metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44⋊4C4, C44.54D4, Dic22⋊4C4, C22.3D44, M4(2)⋊4D11, C11⋊2C4≀C2, C44.6(C2×C4), C4.3(C4×D11), (C2×C22).1D4, (C2×C4).38D22, (C4×Dic11)⋊1C2, C2.11(D22⋊C4), D44⋊5C2.2C2, C4.29(C11⋊D4), (C11×M4(2))⋊8C2, (C2×C44).15C22, C22.10(C22⋊C4), SmallGroup(352,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44⋊4C4
G = < a,b,c | a44=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a31b >
2C2
44C2
22C4
22C22
22C4
22C4
2C22
4D11
2C8
11D4
11Q8
22C2×C4
22C2×C4
22D4
2D22
11C42
11C4○D4
2C88
11C4≀C2
Smallest permutation representation of D44⋊4C4
►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 49)(2 48)(3 47)(4 46)(5 45)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 68)(27 67)(28 66)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)
(1 23)(2 44)(3 21)(4 42)(5 19)(6 40)(7 17)(8 38)(9 15)(10 36)(11 13)(12 34)(14 32)(16 30)(18 28)(20 26)(22 24)(25 43)(27 41)(29 39)(31 37)(33 35)(45 66 67 88)(46 87 68 65)(47 64 69 86)(48 85 70 63)(49 62 71 84)(50 83 72 61)(51 60 73 82)(52 81 74 59)(53 58 75 80)(54 79 76 57)(55 56 77 78)
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,66)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50), (1,23)(2,44)(3,21)(4,42)(5,19)(6,40)(7,17)(8,38)(9,15)(10,36)(11,13)(12,34)(14,32)(16,30)(18,28)(20,26)(22,24)(25,43)(27,41)(29,39)(31,37)(33,35)(45,66,67,88)(46,87,68,65)(47,64,69,86)(48,85,70,63)(49,62,71,84)(50,83,72,61)(51,60,73,82)(52,81,74,59)(53,58,75,80)(54,79,76,57)(55,56,77,78)>;
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,68)(27,67)(28,66)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50), (1,23)(2,44)(3,21)(4,42)(5,19)(6,40)(7,17)(8,38)(9,15)(10,36)(11,13)(12,34)(14,32)(16,30)(18,28)(20,26)(22,24)(25,43)(27,41)(29,39)(31,37)(33,35)(45,66,67,88)(46,87,68,65)(47,64,69,86)(48,85,70,63)(49,62,71,84)(50,83,72,61)(51,60,73,82)(52,81,74,59)(53,58,75,80)(54,79,76,57)(55,56,77,78) );
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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,88),(7,87),(8,86),(9,85),(10,84),(11,83),(12,82),(13,81),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,73),(22,72),(23,71),(24,70),(25,69),(26,68),(27,67),(28,66),(29,65),(30,64),(31,63),(32,62),(33,61),(34,60),(35,59),(36,58),(37,57),(38,56),(39,55),(40,54),(41,53),(42,52),(43,51),(44,50)], [(1,23),(2,44),(3,21),(4,42),(5,19),(6,40),(7,17),(8,38),(9,15),(10,36),(11,13),(12,34),(14,32),(16,30),(18,28),(20,26),(22,24),(25,43),(27,41),(29,39),(31,37),(33,35),(45,66,67,88),(46,87,68,65),(47,64,69,86),(48,85,70,63),(49,62,71,84),(50,83,72,61),(51,60,73,82),(52,81,74,59),(53,58,75,80),(54,79,76,57),(55,56,77,78)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44J | 44K | ··· | 44O | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 44 | 1 | 1 | 2 | 22 | 22 | 22 | 22 | 44 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D11 | C4≀C2 | D22 | C4×D11 | C11⋊D4 | D44 | D44⋊4C4 |
| kernel | D44⋊4C4 | C4×Dic11 | C11×M4(2) | D44⋊5C2 | Dic22 | D44 | C44 | C2×C22 | M4(2) | C11 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 5 | 4 | 5 | 10 | 10 | 10 | 10 |
Matrix representation of D44⋊4C4 ►in GL4(𝔽89) generated by
| 55 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 45 | 33 |
| 0 | 0 | 83 | 51 |
| 0 | 55 | 0 | 0 |
| 34 | 0 | 0 | 0 |
| 0 | 0 | 47 | 78 |
| 0 | 0 | 47 | 42 |
| 88 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 68 | 72 |
G:=sub<GL(4,GF(89))| [55,0,0,0,0,34,0,0,0,0,45,83,0,0,33,51],[0,34,0,0,55,0,0,0,0,0,47,47,0,0,78,42],[88,0,0,0,0,34,0,0,0,0,17,68,0,0,1,72] >;
D44⋊4C4 in GAP, Magma, Sage, TeX
D_{44}\rtimes_4C_4 % in TeX
G:=Group("D44:4C4"); // GroupNames label
G:=SmallGroup(352,31);
// by ID
G=gap.SmallGroup(352,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,86,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^31*b>;
// generators/relations
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