direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×SD16, Q8⋊C22, C8⋊2C22, C88⋊6C2, D4.C22, C22.15D4, C44.18C22, C4.2(C2×C22), (Q8×C11)⋊4C2, C2.4(D4×C11), (D4×C11).2C2, SmallGroup(176,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×SD16
G = < a,b,c | a11=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of C11×SD16
►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 79 59 23 67 40 47 14)(2 80 60 24 68 41 48 15)(3 81 61 25 69 42 49 16)(4 82 62 26 70 43 50 17)(5 83 63 27 71 44 51 18)(6 84 64 28 72 34 52 19)(7 85 65 29 73 35 53 20)(8 86 66 30 74 36 54 21)(9 87 56 31 75 37 55 22)(10 88 57 32 76 38 45 12)(11 78 58 33 77 39 46 13)
(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 34)(20 35)(21 36)(22 37)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 78)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 56)
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), (1,79,59,23,67,40,47,14)(2,80,60,24,68,41,48,15)(3,81,61,25,69,42,49,16)(4,82,62,26,70,43,50,17)(5,83,63,27,71,44,51,18)(6,84,64,28,72,34,52,19)(7,85,65,29,73,35,53,20)(8,86,66,30,74,36,54,21)(9,87,56,31,75,37,55,22)(10,88,57,32,76,38,45,12)(11,78,58,33,77,39,46,13), (12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,34)(20,35)(21,36)(22,37)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,78)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,56)>;
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), (1,79,59,23,67,40,47,14)(2,80,60,24,68,41,48,15)(3,81,61,25,69,42,49,16)(4,82,62,26,70,43,50,17)(5,83,63,27,71,44,51,18)(6,84,64,28,72,34,52,19)(7,85,65,29,73,35,53,20)(8,86,66,30,74,36,54,21)(9,87,56,31,75,37,55,22)(10,88,57,32,76,38,45,12)(11,78,58,33,77,39,46,13), (12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,34)(20,35)(21,36)(22,37)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,78)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,56) );
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)], [(1,79,59,23,67,40,47,14),(2,80,60,24,68,41,48,15),(3,81,61,25,69,42,49,16),(4,82,62,26,70,43,50,17),(5,83,63,27,71,44,51,18),(6,84,64,28,72,34,52,19),(7,85,65,29,73,35,53,20),(8,86,66,30,74,36,54,21),(9,87,56,31,75,37,55,22),(10,88,57,32,76,38,45,12),(11,78,58,33,77,39,46,13)], [(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,34),(20,35),(21,36),(22,37),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,78),(45,57),(46,58),(47,59),(48,60),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,56)]])
C11×SD16 is a maximal subgroup of
D88⋊C2 D4.D22 Q8.D22
77 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 44A | ··· | 44J | 44K | ··· | 44T | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | SD16 | D4×C11 | C11×SD16 |
| kernel | C11×SD16 | C88 | D4×C11 | Q8×C11 | SD16 | C8 | D4 | Q8 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 2 | 10 | 20 |
Matrix representation of C11×SD16 ►in GL2(𝔽89) generated by
| 78 | 0 |
| 0 | 78 |
| 49 | 69 |
| 40 | 0 |
| 1 | 1 |
| 0 | 88 |
G:=sub<GL(2,GF(89))| [78,0,0,78],[49,40,69,0],[1,0,1,88] >;
C11×SD16 in GAP, Magma, Sage, TeX
C_{11}\times {\rm SD}_{16} % in TeX
G:=Group("C11xSD16"); // GroupNames label
G:=SmallGroup(176,25);
// by ID
G=gap.SmallGroup(176,25);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-2,440,461,2643,1328,58]);
// 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^3>;
// generators/relations
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