metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44⋊5C2, C4.16D22, Dic22⋊5C2, C22.4C23, C22.2D22, C44.16C22, D22.1C22, Dic11.2C22, (C2×C44)⋊4C2, (C2×C4)⋊3D11, (C4×D11)⋊4C2, C11⋊1(C4○D4), C11⋊D4⋊3C2, C2.5(C22×D11), (C2×C22).11C22, SmallGroup(176,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44⋊5C2
G = < a,b,c | a44=b2=c2=1, bab=a-1, ac=ca, cbc=a22b >
Smallest permutation representation of D44⋊5C2
►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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 47)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 45)(39 46)(40 47)(41 48)(42 49)(43 50)(44 51)
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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,47)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,45)(39,46)(40,47)(41,48)(42,49)(43,50)(44,51)>;
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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,47)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,45)(39,46)(40,47)(41,48)(42,49)(43,50)(44,51) );
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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(45,47),(48,88),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,80),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,45),(39,46),(40,47),(41,48),(42,49),(43,50),(44,51)]])
D44⋊5C2 is a maximal subgroup of
D44⋊1C4 D44⋊4C4 D44.2C4 D88⋊7C2 D44.C4 C8⋊D22 C8.D22 D44⋊6C22 C44.C23 D4.8D22 D4⋊6D22 Q8.10D22 C4○D4×D11 D4⋊8D22 D4.10D22
D44⋊5C2 is a maximal quotient of
C4×Dic22 C44.6Q8 C42⋊D11 C4×D44 C4.D44 C42⋊2D11 C23.D22 D22.D4 D22⋊D4 Dic11.D4 Dic11.Q8 D22.5D4 D22⋊Q8 C4⋊C4⋊D11 C44.48D4 C23.21D22 C4×C11⋊D4 C23.23D22 C44⋊7D4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 22 | 22 | 1 | 1 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D11 | D22 | D22 | D44⋊5C2 |
| kernel | D44⋊5C2 | Dic22 | C4×D11 | D44 | C11⋊D4 | C2×C44 | C11 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 5 | 10 | 5 | 20 |
Matrix representation of D44⋊5C2 ►in GL2(𝔽89) generated by
| 52 | 54 |
| 60 | 88 |
| 37 | 5 |
| 29 | 52 |
| 80 | 27 |
| 30 | 9 |
G:=sub<GL(2,GF(89))| [52,60,54,88],[37,29,5,52],[80,30,27,9] >;
D44⋊5C2 in GAP, Magma, Sage, TeX
D_{44}\rtimes_5C_2 % in TeX
G:=Group("D44:5C2"); // GroupNames label
G:=SmallGroup(176,30);
// by ID
G=gap.SmallGroup(176,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,46,182,4004]);
// Polycyclic
G:=Group<a,b,c|a^44=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^22*b>;
// generators/relations
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