direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C32⋊C9, C6.1He3, C32⋊2C18, C33.3C6, C6.13- 1+2, (C3×C6)⋊C9, (C3×C9)⋊8C6, (C3×C18)⋊1C3, C6.1(C3×C9), C3.1(C3×C18), C3.1(C2×He3), C32.9(C3×C6), (C3×C6).6C32, (C32×C6).1C3, C3.1(C2×3- 1+2), SmallGroup(162,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊C9
G = < a,b,c,d | a2=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Smallest permutation representation of C2×C32⋊C9
►On 54 points
Generators in S54
(1 51)(2 52)(3 53)(4 54)(5 46)(6 47)(7 48)(8 49)(9 50)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)
(1 4 7)(2 31 42)(3 40 35)(5 34 45)(6 43 29)(8 28 39)(9 37 32)(10 21 46)(11 53 25)(12 15 18)(13 24 49)(14 47 19)(16 27 52)(17 50 22)(20 23 26)(30 33 36)(38 41 44)(48 51 54)
(1 44 36)(2 45 28)(3 37 29)(4 38 30)(5 39 31)(6 40 32)(7 41 33)(8 42 34)(9 43 35)(10 49 27)(11 50 19)(12 51 20)(13 52 21)(14 53 22)(15 54 23)(16 46 24)(17 47 25)(18 48 26)
(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)
G:=sub<Sym(54)| (1,51)(2,52)(3,53)(4,54)(5,46)(6,47)(7,48)(8,49)(9,50)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42), (1,4,7)(2,31,42)(3,40,35)(5,34,45)(6,43,29)(8,28,39)(9,37,32)(10,21,46)(11,53,25)(12,15,18)(13,24,49)(14,47,19)(16,27,52)(17,50,22)(20,23,26)(30,33,36)(38,41,44)(48,51,54), (1,44,36)(2,45,28)(3,37,29)(4,38,30)(5,39,31)(6,40,32)(7,41,33)(8,42,34)(9,43,35)(10,49,27)(11,50,19)(12,51,20)(13,52,21)(14,53,22)(15,54,23)(16,46,24)(17,47,25)(18,48,26), (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)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,46)(6,47)(7,48)(8,49)(9,50)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42), (1,4,7)(2,31,42)(3,40,35)(5,34,45)(6,43,29)(8,28,39)(9,37,32)(10,21,46)(11,53,25)(12,15,18)(13,24,49)(14,47,19)(16,27,52)(17,50,22)(20,23,26)(30,33,36)(38,41,44)(48,51,54), (1,44,36)(2,45,28)(3,37,29)(4,38,30)(5,39,31)(6,40,32)(7,41,33)(8,42,34)(9,43,35)(10,49,27)(11,50,19)(12,51,20)(13,52,21)(14,53,22)(15,54,23)(16,46,24)(17,47,25)(18,48,26), (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) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,46),(6,47),(7,48),(8,49),(9,50),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42)], [(1,4,7),(2,31,42),(3,40,35),(5,34,45),(6,43,29),(8,28,39),(9,37,32),(10,21,46),(11,53,25),(12,15,18),(13,24,49),(14,47,19),(16,27,52),(17,50,22),(20,23,26),(30,33,36),(38,41,44),(48,51,54)], [(1,44,36),(2,45,28),(3,37,29),(4,38,30),(5,39,31),(6,40,32),(7,41,33),(8,42,34),(9,43,35),(10,49,27),(11,50,19),(12,51,20),(13,52,21),(14,53,22),(15,54,23),(16,46,24),(17,47,25),(18,48,26)], [(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)]])
C2×C32⋊C9 is a maximal subgroup of
C32⋊C36 C32⋊Dic9 C32⋊2Dic9 C18×He3 C18×3- 1+2
66 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 6A | ··· | 6H | 6I | ··· | 6N | 9A | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | He3 | 3- 1+2 | C2×He3 | C2×3- 1+2 |
| kernel | C2×C32⋊C9 | C32⋊C9 | C3×C18 | C32×C6 | C3×C9 | C33 | C3×C6 | C32 | C6 | C6 | C3 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 2 | 4 | 2 | 4 |
Matrix representation of C2×C32⋊C9 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 12 |
| 0 | 0 | 11 | 1 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 9 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 6 | 0 | 7 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,11,0,0,12,1,7],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[9,0,0,0,0,12,8,6,0,1,0,0,0,0,0,7] >;
C2×C32⋊C9 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes C_9
% in TeX
G:=Group("C2xC3^2:C9"); // GroupNames label
G:=SmallGroup(162,24);
// by ID
G=gap.SmallGroup(162,24);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,276,187]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
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