metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44⋊4C2, Q8⋊2D11, C4.7D22, C22.8C23, C44.7C22, D22.3C22, Dic11.5C22, (C4×D11)⋊3C2, C11⋊3(C4○D4), (Q8×C11)⋊3C2, C2.9(C22×D11), SmallGroup(176,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44⋊C2
G = < a,b,c | a44=b2=c2=1, bab=a-1, cac=a21, cbc=a42b >
Smallest permutation representation of D44⋊C2
►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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(81 88)(82 87)(83 86)(84 85)
(1 74)(2 51)(3 72)(4 49)(5 70)(6 47)(7 68)(8 45)(9 66)(10 87)(11 64)(12 85)(13 62)(14 83)(15 60)(16 81)(17 58)(18 79)(19 56)(20 77)(21 54)(22 75)(23 52)(24 73)(25 50)(26 71)(27 48)(28 69)(29 46)(30 67)(31 88)(32 65)(33 86)(34 63)(35 84)(36 61)(37 82)(38 59)(39 80)(40 57)(41 78)(42 55)(43 76)(44 53)
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(81,88)(82,87)(83,86)(84,85), (1,74)(2,51)(3,72)(4,49)(5,70)(6,47)(7,68)(8,45)(9,66)(10,87)(11,64)(12,85)(13,62)(14,83)(15,60)(16,81)(17,58)(18,79)(19,56)(20,77)(21,54)(22,75)(23,52)(24,73)(25,50)(26,71)(27,48)(28,69)(29,46)(30,67)(31,88)(32,65)(33,86)(34,63)(35,84)(36,61)(37,82)(38,59)(39,80)(40,57)(41,78)(42,55)(43,76)(44,53)>;
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(81,88)(82,87)(83,86)(84,85), (1,74)(2,51)(3,72)(4,49)(5,70)(6,47)(7,68)(8,45)(9,66)(10,87)(11,64)(12,85)(13,62)(14,83)(15,60)(16,81)(17,58)(18,79)(19,56)(20,77)(21,54)(22,75)(23,52)(24,73)(25,50)(26,71)(27,48)(28,69)(29,46)(30,67)(31,88)(32,65)(33,86)(34,63)(35,84)(36,61)(37,82)(38,59)(39,80)(40,57)(41,78)(42,55)(43,76)(44,53) );
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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(81,88),(82,87),(83,86),(84,85)], [(1,74),(2,51),(3,72),(4,49),(5,70),(6,47),(7,68),(8,45),(9,66),(10,87),(11,64),(12,85),(13,62),(14,83),(15,60),(16,81),(17,58),(18,79),(19,56),(20,77),(21,54),(22,75),(23,52),(24,73),(25,50),(26,71),(27,48),(28,69),(29,46),(30,67),(31,88),(32,65),(33,86),(34,63),(35,84),(36,61),(37,82),(38,59),(39,80),(40,57),(41,78),(42,55),(43,76),(44,53)]])
D44⋊C2 is a maximal subgroup of
D88⋊C2 Q8.D22 Q16⋊D11 D88⋊5C2 Q8.10D22 C4○D4×D11 D4⋊8D22
D44⋊C2 is a maximal quotient of
C44.3Q8 C4⋊C4⋊7D11 D44⋊C4 D22.5D4 C4⋊2D44 C4⋊C4⋊D11 Q8×Dic11 D22⋊3Q8 C44.23D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44O |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 22 | 22 | 22 | 2 | 2 | 2 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D11 | D22 | D44⋊C2 |
| kernel | D44⋊C2 | C4×D11 | D44 | Q8×C11 | C11 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 5 | 15 | 5 |
Matrix representation of D44⋊C2 ►in GL4(𝔽89) generated by
| 54 | 8 | 0 | 0 |
| 35 | 53 | 0 | 0 |
| 0 | 0 | 1 | 53 |
| 0 | 0 | 5 | 88 |
| 26 | 55 | 0 | 0 |
| 12 | 63 | 0 | 0 |
| 0 | 0 | 1 | 53 |
| 0 | 0 | 0 | 88 |
| 13 | 86 | 0 | 0 |
| 56 | 76 | 0 | 0 |
| 0 | 0 | 34 | 22 |
| 0 | 0 | 81 | 55 |
G:=sub<GL(4,GF(89))| [54,35,0,0,8,53,0,0,0,0,1,5,0,0,53,88],[26,12,0,0,55,63,0,0,0,0,1,0,0,0,53,88],[13,56,0,0,86,76,0,0,0,0,34,81,0,0,22,55] >;
D44⋊C2 in GAP, Magma, Sage, TeX
D_{44}\rtimes C_2 % in TeX
G:=Group("D44:C2"); // GroupNames label
G:=SmallGroup(176,34);
// by ID
G=gap.SmallGroup(176,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,46,182,97,42,4004]);
// Polycyclic
G:=Group<a,b,c|a^44=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^21,c*b*c=a^42*b>;
// generators/relations
Export