Aliases: C62.14C32, (C3×C9)⋊2A4, (C6×C18)⋊2C3, C32⋊A4⋊1C3, (C2×C6).2He3, C3.4(C32⋊A4), C32.11(C3×A4), C22⋊1(He3⋊C3), SmallGroup(324,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.14C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, cac-1=ab=ba, ad=da, cbc-1=a3b4, bd=db, dcd-1=a2b2c >
3C2
3C3
36C3
36C3
36C3
3C6
3C6
3C6
3C6
3C9
12C32
12C32
12C32
9A4
9A4
9A4
3C18
3C18
3C18
4He3
4He3
4He3
Smallest permutation representation of C62.14C32
►On 54 points
Generators in S54
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 16 13)(11 17 14)(12 18 15)(28 43 34 40 31 37)(29 44 35 41 32 38)(30 45 36 42 33 39)(46 52 49)(47 53 50)(48 54 51)
(1 20 4 23 7 26)(2 21 5 24 8 27)(3 22 6 25 9 19)(10 46 13 49 16 52)(11 47 14 50 17 53)(12 48 15 51 18 54)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)
(1 52 31)(2 47 32)(3 51 33)(4 46 34)(5 50 35)(6 54 36)(7 49 28)(8 53 29)(9 48 30)(10 40 20)(11 38 27)(12 45 25)(13 43 23)(14 41 21)(15 39 19)(16 37 26)(17 44 24)(18 42 22)
(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)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,16,13)(11,17,14)(12,18,15)(28,43,34,40,31,37)(29,44,35,41,32,38)(30,45,36,42,33,39)(46,52,49)(47,53,50)(48,54,51), (1,20,4,23,7,26)(2,21,5,24,8,27)(3,22,6,25,9,19)(10,46,13,49,16,52)(11,47,14,50,17,53)(12,48,15,51,18,54)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (1,52,31)(2,47,32)(3,51,33)(4,46,34)(5,50,35)(6,54,36)(7,49,28)(8,53,29)(9,48,30)(10,40,20)(11,38,27)(12,45,25)(13,43,23)(14,41,21)(15,39,19)(16,37,26)(17,44,24)(18,42,22), (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( (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,16,13)(11,17,14)(12,18,15)(28,43,34,40,31,37)(29,44,35,41,32,38)(30,45,36,42,33,39)(46,52,49)(47,53,50)(48,54,51), (1,20,4,23,7,26)(2,21,5,24,8,27)(3,22,6,25,9,19)(10,46,13,49,16,52)(11,47,14,50,17,53)(12,48,15,51,18,54)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (1,52,31)(2,47,32)(3,51,33)(4,46,34)(5,50,35)(6,54,36)(7,49,28)(8,53,29)(9,48,30)(10,40,20)(11,38,27)(12,45,25)(13,43,23)(14,41,21)(15,39,19)(16,37,26)(17,44,24)(18,42,22), (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([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,16,13),(11,17,14),(12,18,15),(28,43,34,40,31,37),(29,44,35,41,32,38),(30,45,36,42,33,39),(46,52,49),(47,53,50),(48,54,51)], [(1,20,4,23,7,26),(2,21,5,24,8,27),(3,22,6,25,9,19),(10,46,13,49,16,52),(11,47,14,50,17,53),(12,48,15,51,18,54),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42)], [(1,52,31),(2,47,32),(3,51,33),(4,46,34),(5,50,35),(6,54,36),(7,49,28),(8,53,29),(9,48,30),(10,40,20),(11,38,27),(12,45,25),(13,43,23),(14,41,21),(15,39,19),(16,37,26),(17,44,24),(18,42,22)], [(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)]])
44 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | ··· | 6H | 9A | ··· | 9F | 18A | ··· | 18R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 36 | ··· | 36 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
44 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C3 | A4 | He3 | C3×A4 | He3⋊C3 | C32⋊A4 | C62.14C32 |
| kernel | C62.14C32 | C32⋊A4 | C6×C18 | C3×C9 | C2×C6 | C32 | C22 | C3 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 2 | 6 | 6 | 18 |
Matrix representation of C62.14C32 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 7 |
| 12 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 17 |
G:=sub<GL(3,GF(19))| [18,0,0,0,8,0,0,0,7],[12,0,0,0,7,0,0,0,12],[0,0,1,1,0,0,0,1,0],[16,0,0,0,16,0,0,0,17] >;
C62.14C32 in GAP, Magma, Sage, TeX
C_6^2._{14}C_3^2 % in TeX
G:=Group("C6^2.14C3^2"); // GroupNames label
G:=SmallGroup(324,50);
// by ID
G=gap.SmallGroup(324,50);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,115,1136,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^3=1,d^3=b^2,c*a*c^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=a^2*b^2*c>;
// generators/relations
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