metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D22, D88⋊2C2, C88⋊1C22, C4.14D44, C44.12D4, D44⋊4C22, C22.5D44, M4(2)⋊1D11, C44.32C23, Dic22⋊4C22, (C2×D44)⋊7C2, C8⋊D11⋊1C2, (C2×C22).5D4, C11⋊1(C8⋊C22), C22.13(C2×D4), C2.15(C2×D44), (C2×C4).15D22, D44⋊5C2⋊2C2, (C11×M4(2))⋊1C2, (C2×C44).27C22, C4.30(C22×D11), SmallGroup(352,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D22
G = < a,b,c | a8=b22=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 586 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C11, M4(2), D8, SD16, C2×D4, C4○D4, D11, C22, C22, C8⋊C22, Dic11, C44, D22, C2×C22, C88, Dic22, C4×D11, D44, D44, D44, C11⋊D4, C2×C44, C22×D11, C8⋊D11, D88, C11×M4(2), C2×D44, D44⋊5C2, C8⋊D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8⋊C22, D22, D44, C22×D11, C2×D44, C8⋊D22
►On 88 points
Generators in S88
(1 77 21 64 28 88 44 53)(2 67 22 54 29 78 34 65)(3 79 12 66 30 68 35 55)(4 69 13 56 31 80 36 45)(5 81 14 46 32 70 37 57)(6 71 15 58 33 82 38 47)(7 83 16 48 23 72 39 59)(8 73 17 60 24 84 40 49)(9 85 18 50 25 74 41 61)(10 75 19 62 26 86 42 51)(11 87 20 52 27 76 43 63)
(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 14)(2 13)(3 12)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 44)(33 43)(45 54)(46 53)(47 52)(48 51)(49 50)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(67 69)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)
G:=sub<Sym(88)| (1,77,21,64,28,88,44,53)(2,67,22,54,29,78,34,65)(3,79,12,66,30,68,35,55)(4,69,13,56,31,80,36,45)(5,81,14,46,32,70,37,57)(6,71,15,58,33,82,38,47)(7,83,16,48,23,72,39,59)(8,73,17,60,24,84,40,49)(9,85,18,50,25,74,41,61)(10,75,19,62,26,86,42,51)(11,87,20,52,27,76,43,63), (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), (1,14)(2,13)(3,12)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,44)(33,43)(45,54)(46,53)(47,52)(48,51)(49,50)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(67,69)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)>;
G:=Group( (1,77,21,64,28,88,44,53)(2,67,22,54,29,78,34,65)(3,79,12,66,30,68,35,55)(4,69,13,56,31,80,36,45)(5,81,14,46,32,70,37,57)(6,71,15,58,33,82,38,47)(7,83,16,48,23,72,39,59)(8,73,17,60,24,84,40,49)(9,85,18,50,25,74,41,61)(10,75,19,62,26,86,42,51)(11,87,20,52,27,76,43,63), (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), (1,14)(2,13)(3,12)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,44)(33,43)(45,54)(46,53)(47,52)(48,51)(49,50)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(67,69)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80) );
G=PermutationGroup([[(1,77,21,64,28,88,44,53),(2,67,22,54,29,78,34,65),(3,79,12,66,30,68,35,55),(4,69,13,56,31,80,36,45),(5,81,14,46,32,70,37,57),(6,71,15,58,33,82,38,47),(7,83,16,48,23,72,39,59),(8,73,17,60,24,84,40,49),(9,85,18,50,25,74,41,61),(10,75,19,62,26,86,42,51),(11,87,20,52,27,76,43,63)], [(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)], [(1,14),(2,13),(3,12),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,44),(33,43),(45,54),(46,53),(47,52),(48,51),(49,50),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(67,69),(70,88),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44J | 44K | ··· | 44O | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 44 | 44 | 44 | 2 | 2 | 44 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | D22 | D22 | D44 | D44 | C8⋊C22 | C8⋊D22 |
| kernel | C8⋊D22 | C8⋊D11 | D88 | C11×M4(2) | C2×D44 | D44⋊5C2 | C44 | C2×C22 | M4(2) | C8 | C2×C4 | C4 | C22 | C11 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 5 | 10 | 5 | 10 | 10 | 1 | 10 |
Matrix representation of C8⋊D22 ►in GL6(𝔽89)
| 17 | 30 | 0 | 0 | 0 | 0 |
| 20 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 73 | 0 | 79 | 0 |
| 0 | 0 | 26 | 0 | 48 | 1 |
| 0 | 0 | 66 | 1 | 16 | 0 |
| 0 | 0 | 15 | 0 | 26 | 0 |
| 54 | 53 | 0 | 0 | 0 | 0 |
| 65 | 77 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 0 | 0 | 0 |
| 0 | 0 | 0 | 88 | 0 | 0 |
| 0 | 0 | 21 | 0 | 1 | 0 |
| 0 | 0 | 23 | 0 | 0 | 1 |
| 73 | 11 | 0 | 0 | 0 | 0 |
| 82 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 0 | 0 | 0 |
| 0 | 0 | 63 | 1 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 1 |
| 0 | 0 | 22 | 0 | 1 | 0 |
G:=sub<GL(6,GF(89))| [17,20,0,0,0,0,30,72,0,0,0,0,0,0,73,26,66,15,0,0,0,0,1,0,0,0,79,48,16,26,0,0,0,1,0,0],[54,65,0,0,0,0,53,77,0,0,0,0,0,0,88,0,21,23,0,0,0,88,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[73,82,0,0,0,0,11,16,0,0,0,0,0,0,88,63,22,22,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C8⋊D22 in GAP, Magma, Sage, TeX
C_8\rtimes D_{22} % in TeX
G:=Group("C8:D22"); // GroupNames label
G:=SmallGroup(352,103);
// by ID
G=gap.SmallGroup(352,103);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,188,50,579,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^8=b^22=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations