Aliases: C62.25C32, C9⋊A4⋊3C3, (C3×C9)⋊5A4, (C9×A4)⋊1C3, (C6×C18)⋊9C3, C9.6(C3×A4), C32⋊A4.4C3, (C2×C6).3C33, C3.4(C32×A4), C32.A4⋊10C3, (C3×A4).2C32, C32.14(C3×A4), (C2×C18).6C32, C22⋊1(C9○He3), C3.A4.1C32, SmallGroup(324,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.25C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, ab=ba, cac-1=ab-1, ad=da, cbc-1=a3b4, bd=db, cd=dc >
Subgroups: 205 in 76 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C3, C3, C22, C6, C9, C9, C9, C32, C32, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, C3.A4, C2×C18, C2×C18, C3×A4, C62, C3×C18, C9○He3, C9×A4, C9⋊A4, C32.A4, C32⋊A4, C6×C18, C62.25C32
Quotients: C1, C3, C32, A4, C33, C3×A4, C9○He3, C32×A4, C62.25C32
►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 46 16 52 13 49)(11 47 17 53 14 50)(12 48 18 54 15 51)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)
(1 7 4)(2 8 5)(3 9 6)(10 49 13 52 16 46)(11 50 14 53 17 47)(12 51 15 54 18 48)(19 25 22)(20 26 23)(21 27 24)(28 42 31 45 34 39)(29 43 32 37 35 40)(30 44 33 38 36 41)
(1 49 31)(2 50 32)(3 51 33)(4 52 34)(5 53 35)(6 54 36)(7 46 28)(8 47 29)(9 48 30)(10 42 24)(11 43 25)(12 44 26)(13 45 27)(14 37 19)(15 38 20)(16 39 21)(17 40 22)(18 41 23)
(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)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(10,46,16,52,13,49)(11,47,17,53,14,50)(12,48,18,54,15,51)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (1,7,4)(2,8,5)(3,9,6)(10,49,13,52,16,46)(11,50,14,53,17,47)(12,51,15,54,18,48)(19,25,22)(20,26,23)(21,27,24)(28,42,31,45,34,39)(29,43,32,37,35,40)(30,44,33,38,36,41), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,42,24)(11,43,25)(12,44,26)(13,45,27)(14,37,19)(15,38,20)(16,39,21)(17,40,22)(18,41,23), (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( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(10,46,16,52,13,49)(11,47,17,53,14,50)(12,48,18,54,15,51)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (1,7,4)(2,8,5)(3,9,6)(10,49,13,52,16,46)(11,50,14,53,17,47)(12,51,15,54,18,48)(19,25,22)(20,26,23)(21,27,24)(28,42,31,45,34,39)(29,43,32,37,35,40)(30,44,33,38,36,41), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,42,24)(11,43,25)(12,44,26)(13,45,27)(14,37,19)(15,38,20)(16,39,21)(17,40,22)(18,41,23), (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([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,19),(9,20),(10,46,16,52,13,49),(11,47,17,53,14,50),(12,48,18,54,15,51),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42)], [(1,7,4),(2,8,5),(3,9,6),(10,49,13,52,16,46),(11,50,14,53,17,47),(12,51,15,54,18,48),(19,25,22),(20,26,23),(21,27,24),(28,42,31,45,34,39),(29,43,32,37,35,40),(30,44,33,38,36,41)], [(1,49,31),(2,50,32),(3,51,33),(4,52,34),(5,53,35),(6,54,36),(7,46,28),(8,47,29),(9,48,30),(10,42,24),(11,43,25),(12,44,26),(13,45,27),(14,37,19),(15,38,20),(16,39,21),(17,40,22),(18,41,23)], [(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)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | ··· | 6H | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 9K | ··· | 9V | 18A | ··· | 18R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 12 | ··· | 12 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||
| image | C1 | C3 | C3 | C3 | C3 | C3 | A4 | C3×A4 | C3×A4 | C9○He3 | C62.25C32 |
| kernel | C62.25C32 | C9×A4 | C9⋊A4 | C32.A4 | C32⋊A4 | C6×C18 | C3×C9 | C9 | C32 | C22 | C1 |
| # reps | 1 | 6 | 12 | 4 | 2 | 2 | 1 | 6 | 2 | 6 | 18 |
Matrix representation of C62.25C32 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 8 |
| 7 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(19))| [18,0,0,0,7,0,0,0,8],[7,0,0,0,12,0,0,0,12],[0,0,1,1,0,0,0,1,0],[16,0,0,0,16,0,0,0,16] >;
C62.25C32 in GAP, Magma, Sage, TeX
C_6^2._{25}C_3^2 % in TeX
G:=Group("C6^2.25C3^2"); // GroupNames label
G:=SmallGroup(324,128);
// by ID
G=gap.SmallGroup(324,128);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,115,650,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^3=1,d^3=b^2,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,c*d=d*c>;
// generators/relations