direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D11, C11⋊C14, C77⋊2C2, SmallGroup(154,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C7×D11 |
Generators and relations for C7×D11
G = < a,b,c | a7=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×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 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), (1,2,3,4,5,6,7,8,9,10,11)(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), (1,2,3,4,5,6,7,8,9,10,11)(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)], [(1,2,3,4,5,6,7,8,9,10,11),(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)]])
49 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 11A | ··· | 11E | 14A | ··· | 14F | 77A | ··· | 77AD |
| order | 1 | 2 | 7 | ··· | 7 | 11 | ··· | 11 | 14 | ··· | 14 | 77 | ··· | 77 |
| size | 1 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D11 | C7×D11 |
| kernel | C7×D11 | C77 | D11 | C11 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 5 | 30 |
Matrix representation of C7×D11 ►in GL2(𝔽43) generated by
| 4 | 0 |
| 0 | 4 |
| 42 | 2 |
| 35 | 15 |
| 15 | 15 |
| 8 | 28 |
G:=sub<GL(2,GF(43))| [4,0,0,4],[42,35,2,15],[15,8,15,28] >;
C7×D11 in GAP, Magma, Sage, TeX
C_7\times D_{11} % in TeX
G:=Group("C7xD11"); // GroupNames label
G:=SmallGroup(154,2);
// by ID
G=gap.SmallGroup(154,2);
# by ID
G:=PCGroup([3,-2,-7,-11,1262]);
// Polycyclic
G:=Group<a,b,c|a^7=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export