metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22⋊4D4, C22⋊2D44, C23.15D22, (C2×C22)⋊1D4, (C2×C4)⋊1D22, D22⋊C4⋊4C2, C11⋊1C22≀C2, (C2×D44)⋊2C2, C2.7(D4×D11), C2.7(C2×D44), C22.5(C2×D4), (C2×C44)⋊1C22, C22⋊C4⋊2D11, (C23×D11)⋊1C2, (C2×C22).23C23, (C2×Dic11)⋊1C22, (C22×D11)⋊1C22, (C22×C22).12C22, C22.41(C22×D11), (C2×C11⋊D4)⋊1C2, (C11×C22⋊C4)⋊3C2, SmallGroup(352,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊D44
G = < a,b,c,d | a2=b2=c44=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1042 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C22⋊C4, C2×D4, C24, D11, C22, C22, C22, C22≀C2, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, D44, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×D11, C22×D11, C22×C22, D22⋊C4, C11×C22⋊C4, C2×D44, C2×C11⋊D4, C23×D11, C22⋊D44
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C22≀C2, D22, D44, C22×D11, C2×D44, D4×D11, C22⋊D44
►On 88 points
Generators in S88
(2 54)(4 56)(6 58)(8 60)(10 62)(12 64)(14 66)(16 68)(18 70)(20 72)(22 74)(24 76)(26 78)(28 80)(30 82)(32 84)(34 86)(36 88)(38 46)(40 48)(42 50)(44 52)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)
G:=sub<Sym(88)| (2,54)(4,56)(6,58)(8,60)(10,62)(12,64)(14,66)(16,68)(18,70)(20,72)(22,74)(24,76)(26,78)(28,80)(30,82)(32,84)(34,86)(36,88)(38,46)(40,48)(42,50)(44,52), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)>;
G:=Group( (2,54)(4,56)(6,58)(8,60)(10,62)(12,64)(14,66)(16,68)(18,70)(20,72)(22,74)(24,76)(26,78)(28,80)(30,82)(32,84)(34,86)(36,88)(38,46)(40,48)(42,50)(44,52), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53) );
G=PermutationGroup([[(2,54),(4,56),(6,58),(8,60),(10,62),(12,64),(14,66),(16,68),(18,70),(20,72),(22,74),(24,76),(26,78),(28,80),(30,82),(32,84),(34,86),(36,88),(38,46),(40,48),(42,50),(44,52)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,45),(38,46),(39,47),(40,48),(41,49),(42,50),(43,51),(44,52)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,81),(17,80),(18,79),(19,78),(20,77),(21,76),(22,75),(23,74),(24,73),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,54),(44,53)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 44 | 4 | 4 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | D22 | D22 | D44 | D4×D11 |
| kernel | C22⋊D44 | D22⋊C4 | C11×C22⋊C4 | C2×D44 | C2×C11⋊D4 | C23×D11 | D22 | C2×C22 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 5 | 10 | 5 | 20 | 10 |
Matrix representation of C22⋊D44 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 85 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 36 | 68 | 0 | 0 |
| 52 | 76 | 0 | 0 |
| 0 | 0 | 79 | 84 |
| 0 | 0 | 2 | 10 |
| 16 | 29 | 0 | 0 |
| 71 | 73 | 0 | 0 |
| 0 | 0 | 10 | 5 |
| 0 | 0 | 87 | 79 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,85,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[36,52,0,0,68,76,0,0,0,0,79,2,0,0,84,10],[16,71,0,0,29,73,0,0,0,0,10,87,0,0,5,79] >;
C22⋊D44 in GAP, Magma, Sage, TeX
C_2^2\rtimes D_{44} % in TeX
G:=Group("C2^2:D44"); // GroupNames label
G:=SmallGroup(352,77);
// by ID
G=gap.SmallGroup(352,77);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,188,50,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^44=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations