direct product, metabelian, soluble, monomial
Aliases: C2×C9⋊A4, C18⋊A4, C23⋊13- 1+2, C9⋊2(C2×A4), (C6×A4).C3, (C3×A4).C6, (C2×C18)⋊7C6, C3.A4⋊2C6, C3.3(C6×A4), C6.4(C3×A4), (C22×C18)⋊2C3, (C22×C6).2C32, C22⋊1(C2×3- 1+2), (C2×C3.A4)⋊1C3, (C2×C6).2(C3×C6), SmallGroup(216,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C9⋊A4
G = < a,b,c,d,e | a2=b9=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b7, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C2×C9⋊A4
►On 54 points
Generators in S54
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 46)(17 47)(18 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(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)
(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 46)(17 47)(18 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 52 28)(2 47 35)(3 51 33)(4 46 31)(5 50 29)(6 54 36)(7 49 34)(8 53 32)(9 48 30)(10 43 27)(11 38 25)(12 42 23)(13 37 21)(14 41 19)(15 45 26)(16 40 24)(17 44 22)(18 39 20)
G:=sub<Sym(54)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,46)(17,47)(18,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (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), (10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,46)(17,47)(18,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (1,52,28)(2,47,35)(3,51,33)(4,46,31)(5,50,29)(6,54,36)(7,49,34)(8,53,32)(9,48,30)(10,43,27)(11,38,25)(12,42,23)(13,37,21)(14,41,19)(15,45,26)(16,40,24)(17,44,22)(18,39,20)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,46)(17,47)(18,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (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), (10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,46)(17,47)(18,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (1,52,28)(2,47,35)(3,51,33)(4,46,31)(5,50,29)(6,54,36)(7,49,34)(8,53,32)(9,48,30)(10,43,27)(11,38,25)(12,42,23)(13,37,21)(14,41,19)(15,45,26)(16,40,24)(17,44,22)(18,39,20) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,19),(9,20),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,46),(17,47),(18,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45)], [(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)], [(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,46),(17,47),(18,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,19),(9,20),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45)], [(1,52,28),(2,47,35),(3,51,33),(4,46,31),(5,50,29),(6,54,36),(7,49,34),(8,53,32),(9,48,30),(10,43,27),(11,38,25),(12,42,23),(13,37,21),(14,41,19),(15,45,26),(16,40,24),(17,44,22),(18,39,20)]])
C2×C9⋊A4 is a maximal subgroup of
Dic9⋊A4
C2×C9⋊A4 is a maximal quotient of C36.A4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | 9B | 9C | 9D | 9E | 9F | 18A | ··· | 18N | 18O | 18P | 18Q | 18R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 18 | 18 | 18 | 18 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 12 | 12 | 1 | 1 | 3 | 3 | 3 | 3 | 12 | 12 | 3 | 3 | 12 | 12 | 12 | 12 | 3 | ··· | 3 | 12 | 12 | 12 | 12 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | A4 | C2×A4 | 3- 1+2 | C3×A4 | C2×3- 1+2 | C6×A4 | C9⋊A4 | C2×C9⋊A4 |
| kernel | C2×C9⋊A4 | C9⋊A4 | C2×C3.A4 | C22×C18 | C6×A4 | C3.A4 | C2×C18 | C3×A4 | C18 | C9 | C23 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 |
Matrix representation of C2×C9⋊A4 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 9 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[9,0,0,0,6,0,0,0,4],[1,0,0,0,18,0,0,0,18],[18,0,0,0,18,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;
C2×C9⋊A4 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes A_4
% in TeX
G:=Group("C2xC9:A4"); // GroupNames label
G:=SmallGroup(216,104);
// by ID
G=gap.SmallGroup(216,104);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,224,68,1630,2927]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^9=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^7,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export