direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×D11, D77⋊C2, C7⋊1D22, C77⋊C22, C11⋊1D14, (C7×D11)⋊C2, (C11×D7)⋊C2, SmallGroup(308,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C77 — D7×D11 |
Generators and relations for D7×D11
G = < a,b,c,d | a7=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D7×D11
►On 77 points
Generators in S77
(1 76 65 54 43 32 21)(2 77 66 55 44 33 22)(3 67 56 45 34 23 12)(4 68 57 46 35 24 13)(5 69 58 47 36 25 14)(6 70 59 48 37 26 15)(7 71 60 49 38 27 16)(8 72 61 50 39 28 17)(9 73 62 51 40 29 18)(10 74 63 52 41 30 19)(11 75 64 53 42 31 20)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)
G:=sub<Sym(77)| (1,76,65,54,43,32,21)(2,77,66,55,44,33,22)(3,67,56,45,34,23,12)(4,68,57,46,35,24,13)(5,69,58,47,36,25,14)(6,70,59,48,37,26,15)(7,71,60,49,38,27,16)(8,72,61,50,39,28,17)(9,73,62,51,40,29,18)(10,74,63,52,41,30,19)(11,75,64,53,42,31,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)>;
G:=Group( (1,76,65,54,43,32,21)(2,77,66,55,44,33,22)(3,67,56,45,34,23,12)(4,68,57,46,35,24,13)(5,69,58,47,36,25,14)(6,70,59,48,37,26,15)(7,71,60,49,38,27,16)(8,72,61,50,39,28,17)(9,73,62,51,40,29,18)(10,74,63,52,41,30,19)(11,75,64,53,42,31,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76) );
G=PermutationGroup([[(1,76,65,54,43,32,21),(2,77,66,55,44,33,22),(3,67,56,45,34,23,12),(4,68,57,46,35,24,13),(5,69,58,47,36,25,14),(6,70,59,48,37,26,15),(7,71,60,49,38,27,16),(8,72,61,50,39,28,17),(9,73,62,51,40,29,18),(10,74,63,52,41,30,19),(11,75,64,53,42,31,20)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 11A | ··· | 11E | 14A | 14B | 14C | 22A | ··· | 22E | 77A | ··· | 77O |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 11 | ··· | 11 | 14 | 14 | 14 | 22 | ··· | 22 | 77 | ··· | 77 |
| size | 1 | 7 | 11 | 77 | 2 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 14 | ··· | 14 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D7 | D11 | D14 | D22 | D7×D11 |
| kernel | D7×D11 | C11×D7 | C7×D11 | D77 | D11 | D7 | C11 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 3 | 5 | 3 | 5 | 15 |
Matrix representation of D7×D11 ►in GL4(𝔽463) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 462 | 320 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 391 | 1 | 0 | 0 |
| 404 | 168 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 215 | 45 | 0 | 0 |
| 84 | 248 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(463))| [1,0,0,0,0,1,0,0,0,0,0,462,0,0,1,320],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[391,404,0,0,1,168,0,0,0,0,1,0,0,0,0,1],[215,84,0,0,45,248,0,0,0,0,1,0,0,0,0,1] >;
D7×D11 in GAP, Magma, Sage, TeX
D_7\times D_{11} % in TeX
G:=Group("D7xD11"); // GroupNames label
G:=SmallGroup(308,5);
// by ID
G=gap.SmallGroup(308,5);
# by ID
G:=PCGroup([4,-2,-2,-7,-11,150,4483]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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