metacyclic, supersoluble, monomial, Z-group
Aliases: C11⋊F7, D77⋊C3, C77⋊1C6, C7⋊C3⋊D11, C7⋊(C3×D11), (C11×C7⋊C3)⋊1C2, SmallGroup(462,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C77 — C11⋊F7 |
Generators and relations for C11⋊F7
G = < a,b,c | a11=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >
Smallest permutation representation of C11⋊F7
►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 21 32 43 54 65 76)(2 22 33 44 55 66 77)(3 12 23 34 45 56 67)(4 13 24 35 46 57 68)(5 14 25 36 47 58 69)(6 15 26 37 48 59 70)(7 16 27 38 49 60 71)(8 17 28 39 50 61 72)(9 18 29 40 51 62 73)(10 19 30 41 52 63 74)(11 20 31 42 53 64 75)
(2 11)(3 10)(4 9)(5 8)(6 7)(12 41 23 74 45 63)(13 40 24 73 46 62)(14 39 25 72 47 61)(15 38 26 71 48 60)(16 37 27 70 49 59)(17 36 28 69 50 58)(18 35 29 68 51 57)(19 34 30 67 52 56)(20 44 31 77 53 66)(21 43 32 76 54 65)(22 42 33 75 55 64)
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), (1,21,32,43,54,65,76)(2,22,33,44,55,66,77)(3,12,23,34,45,56,67)(4,13,24,35,46,57,68)(5,14,25,36,47,58,69)(6,15,26,37,48,59,70)(7,16,27,38,49,60,71)(8,17,28,39,50,61,72)(9,18,29,40,51,62,73)(10,19,30,41,52,63,74)(11,20,31,42,53,64,75), (2,11)(3,10)(4,9)(5,8)(6,7)(12,41,23,74,45,63)(13,40,24,73,46,62)(14,39,25,72,47,61)(15,38,26,71,48,60)(16,37,27,70,49,59)(17,36,28,69,50,58)(18,35,29,68,51,57)(19,34,30,67,52,56)(20,44,31,77,53,66)(21,43,32,76,54,65)(22,42,33,75,55,64)>;
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), (1,21,32,43,54,65,76)(2,22,33,44,55,66,77)(3,12,23,34,45,56,67)(4,13,24,35,46,57,68)(5,14,25,36,47,58,69)(6,15,26,37,48,59,70)(7,16,27,38,49,60,71)(8,17,28,39,50,61,72)(9,18,29,40,51,62,73)(10,19,30,41,52,63,74)(11,20,31,42,53,64,75), (2,11)(3,10)(4,9)(5,8)(6,7)(12,41,23,74,45,63)(13,40,24,73,46,62)(14,39,25,72,47,61)(15,38,26,71,48,60)(16,37,27,70,49,59)(17,36,28,69,50,58)(18,35,29,68,51,57)(19,34,30,67,52,56)(20,44,31,77,53,66)(21,43,32,76,54,65)(22,42,33,75,55,64) );
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)], [(1,21,32,43,54,65,76),(2,22,33,44,55,66,77),(3,12,23,34,45,56,67),(4,13,24,35,46,57,68),(5,14,25,36,47,58,69),(6,15,26,37,48,59,70),(7,16,27,38,49,60,71),(8,17,28,39,50,61,72),(9,18,29,40,51,62,73),(10,19,30,41,52,63,74),(11,20,31,42,53,64,75)], [(2,11),(3,10),(4,9),(5,8),(6,7),(12,41,23,74,45,63),(13,40,24,73,46,62),(14,39,25,72,47,61),(15,38,26,71,48,60),(16,37,27,70,49,59),(17,36,28,69,50,58),(18,35,29,68,51,57),(19,34,30,67,52,56),(20,44,31,77,53,66),(21,43,32,76,54,65),(22,42,33,75,55,64)]])
32 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7 | 11A | ··· | 11E | 33A | ··· | 33J | 77A | ··· | 77J |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 11 | ··· | 11 | 33 | ··· | 33 | 77 | ··· | 77 |
| size | 1 | 77 | 7 | 7 | 77 | 77 | 6 | 2 | ··· | 2 | 14 | ··· | 14 | 6 | ··· | 6 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D11 | C3×D11 | F7 | C11⋊F7 |
| kernel | C11⋊F7 | C11×C7⋊C3 | D77 | C77 | C7⋊C3 | C7 | C11 | C1 |
| # reps | 1 | 1 | 2 | 2 | 5 | 10 | 1 | 10 |
Matrix representation of C11⋊F7 ►in GL8(𝔽463)
| 367 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 212 | 128 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 462 | 0 | 0 | 0 | 208 |
| 0 | 0 | 14 | 462 | 0 | 0 | 0 | 208 |
| 0 | 0 | 320 | 11 | 0 | 0 | 0 | 26 |
| 0 | 0 | 22 | 227 | 1 | 0 | 0 | 460 |
| 0 | 0 | 433 | 216 | 0 | 1 | 0 | 447 |
| 0 | 0 | 411 | 452 | 0 | 0 | 1 | 450 |
| 340 | 254 | 0 | 0 | 0 | 0 | 0 | 0 |
| 88 | 123 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 462 | 449 | 0 | 0 | 255 | 0 |
| 0 | 0 | 14 | 13 | 255 | 0 | 0 | 0 |
| 0 | 0 | 52 | 11 | 450 | 0 | 449 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 422 | 411 | 14 | 1 | 1 | 0 |
| 0 | 0 | 452 | 41 | 462 | 0 | 13 | 0 |
G:=sub<GL(8,GF(463))| [367,212,0,0,0,0,0,0,1,128,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,14,320,22,433,411,0,0,462,462,11,227,216,452,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,208,208,26,460,447,450],[340,88,0,0,0,0,0,0,254,123,0,0,0,0,0,0,0,0,462,14,52,0,422,452,0,0,449,13,11,0,411,41,0,0,0,255,450,1,14,462,0,0,0,0,0,0,1,0,0,0,255,0,449,1,1,13,0,0,0,0,0,1,0,0] >;
C11⋊F7 in GAP, Magma, Sage, TeX
C_{11}\rtimes F_7 % in TeX
G:=Group("C11:F7"); // GroupNames label
G:=SmallGroup(462,3);
// by ID
G=gap.SmallGroup(462,3);
# by ID
G:=PCGroup([4,-2,-3,-7,-11,434,78,6723]);
// Polycyclic
G:=Group<a,b,c|a^11=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
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