direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D11, C88⋊3C2, D22.2C4, C4.12D22, C44.12C22, Dic11.2C4, C11⋊C8⋊6C2, C11⋊1(C2×C8), C22.1(C2×C4), C2.1(C4×D11), (C4×D11).3C2, SmallGroup(176,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C8×D11 |
Generators and relations for C8×D11
G = < a,b,c | a8=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D11
►On 88 points
Generators in S88
(1 87 43 65 21 76 32 54)(2 88 44 66 22 77 33 55)(3 78 34 56 12 67 23 45)(4 79 35 57 13 68 24 46)(5 80 36 58 14 69 25 47)(6 81 37 59 15 70 26 48)(7 82 38 60 16 71 27 49)(8 83 39 61 17 72 28 50)(9 84 40 62 18 73 29 51)(10 85 41 63 19 74 30 52)(11 86 42 64 20 75 31 53)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)
G:=sub<Sym(88)| (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)>;
G:=Group( (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85) );
G=PermutationGroup([[(1,87,43,65,21,76,32,54),(2,88,44,66,22,77,33,55),(3,78,34,56,12,67,23,45),(4,79,35,57,13,68,24,46),(5,80,36,58,14,69,25,47),(6,81,37,59,15,70,26,48),(7,82,38,60,16,71,27,49),(8,83,39,61,17,72,28,50),(9,84,40,62,18,73,29,51),(10,85,41,63,19,74,30,52),(11,86,42,64,20,75,31,53)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,44),(31,43),(32,42),(33,41),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,66),(53,65),(54,64),(55,63),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,88),(75,87),(76,86),(77,85)]])
C8×D11 is a maximal subgroup of
D22.C8 D44.2C4 D44.C4 D8⋊3D11 Q8.D22 D88⋊5C2
C8×D11 is a maximal quotient of D22.C8 Dic11⋊C8 D22⋊C8
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D11 | D22 | C4×D11 | C8×D11 |
| kernel | C8×D11 | C11⋊C8 | C88 | C4×D11 | Dic11 | D22 | D11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 5 | 5 | 10 | 20 |
Matrix representation of C8×D11 ►in GL3(𝔽89) generated by
| 12 | 0 | 0 |
| 0 | 34 | 0 |
| 0 | 0 | 34 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 88 | 55 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(89))| [12,0,0,0,34,0,0,0,34],[1,0,0,0,0,88,0,1,55],[1,0,0,0,0,1,0,1,0] >;
C8×D11 in GAP, Magma, Sage, TeX
C_8\times D_{11} % in TeX
G:=Group("C8xD11"); // GroupNames label
G:=SmallGroup(176,3);
// by ID
G=gap.SmallGroup(176,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,26,42,4004]);
// Polycyclic
G:=Group<a,b,c|a^8=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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