direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D7, C7⋊C22, C77⋊3C2, SmallGroup(154,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C11×D7 |
Generators and relations for C11×D7
G = < a,b,c | a11=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D7
►On 77 points
Generators in S77
(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)
(1 70 60 54 41 27 17)(2 71 61 55 42 28 18)(3 72 62 45 43 29 19)(4 73 63 46 44 30 20)(5 74 64 47 34 31 21)(6 75 65 48 35 32 22)(7 76 66 49 36 33 12)(8 77 56 50 37 23 13)(9 67 57 51 38 24 14)(10 68 58 52 39 25 15)(11 69 59 53 40 26 16)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 12)(8 13)(9 14)(10 15)(11 16)(23 77)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 64)(35 65)(36 66)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)
G:=sub<Sym(77)| (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), (1,70,60,54,41,27,17)(2,71,61,55,42,28,18)(3,72,62,45,43,29,19)(4,73,63,46,44,30,20)(5,74,64,47,34,31,21)(6,75,65,48,35,32,22)(7,76,66,49,36,33,12)(8,77,56,50,37,23,13)(9,67,57,51,38,24,14)(10,68,58,52,39,25,15)(11,69,59,53,40,26,16), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,12)(8,13)(9,14)(10,15)(11,16)(23,77)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,64)(35,65)(36,66)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)>;
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), (1,70,60,54,41,27,17)(2,71,61,55,42,28,18)(3,72,62,45,43,29,19)(4,73,63,46,44,30,20)(5,74,64,47,34,31,21)(6,75,65,48,35,32,22)(7,76,66,49,36,33,12)(8,77,56,50,37,23,13)(9,67,57,51,38,24,14)(10,68,58,52,39,25,15)(11,69,59,53,40,26,16), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,12)(8,13)(9,14)(10,15)(11,16)(23,77)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,64)(35,65)(36,66)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63) );
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)], [(1,70,60,54,41,27,17),(2,71,61,55,42,28,18),(3,72,62,45,43,29,19),(4,73,63,46,44,30,20),(5,74,64,47,34,31,21),(6,75,65,48,35,32,22),(7,76,66,49,36,33,12),(8,77,56,50,37,23,13),(9,67,57,51,38,24,14),(10,68,58,52,39,25,15),(11,69,59,53,40,26,16)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,12),(8,13),(9,14),(10,15),(11,16),(23,77),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,64),(35,65),(36,66),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63)]])
55 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 11A | ··· | 11J | 22A | ··· | 22J | 77A | ··· | 77AD |
| order | 1 | 2 | 7 | 7 | 7 | 11 | ··· | 11 | 22 | ··· | 22 | 77 | ··· | 77 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C11 | C22 | D7 | C11×D7 |
| kernel | C11×D7 | C77 | D7 | C7 | C11 | C1 |
| # reps | 1 | 1 | 10 | 10 | 3 | 30 |
Matrix representation of C11×D7 ►in GL2(𝔽463) generated by
| 225 | 0 |
| 0 | 225 |
| 320 | 1 |
| 462 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(463))| [225,0,0,225],[320,462,1,0],[0,1,1,0] >;
C11×D7 in GAP, Magma, Sage, TeX
C_{11}\times D_7 % in TeX
G:=Group("C11xD7"); // GroupNames label
G:=SmallGroup(154,1);
// by ID
G=gap.SmallGroup(154,1);
# by ID
G:=PCGroup([3,-2,-11,-7,1190]);
// Polycyclic
G:=Group<a,b,c|a^11=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export