direct product, non-abelian, soluble, monomial
Aliases: C2×C9⋊S4, C18⋊S4, A4⋊2D18, (C2×A4)⋊D9, C9⋊2(C2×S4), (C2×C18)⋊3D6, C3.A4⋊2D6, (A4×C18)⋊2C2, C6.7(C3⋊S4), (C6×A4).7S3, (C3×A4).3D6, C23⋊1(C9⋊S3), (C9×A4)⋊3C22, (C22×C18)⋊2S3, C3.1(C2×C3⋊S4), (C2×C3.A4)⋊1S3, C22⋊1(C2×C9⋊S3), (C22×C6).6(C3⋊S3), (C2×C6).2(C2×C3⋊S3), SmallGroup(432,536)
Series: Derived ►Chief ►Lower central ►Upper central
| C9×A4 — C2×C9⋊S4 |
Generators and relations for C2×C9⋊S4
G = < a,b,c,d,e,f | a2=b9=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 1414 in 139 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, C9, C32, Dic3, A4, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C3⋊S3, C3×C6, C2×Dic3, C3⋊D4, S4, C2×A4, C22×S3, C22×C6, C3×C9, Dic9, C3.A4, D18, C2×C18, C2×C18, C3×A4, C2×C3⋊S3, C2×C3⋊D4, C2×S4, C9⋊S3, C3×C18, C2×Dic9, C9⋊D4, C3.S4, C2×C3.A4, C22×D9, C22×C18, C3⋊S4, C6×A4, C9×A4, C2×C9⋊S3, C2×C9⋊D4, C2×C3.S4, C2×C3⋊S4, C9⋊S4, A4×C18, C2×C9⋊S4
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, S4, D18, C2×C3⋊S3, C2×S4, C9⋊S3, C3⋊S4, C2×C9⋊S3, C2×C3⋊S4, C9⋊S4, C2×C9⋊S4
►On 54 points
Generators in S54
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 52)(11 53)(12 54)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(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)
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(10 52)(11 53)(12 54)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(1 31 49)(2 32 50)(3 33 51)(4 34 52)(5 35 53)(6 36 54)(7 28 46)(8 29 47)(9 30 48)(10 19 37)(11 20 38)(12 21 39)(13 22 40)(14 23 41)(15 24 42)(16 25 43)(17 26 44)(18 27 45)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 27)(9 26)(10 28)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(37 46)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)
G:=sub<Sym(54)| (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (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), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,31,49)(2,32,50)(3,33,51)(4,34,52)(5,35,53)(6,36,54)(7,28,46)(8,29,47)(9,30,48)(10,19,37)(11,20,38)(12,21,39)(13,22,40)(14,23,41)(15,24,42)(16,25,43)(17,26,44)(18,27,45), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,27)(9,26)(10,28)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(37,46)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)>;
G:=Group( (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (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), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,31,49)(2,32,50)(3,33,51)(4,34,52)(5,35,53)(6,36,54)(7,28,46)(8,29,47)(9,30,48)(10,19,37)(11,20,38)(12,21,39)(13,22,40)(14,23,41)(15,24,42)(16,25,43)(17,26,44)(18,27,45), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,27)(9,26)(10,28)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(37,46)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,52),(11,53),(12,54),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)], [(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)], [(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)], [(10,52),(11,53),(12,54),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)], [(1,31,49),(2,32,50),(3,33,51),(4,34,52),(5,35,53),(6,36,54),(7,28,46),(8,29,47),(9,30,48),(10,19,37),(11,20,38),(12,21,39),(13,22,40),(14,23,41),(15,24,42),(16,25,43),(17,26,44),(18,27,45)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,27),(9,26),(10,28),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(37,46),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | 9B | 9C | 9D | ··· | 9I | 18A | 18B | 18C | 18D | ··· | 18I | 18J | ··· | 18O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | ··· | 9 | 18 | 18 | 18 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 54 | 54 | 2 | 8 | 8 | 8 | 54 | 54 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 6 | ··· | 6 | 8 | ··· | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | S3 | S3 | D6 | D6 | D6 | D9 | D18 | S4 | C2×S4 | C3⋊S4 | C2×C3⋊S4 | C9⋊S4 | C2×C9⋊S4 |
| kernel | C2×C9⋊S4 | C9⋊S4 | A4×C18 | C2×C3.A4 | C22×C18 | C6×A4 | C3.A4 | C2×C18 | C3×A4 | C2×A4 | A4 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 9 | 9 | 2 | 2 | 1 | 1 | 3 | 3 |
Matrix representation of C2×C9⋊S4 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 11 | 20 | 0 | 0 | 0 |
| 17 | 31 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 0 | 36 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 17 | 31 | 0 | 0 | 0 |
| 11 | 20 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 36 | 0 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[11,17,0,0,0,20,31,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[0,1,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[17,11,0,0,0,31,20,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,36,0] >;
C2×C9⋊S4 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes S_4
% in TeX
G:=Group("C2xC9:S4"); // GroupNames label
G:=SmallGroup(432,536);
// by ID
G=gap.SmallGroup(432,536);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,1683,192,2524,9077,2287,5298,3989]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^9=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations