direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: C11×F7, C7⋊C66, D7⋊C33, C77⋊3C6, C7⋊C3⋊C22, (C11×D7)⋊C3, (C11×C7⋊C3)⋊3C2, SmallGroup(462,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C11×F7 |
Generators and relations for C11×F7
G = < a,b,c | a11=b7=c6=1, ab=ba, ac=ca, 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 38 48 28 15 60 70)(2 39 49 29 16 61 71)(3 40 50 30 17 62 72)(4 41 51 31 18 63 73)(5 42 52 32 19 64 74)(6 43 53 33 20 65 75)(7 44 54 23 21 66 76)(8 34 55 24 22 56 77)(9 35 45 25 12 57 67)(10 36 46 26 13 58 68)(11 37 47 27 14 59 69)
(12 57 35 25 45 67)(13 58 36 26 46 68)(14 59 37 27 47 69)(15 60 38 28 48 70)(16 61 39 29 49 71)(17 62 40 30 50 72)(18 63 41 31 51 73)(19 64 42 32 52 74)(20 65 43 33 53 75)(21 66 44 23 54 76)(22 56 34 24 55 77)
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), (1,38,48,28,15,60,70)(2,39,49,29,16,61,71)(3,40,50,30,17,62,72)(4,41,51,31,18,63,73)(5,42,52,32,19,64,74)(6,43,53,33,20,65,75)(7,44,54,23,21,66,76)(8,34,55,24,22,56,77)(9,35,45,25,12,57,67)(10,36,46,26,13,58,68)(11,37,47,27,14,59,69), (12,57,35,25,45,67)(13,58,36,26,46,68)(14,59,37,27,47,69)(15,60,38,28,48,70)(16,61,39,29,49,71)(17,62,40,30,50,72)(18,63,41,31,51,73)(19,64,42,32,52,74)(20,65,43,33,53,75)(21,66,44,23,54,76)(22,56,34,24,55,77)>;
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), (1,38,48,28,15,60,70)(2,39,49,29,16,61,71)(3,40,50,30,17,62,72)(4,41,51,31,18,63,73)(5,42,52,32,19,64,74)(6,43,53,33,20,65,75)(7,44,54,23,21,66,76)(8,34,55,24,22,56,77)(9,35,45,25,12,57,67)(10,36,46,26,13,58,68)(11,37,47,27,14,59,69), (12,57,35,25,45,67)(13,58,36,26,46,68)(14,59,37,27,47,69)(15,60,38,28,48,70)(16,61,39,29,49,71)(17,62,40,30,50,72)(18,63,41,31,51,73)(19,64,42,32,52,74)(20,65,43,33,53,75)(21,66,44,23,54,76)(22,56,34,24,55,77) );
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)], [(1,38,48,28,15,60,70),(2,39,49,29,16,61,71),(3,40,50,30,17,62,72),(4,41,51,31,18,63,73),(5,42,52,32,19,64,74),(6,43,53,33,20,65,75),(7,44,54,23,21,66,76),(8,34,55,24,22,56,77),(9,35,45,25,12,57,67),(10,36,46,26,13,58,68),(11,37,47,27,14,59,69)], [(12,57,35,25,45,67),(13,58,36,26,46,68),(14,59,37,27,47,69),(15,60,38,28,48,70),(16,61,39,29,49,71),(17,62,40,30,50,72),(18,63,41,31,51,73),(19,64,42,32,52,74),(20,65,43,33,53,75),(21,66,44,23,54,76),(22,56,34,24,55,77)]])
77 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7 | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33T | 66A | ··· | 66T | 77A | ··· | 77J |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 | 77 | ··· | 77 |
| size | 1 | 7 | 7 | 7 | 7 | 7 | 6 | 1 | ··· | 1 | 7 | ··· | 7 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | ··· | 6 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 |
| type | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 | F7 | C11×F7 |
| kernel | C11×F7 | C11×C7⋊C3 | C11×D7 | C77 | F7 | C7⋊C3 | D7 | C7 | C11 | C1 |
| # reps | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 | 1 | 10 |
Matrix representation of C11×F7 ►in GL6(𝔽463)
| 425 | 0 | 0 | 0 | 0 | 0 |
| 0 | 425 | 0 | 0 | 0 | 0 |
| 0 | 0 | 425 | 0 | 0 | 0 |
| 0 | 0 | 0 | 425 | 0 | 0 |
| 0 | 0 | 0 | 0 | 425 | 0 |
| 0 | 0 | 0 | 0 | 0 | 425 |
| 0 | 0 | 0 | 0 | 0 | 462 |
| 1 | 0 | 0 | 0 | 0 | 462 |
| 0 | 1 | 0 | 0 | 0 | 462 |
| 0 | 0 | 1 | 0 | 0 | 462 |
| 0 | 0 | 0 | 1 | 0 | 462 |
| 0 | 0 | 0 | 0 | 1 | 462 |
| 1 | 0 | 462 | 0 | 0 | 0 |
| 0 | 0 | 462 | 0 | 0 | 1 |
| 0 | 0 | 462 | 1 | 0 | 0 |
| 0 | 1 | 462 | 0 | 0 | 0 |
| 0 | 0 | 462 | 0 | 0 | 0 |
| 0 | 0 | 462 | 0 | 1 | 0 |
G:=sub<GL(6,GF(463))| [425,0,0,0,0,0,0,425,0,0,0,0,0,0,425,0,0,0,0,0,0,425,0,0,0,0,0,0,425,0,0,0,0,0,0,425],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,462,462,462,462,462,462],[1,0,0,0,0,0,0,0,0,1,0,0,462,462,462,462,462,462,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0] >;
C11×F7 in GAP, Magma, Sage, TeX
C_{11}\times F_7 % in TeX
G:=Group("C11xF7"); // GroupNames label
G:=SmallGroup(462,2);
// by ID
G=gap.SmallGroup(462,2);
# by ID
G:=PCGroup([4,-2,-3,-11,-7,6339,2119]);
// Polycyclic
G:=Group<a,b,c|a^11=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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