metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44⋊1C4, C44.33D4, C4.17D44, C42⋊3D11, Dic22⋊1C4, C11⋊1C4≀C2, (C4×C44)⋊6C2, C4.6(C4×D11), C44.16(C2×C4), (C2×C22).26D4, (C2×C4).66D22, C44.C4⋊1C2, C2.3(D22⋊C4), D44⋊5C2.1C2, C22.1(C22⋊C4), (C2×C44).96C22, C22.7(C11⋊D4), SmallGroup(352,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44⋊1C4
G = < a,b,c | a44=c4=1, b2=a22, bab-1=a-1, ac=ca, cbc-1=a11b >
2C2
44C2
2C4
2C4
22C22
22C4
2C22
4D11
11D4
11Q8
22C8
22D4
22C2×C4
2C44
2C44
2D22
11M4(2)
11C4○D4
11C4≀C2
Smallest permutation representation of D44⋊1C4
►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 67 23 45)(2 66 24 88)(3 65 25 87)(4 64 26 86)(5 63 27 85)(6 62 28 84)(7 61 29 83)(8 60 30 82)(9 59 31 81)(10 58 32 80)(11 57 33 79)(12 56 34 78)(13 55 35 77)(14 54 36 76)(15 53 37 75)(16 52 38 74)(17 51 39 73)(18 50 40 72)(19 49 41 71)(20 48 42 70)(21 47 43 69)(22 46 44 68)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 78 67 56)(46 79 68 57)(47 80 69 58)(48 81 70 59)(49 82 71 60)(50 83 72 61)(51 84 73 62)(52 85 74 63)(53 86 75 64)(54 87 76 65)(55 88 77 66)
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), (1,67,23,45)(2,66,24,88)(3,65,25,87)(4,64,26,86)(5,63,27,85)(6,62,28,84)(7,61,29,83)(8,60,30,82)(9,59,31,81)(10,58,32,80)(11,57,33,79)(12,56,34,78)(13,55,35,77)(14,54,36,76)(15,53,37,75)(16,52,38,74)(17,51,39,73)(18,50,40,72)(19,49,41,71)(20,48,42,70)(21,47,43,69)(22,46,44,68), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,78,67,56)(46,79,68,57)(47,80,69,58)(48,81,70,59)(49,82,71,60)(50,83,72,61)(51,84,73,62)(52,85,74,63)(53,86,75,64)(54,87,76,65)(55,88,77,66)>;
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), (1,67,23,45)(2,66,24,88)(3,65,25,87)(4,64,26,86)(5,63,27,85)(6,62,28,84)(7,61,29,83)(8,60,30,82)(9,59,31,81)(10,58,32,80)(11,57,33,79)(12,56,34,78)(13,55,35,77)(14,54,36,76)(15,53,37,75)(16,52,38,74)(17,51,39,73)(18,50,40,72)(19,49,41,71)(20,48,42,70)(21,47,43,69)(22,46,44,68), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,78,67,56)(46,79,68,57)(47,80,69,58)(48,81,70,59)(49,82,71,60)(50,83,72,61)(51,84,73,62)(52,85,74,63)(53,86,75,64)(54,87,76,65)(55,88,77,66) );
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)], [(1,67,23,45),(2,66,24,88),(3,65,25,87),(4,64,26,86),(5,63,27,85),(6,62,28,84),(7,61,29,83),(8,60,30,82),(9,59,31,81),(10,58,32,80),(11,57,33,79),(12,56,34,78),(13,55,35,77),(14,54,36,76),(15,53,37,75),(16,52,38,74),(17,51,39,73),(18,50,40,72),(19,49,41,71),(20,48,42,70),(21,47,43,69),(22,46,44,68)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,78,67,56),(46,79,68,57),(47,80,69,58),(48,81,70,59),(49,82,71,60),(50,83,72,61),(51,84,73,62),(52,85,74,63),(53,86,75,64),(54,87,76,65),(55,88,77,66)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 44 | 1 | 1 | 2 | ··· | 2 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D11 | C4≀C2 | D22 | C4×D11 | D44 | C11⋊D4 | D44⋊1C4 |
| kernel | D44⋊1C4 | C44.C4 | C4×C44 | D44⋊5C2 | Dic22 | D44 | C44 | C2×C22 | C42 | C11 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 5 | 4 | 5 | 10 | 10 | 10 | 40 |
Matrix representation of D44⋊1C4 ►in GL2(𝔽89) generated by
| 53 | 0 |
| 0 | 42 |
| 0 | 1 |
| 88 | 0 |
| 88 | 0 |
| 0 | 55 |
G:=sub<GL(2,GF(89))| [53,0,0,42],[0,88,1,0],[88,0,0,55] >;
D44⋊1C4 in GAP, Magma, Sage, TeX
D_{44}\rtimes_1C_4 % in TeX
G:=Group("D44:1C4"); // GroupNames label
G:=SmallGroup(352,11);
// by ID
G=gap.SmallGroup(352,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,362,579,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=c^4=1,b^2=a^22,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations
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