direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C3.A4, C32.2A4, C62.1C3, (C2×C6)⋊C9, C3.2(C3×A4), C22⋊2(C3×C9), (C2×C6).3C32, SmallGroup(108,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C3×C3.A4 |
Generators and relations for C3×C3.A4
G = < a,b,c,d,e | a3=b3=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 C3×C3.A4
►On 54 points
Generators in S54
(1 28 13)(2 29 14)(3 30 15)(4 31 16)(5 32 17)(6 33 18)(7 34 10)(8 35 11)(9 36 12)(19 51 37)(20 52 38)(21 53 39)(22 54 40)(23 46 41)(24 47 42)(25 48 43)(26 49 44)(27 50 45)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(2 53)(3 54)(5 47)(6 48)(8 50)(9 51)(11 27)(12 19)(14 21)(15 22)(17 24)(18 25)(29 39)(30 40)(32 42)(33 43)(35 45)(36 37)
(1 52)(3 54)(4 46)(6 48)(7 49)(9 51)(10 26)(12 19)(13 20)(15 22)(16 23)(18 25)(28 38)(30 40)(31 41)(33 43)(34 44)(36 37)
(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)| (1,28,13)(2,29,14)(3,30,15)(4,31,16)(5,32,17)(6,33,18)(7,34,10)(8,35,11)(9,36,12)(19,51,37)(20,52,38)(21,53,39)(22,54,40)(23,46,41)(24,47,42)(25,48,43)(26,49,44)(27,50,45), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(11,27)(12,19)(14,21)(15,22)(17,24)(18,25)(29,39)(30,40)(32,42)(33,43)(35,45)(36,37), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(10,26)(12,19)(13,20)(15,22)(16,23)(18,25)(28,38)(30,40)(31,41)(33,43)(34,44)(36,37), (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( (1,28,13)(2,29,14)(3,30,15)(4,31,16)(5,32,17)(6,33,18)(7,34,10)(8,35,11)(9,36,12)(19,51,37)(20,52,38)(21,53,39)(22,54,40)(23,46,41)(24,47,42)(25,48,43)(26,49,44)(27,50,45), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(11,27)(12,19)(14,21)(15,22)(17,24)(18,25)(29,39)(30,40)(32,42)(33,43)(35,45)(36,37), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(10,26)(12,19)(13,20)(15,22)(16,23)(18,25)(28,38)(30,40)(31,41)(33,43)(34,44)(36,37), (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([[(1,28,13),(2,29,14),(3,30,15),(4,31,16),(5,32,17),(6,33,18),(7,34,10),(8,35,11),(9,36,12),(19,51,37),(20,52,38),(21,53,39),(22,54,40),(23,46,41),(24,47,42),(25,48,43),(26,49,44),(27,50,45)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(2,53),(3,54),(5,47),(6,48),(8,50),(9,51),(11,27),(12,19),(14,21),(15,22),(17,24),(18,25),(29,39),(30,40),(32,42),(33,43),(35,45),(36,37)], [(1,52),(3,54),(4,46),(6,48),(7,49),(9,51),(10,26),(12,19),(13,20),(15,22),(16,23),(18,25),(28,38),(30,40),(31,41),(33,43),(34,44),(36,37)], [(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)]])
C3×C3.A4 is a maximal subgroup of
C32.3S4 C62.11C32 C62.12C32 C62.16C32 He3.A4 He3⋊A4 C62.C32 3- 1+2⋊A4 C62⋊C9 A4×C3×C9 He3.2A4 C62.9C32
C3×C3.A4 is a maximal quotient of
C62.C9 C62.12C32 C62⋊C9
36 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9R |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
| size | 1 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||
| image | C1 | C3 | C3 | C9 | A4 | C3.A4 | C3×A4 |
| kernel | C3×C3.A4 | C3.A4 | C62 | C2×C6 | C32 | C3 | C3 |
| # reps | 1 | 6 | 2 | 18 | 1 | 6 | 2 |
Matrix representation of C3×C3.A4 ►in GL4(𝔽19) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 6 | 18 | 0 |
| 0 | 17 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 2 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 17 | 0 |
| 0 | 0 | 13 | 1 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,6,17,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,2,0,0,18,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,17,13,2,0,0,1,0] >;
C3×C3.A4 in GAP, Magma, Sage, TeX
C_3\times C_3.A_4
% in TeX
G:=Group("C3xC3.A4"); // GroupNames label
G:=SmallGroup(108,20);
// by ID
G=gap.SmallGroup(108,20);
# by ID
G:=PCGroup([5,-3,-3,-3,-2,2,45,1083,2029]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=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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