direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×D8, D4⋊C22, C8⋊1C22, C88⋊5C2, C22.14D4, C44.17C22, (D4×C11)⋊4C2, C4.1(C2×C22), C2.3(D4×C11), SmallGroup(176,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×D8
G = < a,b,c | a11=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D8
►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 75 30 50 79 66 21)(2 41 76 31 51 80 56 22)(3 42 77 32 52 81 57 12)(4 43 67 33 53 82 58 13)(5 44 68 23 54 83 59 14)(6 34 69 24 55 84 60 15)(7 35 70 25 45 85 61 16)(8 36 71 26 46 86 62 17)(9 37 72 27 47 87 63 18)(10 38 73 28 48 88 64 19)(11 39 74 29 49 78 65 20)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 54)(24 55)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 56)(42 57)(43 58)(44 59)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 78)(75 79)(76 80)(77 81)
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), (1,40,75,30,50,79,66,21)(2,41,76,31,51,80,56,22)(3,42,77,32,52,81,57,12)(4,43,67,33,53,82,58,13)(5,44,68,23,54,83,59,14)(6,34,69,24,55,84,60,15)(7,35,70,25,45,85,61,16)(8,36,71,26,46,86,62,17)(9,37,72,27,47,87,63,18)(10,38,73,28,48,88,64,19)(11,39,74,29,49,78,65,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,54)(24,55)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,56)(42,57)(43,58)(44,59)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,78)(75,79)(76,80)(77,81)>;
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), (1,40,75,30,50,79,66,21)(2,41,76,31,51,80,56,22)(3,42,77,32,52,81,57,12)(4,43,67,33,53,82,58,13)(5,44,68,23,54,83,59,14)(6,34,69,24,55,84,60,15)(7,35,70,25,45,85,61,16)(8,36,71,26,46,86,62,17)(9,37,72,27,47,87,63,18)(10,38,73,28,48,88,64,19)(11,39,74,29,49,78,65,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,54)(24,55)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,56)(42,57)(43,58)(44,59)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,78)(75,79)(76,80)(77,81) );
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)], [(1,40,75,30,50,79,66,21),(2,41,76,31,51,80,56,22),(3,42,77,32,52,81,57,12),(4,43,67,33,53,82,58,13),(5,44,68,23,54,83,59,14),(6,34,69,24,55,84,60,15),(7,35,70,25,45,85,61,16),(8,36,71,26,46,86,62,17),(9,37,72,27,47,87,63,18),(10,38,73,28,48,88,64,19),(11,39,74,29,49,78,65,20)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,54),(24,55),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,56),(42,57),(43,58),(44,59),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,78),(75,79),(76,80),(77,81)]])
C11×D8 is a maximal subgroup of
C11⋊D16 D8.D11 D4⋊D22 D8⋊3D11
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D4 | D8 | D4×C11 | C11×D8 |
| kernel | C11×D8 | C88 | D4×C11 | D8 | C8 | D4 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 2 | 10 | 20 |
Matrix representation of C11×D8 ►in GL2(𝔽23) generated by
| 2 | 0 |
| 0 | 2 |
| 5 | 14 |
| 18 | 0 |
| 0 | 9 |
| 18 | 0 |
G:=sub<GL(2,GF(23))| [2,0,0,2],[5,18,14,0],[0,18,9,0] >;
C11×D8 in GAP, Magma, Sage, TeX
C_{11}\times D_8 % in TeX
G:=Group("C11xD8"); // GroupNames label
G:=SmallGroup(176,24);
// by ID
G=gap.SmallGroup(176,24);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-2,461,2643,1328,58]);
// Polycyclic
G:=Group<a,b,c|a^11=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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