direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C9.A4, C23⋊C27, C22⋊C54, C18.3A4, C9.(C2×A4), (C2×C6).C18, (C22×C6).C9, (C2×C18).2C6, (C22×C18).C3, C6.2(C3.A4), C3.(C2×C3.A4), SmallGroup(216,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C2×C9.A4 |
Generators and relations for C2×C9.A4
G = < a,b,c,d,e | a2=b9=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C2×C9.A4
►On 54 points
Generators in S54
(1 52)(2 53)(3 54)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)
(2 53)(3 54)(5 29)(6 30)(8 32)(9 33)(11 35)(12 36)(14 38)(15 39)(17 41)(18 42)(20 44)(21 45)(23 47)(24 48)(26 50)(27 51)
(1 52)(3 54)(4 28)(6 30)(7 31)(9 33)(10 34)(12 36)(13 37)(15 39)(16 40)(18 42)(19 43)(21 45)(22 46)(24 48)(25 49)(27 51)
(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)| (1,52)(2,53)(3,54)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (2,53)(3,54)(5,29)(6,30)(8,32)(9,33)(11,35)(12,36)(14,38)(15,39)(17,41)(18,42)(20,44)(21,45)(23,47)(24,48)(26,50)(27,51), (1,52)(3,54)(4,28)(6,30)(7,31)(9,33)(10,34)(12,36)(13,37)(15,39)(16,40)(18,42)(19,43)(21,45)(22,46)(24,48)(25,49)(27,51), (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( (1,52)(2,53)(3,54)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (2,53)(3,54)(5,29)(6,30)(8,32)(9,33)(11,35)(12,36)(14,38)(15,39)(17,41)(18,42)(20,44)(21,45)(23,47)(24,48)(26,50)(27,51), (1,52)(3,54)(4,28)(6,30)(7,31)(9,33)(10,34)(12,36)(13,37)(15,39)(16,40)(18,42)(19,43)(21,45)(22,46)(24,48)(25,49)(27,51), (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([[(1,52),(2,53),(3,54),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54)], [(2,53),(3,54),(5,29),(6,30),(8,32),(9,33),(11,35),(12,36),(14,38),(15,39),(17,41),(18,42),(20,44),(21,45),(23,47),(24,48),(26,50),(27,51)], [(1,52),(3,54),(4,28),(6,30),(7,31),(9,33),(10,34),(12,36),(13,37),(15,39),(16,40),(18,42),(19,43),(21,45),(22,46),(24,48),(25,49),(27,51)], [(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×C9.A4 is a maximal subgroup of
C18.S4
C2×C9.A4 is a maximal quotient of Q8.C54
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 18A | ··· | 18F | 18G | ··· | 18R | 27A | ··· | 27R | 54A | ··· | 54R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | C9.A4 | C2×C9.A4 |
| kernel | C2×C9.A4 | C9.A4 | C22×C18 | C2×C18 | C22×C6 | C2×C6 | C23 | C22 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of C2×C9.A4 ►in GL3(𝔽109) generated by
| 108 | 0 | 0 |
| 0 | 108 | 0 |
| 0 | 0 | 108 |
| 27 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 16 | 108 | 0 |
| 22 | 0 | 108 |
| 108 | 0 | 0 |
| 0 | 108 | 0 |
| 87 | 0 | 1 |
| 16 | 107 | 0 |
| 8 | 93 | 1 |
| 108 | 87 | 0 |
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[27,0,0,0,27,0,0,0,27],[1,16,22,0,108,0,0,0,108],[108,0,87,0,108,0,0,0,1],[16,8,108,107,93,87,0,1,0] >;
C2×C9.A4 in GAP, Magma, Sage, TeX
C_2\times C_9.A_4
% in TeX
G:=Group("C2xC9.A4"); // GroupNames label
G:=SmallGroup(216,22);
// by ID
G=gap.SmallGroup(216,22);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,43,68,1630,2927]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^9=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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