direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D9, C9⋊C14, C63⋊2C2, C21.2S3, C3.(S3×C7), SmallGroup(126,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C7×D9 |
Generators and relations for C7×D9
G = < a,b,c | a7=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D9
►On 63 points
Generators in S63
(1 56 47 38 29 20 11)(2 57 48 39 30 21 12)(3 58 49 40 31 22 13)(4 59 50 41 32 23 14)(5 60 51 42 33 24 15)(6 61 52 43 34 25 16)(7 62 53 44 35 26 17)(8 63 54 45 36 27 18)(9 55 46 37 28 19 10)
(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)
(1 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 20)(21 27)(22 26)(23 25)(28 29)(30 36)(31 35)(32 34)(37 38)(39 45)(40 44)(41 43)(46 47)(48 54)(49 53)(50 52)(55 56)(57 63)(58 62)(59 61)
G:=sub<Sym(63)| (1,56,47,38,29,20,11)(2,57,48,39,30,21,12)(3,58,49,40,31,22,13)(4,59,50,41,32,23,14)(5,60,51,42,33,24,15)(6,61,52,43,34,25,16)(7,62,53,44,35,26,17)(8,63,54,45,36,27,18)(9,55,46,37,28,19,10), (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), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,20)(21,27)(22,26)(23,25)(28,29)(30,36)(31,35)(32,34)(37,38)(39,45)(40,44)(41,43)(46,47)(48,54)(49,53)(50,52)(55,56)(57,63)(58,62)(59,61)>;
G:=Group( (1,56,47,38,29,20,11)(2,57,48,39,30,21,12)(3,58,49,40,31,22,13)(4,59,50,41,32,23,14)(5,60,51,42,33,24,15)(6,61,52,43,34,25,16)(7,62,53,44,35,26,17)(8,63,54,45,36,27,18)(9,55,46,37,28,19,10), (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), (1,9)(2,8)(3,7)(4,6)(10,11)(12,18)(13,17)(14,16)(19,20)(21,27)(22,26)(23,25)(28,29)(30,36)(31,35)(32,34)(37,38)(39,45)(40,44)(41,43)(46,47)(48,54)(49,53)(50,52)(55,56)(57,63)(58,62)(59,61) );
G=PermutationGroup([[(1,56,47,38,29,20,11),(2,57,48,39,30,21,12),(3,58,49,40,31,22,13),(4,59,50,41,32,23,14),(5,60,51,42,33,24,15),(6,61,52,43,34,25,16),(7,62,53,44,35,26,17),(8,63,54,45,36,27,18),(9,55,46,37,28,19,10)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,11),(12,18),(13,17),(14,16),(19,20),(21,27),(22,26),(23,25),(28,29),(30,36),(31,35),(32,34),(37,38),(39,45),(40,44),(41,43),(46,47),(48,54),(49,53),(50,52),(55,56),(57,63),(58,62),(59,61)]])
C7×D9 is a maximal subgroup of
C63⋊C6 C63⋊6C6
42 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7F | 9A | 9B | 9C | 14A | ··· | 14F | 21A | ··· | 21F | 63A | ··· | 63R |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 9 | 9 | 9 | 14 | ··· | 14 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 9 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C7 | C14 | S3 | D9 | S3×C7 | C7×D9 |
| kernel | C7×D9 | C63 | D9 | C9 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 3 | 6 | 18 |
Matrix representation of C7×D9 ►in GL2(𝔽127) generated by
| 32 | 0 |
| 0 | 32 |
| 96 | 9 |
| 118 | 105 |
| 105 | 31 |
| 9 | 22 |
G:=sub<GL(2,GF(127))| [32,0,0,32],[96,118,9,105],[105,9,31,22] >;
C7×D9 in GAP, Magma, Sage, TeX
C_7\times D_9
% in TeX
G:=Group("C7xD9"); // GroupNames label
G:=SmallGroup(126,3);
// by ID
G=gap.SmallGroup(126,3);
# by ID
G:=PCGroup([4,-2,-7,-3,-3,842,82,1347]);
// Polycyclic
G:=Group<a,b,c|a^7=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export