metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22⋊C4, C22.6D4, C2.2D44, C22.6D22, (C2×C44)⋊1C2, (C2×C4)⋊1D11, C22.5(C2×C4), C2.5(C4×D11), C11⋊1(C22⋊C4), (C22×D11).C2, (C2×Dic11)⋊1C2, C2.2(C11⋊D4), (C2×C22).6C22, SmallGroup(176,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D22⋊C4
G = < a,b,c | a22=b2=c4=1, bab=a-1, ac=ca, cbc-1=a11b >
Smallest permutation representation of D22⋊C4
►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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 44)(20 43)(21 42)(22 41)(45 70)(46 69)(47 68)(48 67)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)
(1 67 41 60)(2 68 42 61)(3 69 43 62)(4 70 44 63)(5 71 23 64)(6 72 24 65)(7 73 25 66)(8 74 26 45)(9 75 27 46)(10 76 28 47)(11 77 29 48)(12 78 30 49)(13 79 31 50)(14 80 32 51)(15 81 33 52)(16 82 34 53)(17 83 35 54)(18 84 36 55)(19 85 37 56)(20 86 38 57)(21 87 39 58)(22 88 40 59)
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,44)(20,43)(21,42)(22,41)(45,70)(46,69)(47,68)(48,67)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71), (1,67,41,60)(2,68,42,61)(3,69,43,62)(4,70,44,63)(5,71,23,64)(6,72,24,65)(7,73,25,66)(8,74,26,45)(9,75,27,46)(10,76,28,47)(11,77,29,48)(12,78,30,49)(13,79,31,50)(14,80,32,51)(15,81,33,52)(16,82,34,53)(17,83,35,54)(18,84,36,55)(19,85,37,56)(20,86,38,57)(21,87,39,58)(22,88,40,59)>;
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,44)(20,43)(21,42)(22,41)(45,70)(46,69)(47,68)(48,67)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71), (1,67,41,60)(2,68,42,61)(3,69,43,62)(4,70,44,63)(5,71,23,64)(6,72,24,65)(7,73,25,66)(8,74,26,45)(9,75,27,46)(10,76,28,47)(11,77,29,48)(12,78,30,49)(13,79,31,50)(14,80,32,51)(15,81,33,52)(16,82,34,53)(17,83,35,54)(18,84,36,55)(19,85,37,56)(20,86,38,57)(21,87,39,58)(22,88,40,59) );
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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,44),(20,43),(21,42),(22,41),(45,70),(46,69),(47,68),(48,67),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71)], [(1,67,41,60),(2,68,42,61),(3,69,43,62),(4,70,44,63),(5,71,23,64),(6,72,24,65),(7,73,25,66),(8,74,26,45),(9,75,27,46),(10,76,28,47),(11,77,29,48),(12,78,30,49),(13,79,31,50),(14,80,32,51),(15,81,33,52),(16,82,34,53),(17,83,35,54),(18,84,36,55),(19,85,37,56),(20,86,38,57),(21,87,39,58),(22,88,40,59)]])
D22⋊C4 is a maximal subgroup of
C42⋊D11 C4×D44 C4.D44 C42⋊2D11 C22⋊C4×D11 Dic11⋊4D4 C22⋊D44 D22.D4 D22⋊D4 Dic11.D4 C22.D44 C4⋊C4⋊7D11 D44⋊C4 D22.5D4 C4⋊2D44 D22⋊Q8 D22⋊2Q8 C4⋊C4⋊D11 C4×C11⋊D4 C23.23D22 C44⋊7D4 C23⋊D22 Dic11⋊D4 D22⋊3Q8 C44.23D4
D22⋊C4 is a maximal quotient of
D44⋊1C4 C22.2D44 C22.D8 C22.Q16 C44.44D4 D22⋊C8 C2.D88 C44.46D4 C44.47D4 D44⋊4C4 C22.C42
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 22 | 22 | 2 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D11 | D22 | C4×D11 | D44 | C11⋊D4 |
| kernel | D22⋊C4 | C2×Dic11 | C2×C44 | C22×D11 | D22 | C22 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 5 | 5 | 10 | 10 | 10 |
Matrix representation of D22⋊C4 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 36 | 30 |
| 0 | 82 | 56 |
| 1 | 0 | 0 |
| 0 | 0 | 62 |
| 0 | 56 | 0 |
| 55 | 0 | 0 |
| 0 | 86 | 9 |
| 0 | 78 | 3 |
G:=sub<GL(3,GF(89))| [1,0,0,0,36,82,0,30,56],[1,0,0,0,0,56,0,62,0],[55,0,0,0,86,78,0,9,3] >;
D22⋊C4 in GAP, Magma, Sage, TeX
D_{22}\rtimes C_4 % in TeX
G:=Group("D22:C4"); // GroupNames label
G:=SmallGroup(176,13);
// by ID
G=gap.SmallGroup(176,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,101,26,4004]);
// Polycyclic
G:=Group<a,b,c|a^22=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations
Export