direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×D8, D4⋊C18, C8⋊1C18, C72⋊5C2, C24.2C6, C18.14D4, C36.17C22, C3.(C3×D8), (C3×D8).C3, (D4×C9)⋊4C2, C2.3(D4×C9), C4.1(C2×C18), (C3×D4).2C6, C6.14(C3×D4), C12.17(C2×C6), SmallGroup(144,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D8
G = < a,b,c | a9=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D8
►On 72 points
Generators in S72
(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)
(1 28 42 15 50 66 59 23)(2 29 43 16 51 67 60 24)(3 30 44 17 52 68 61 25)(4 31 45 18 53 69 62 26)(5 32 37 10 54 70 63 27)(6 33 38 11 46 71 55 19)(7 34 39 12 47 72 56 20)(8 35 40 13 48 64 57 21)(9 36 41 14 49 65 58 22)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 54)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(28 59)(29 60)(30 61)(31 62)(32 63)(33 55)(34 56)(35 57)(36 58)(37 70)(38 71)(39 72)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)
G:=sub<Sym(72)| (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), (1,28,42,15,50,66,59,23)(2,29,43,16,51,67,60,24)(3,30,44,17,52,68,61,25)(4,31,45,18,53,69,62,26)(5,32,37,10,54,70,63,27)(6,33,38,11,46,71,55,19)(7,34,39,12,47,72,56,20)(8,35,40,13,48,64,57,21)(9,36,41,14,49,65,58,22), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,54)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(28,59)(29,60)(30,61)(31,62)(32,63)(33,55)(34,56)(35,57)(36,58)(37,70)(38,71)(39,72)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)>;
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), (1,28,42,15,50,66,59,23)(2,29,43,16,51,67,60,24)(3,30,44,17,52,68,61,25)(4,31,45,18,53,69,62,26)(5,32,37,10,54,70,63,27)(6,33,38,11,46,71,55,19)(7,34,39,12,47,72,56,20)(8,35,40,13,48,64,57,21)(9,36,41,14,49,65,58,22), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,54)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(28,59)(29,60)(30,61)(31,62)(32,63)(33,55)(34,56)(35,57)(36,58)(37,70)(38,71)(39,72)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69) );
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)], [(1,28,42,15,50,66,59,23),(2,29,43,16,51,67,60,24),(3,30,44,17,52,68,61,25),(4,31,45,18,53,69,62,26),(5,32,37,10,54,70,63,27),(6,33,38,11,46,71,55,19),(7,34,39,12,47,72,56,20),(8,35,40,13,48,64,57,21),(9,36,41,14,49,65,58,22)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,54),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(28,59),(29,60),(30,61),(31,62),(32,63),(33,55),(34,56),(35,57),(36,58),(37,70),(38,71),(39,72),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69)]])
C9×D8 is a maximal subgroup of
C9⋊D16 D8.D9 D8⋊D9 D8⋊3D9
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 18A | ··· | 18F | 18G | ··· | 18R | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | D8 | C3×D4 | C3×D8 | D4×C9 | C9×D8 |
| kernel | C9×D8 | C72 | D4×C9 | C3×D8 | C24 | C3×D4 | D8 | C8 | D4 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 2 | 2 | 4 | 6 | 12 |
Matrix representation of C9×D8 ►in GL2(𝔽73) generated by
| 32 | 0 |
| 0 | 32 |
| 16 | 57 |
| 16 | 16 |
| 16 | 57 |
| 57 | 57 |
G:=sub<GL(2,GF(73))| [32,0,0,32],[16,16,57,16],[16,57,57,57] >;
C9×D8 in GAP, Magma, Sage, TeX
C_9\times D_8
% in TeX
G:=Group("C9xD8"); // GroupNames label
G:=SmallGroup(144,25);
// by ID
G=gap.SmallGroup(144,25);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-2,169,122,2019,1017,165]);
// Polycyclic
G:=Group<a,b,c|a^9=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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