(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), (1,87)(2,88)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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), (1,87)(2,88)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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)], [(1,87),(2,88),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66)], [(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)]])
D4⋊D11 is a maximal subgroup of
D8×D11 D4⋊D22 D88⋊C2 Q8.D22 D44⋊6C22 Q8⋊D22 D4.8D22
D4⋊D11 is a maximal quotient of C44.Q8 C22.D8 C11⋊D16 D8.D11 C8.6D22 C11⋊Q32 D4⋊Dic11
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22O | 44A | ··· | 44E |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 4 | 44 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D11 | D22 | C11⋊D4 | D4⋊D11 |
| kernel | D4⋊D11 | C11⋊C8 | D44 | D4×C11 | C22 | C11 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 10 | 5 |
Matrix representation of D4⋊D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 36 |
| 0 | 0 | 84 | 1 |
| 16 | 39 | 0 | 0 |
| 14 | 73 | 0 | 0 |
| 0 | 0 | 25 | 84 |
| 0 | 0 | 18 | 64 |
| 37 | 1 | 0 | 0 |
| 46 | 59 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 66 | 41 | 0 | 0 |
| 11 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 88 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,84,0,0,36,1],[16,14,0,0,39,73,0,0,0,0,25,18,0,0,84,64],[37,46,0,0,1,59,0,0,0,0,1,0,0,0,0,1],[66,11,0,0,41,23,0,0,0,0,1,5,0,0,0,88] >;
D4⋊D11 in GAP, Magma, Sage, TeX
D_4\rtimes D_{11} % in TeX
G:=Group("D4:D11"); // GroupNames label
G:=SmallGroup(176,14);
// by ID
G=gap.SmallGroup(176,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,61,182,97,42,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^11=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export