direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D7, C63⋊3C2, C7⋊3C18, C21.3C6, C3.(C3×D7), (C3×D7).2C3, SmallGroup(126,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C9×D7 |
Generators and relations for C9×D7
G = < a,b,c | a9=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D7
►On 63 points
Generators in S63
(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 56 16 39 21 29 48)(2 57 17 40 22 30 49)(3 58 18 41 23 31 50)(4 59 10 42 24 32 51)(5 60 11 43 25 33 52)(6 61 12 44 26 34 53)(7 62 13 45 27 35 54)(8 63 14 37 19 36 46)(9 55 15 38 20 28 47)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 46)(9 47)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)
G:=sub<Sym(63)| (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), (1,56,16,39,21,29,48)(2,57,17,40,22,30,49)(3,58,18,41,23,31,50)(4,59,10,42,24,32,51)(5,60,11,43,25,33,52)(6,61,12,44,26,34,53)(7,62,13,45,27,35,54)(8,63,14,37,19,36,46)(9,55,15,38,20,28,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)>;
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), (1,56,16,39,21,29,48)(2,57,17,40,22,30,49)(3,58,18,41,23,31,50)(4,59,10,42,24,32,51)(5,60,11,43,25,33,52)(6,61,12,44,26,34,53)(7,62,13,45,27,35,54)(8,63,14,37,19,36,46)(9,55,15,38,20,28,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63) );
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)], [(1,56,16,39,21,29,48),(2,57,17,40,22,30,49),(3,58,18,41,23,31,50),(4,59,10,42,24,32,51),(5,60,11,43,25,33,52),(6,61,12,44,26,34,53),(7,62,13,45,27,35,54),(8,63,14,37,19,36,46),(9,55,15,38,20,28,47)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,46),(9,47),(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63)]])
C9×D7 is a maximal subgroup of
C7⋊C54 C9⋊3F7 C9⋊4F7
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 9A | ··· | 9F | 18A | ··· | 18F | 21A | ··· | 21F | 63A | ··· | 63R |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 9 | ··· | 9 | 18 | ··· | 18 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D7 | C3×D7 | C9×D7 |
| kernel | C9×D7 | C63 | C3×D7 | C21 | D7 | C7 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 3 | 6 | 18 |
Matrix representation of C9×D7 ►in GL2(𝔽127) generated by
| 103 | 0 |
| 0 | 103 |
| 126 | 1 |
| 89 | 37 |
| 126 | 0 |
| 89 | 1 |
G:=sub<GL(2,GF(127))| [103,0,0,103],[126,89,1,37],[126,89,0,1] >;
C9×D7 in GAP, Magma, Sage, TeX
C_9\times D_7
% in TeX
G:=Group("C9xD7"); // GroupNames label
G:=SmallGroup(126,4);
// by ID
G=gap.SmallGroup(126,4);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,29,1731]);
// Polycyclic
G:=Group<a,b,c|a^9=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export