direct product, metabelian, supersoluble, monomial
Aliases: C3×C62⋊5C4, C63.3C2, C62⋊15C12, C62⋊14Dic3, C62.144D6, (C3×C62)⋊7C4, C62.68(C2×C6), (C2×C62).27C6, (C2×C62).27S3, C6.28(C6×Dic3), (C32×C6).77D4, C33⋊18(C22⋊C4), C6.39(C32⋊7D4), (C3×C62).50C22, C32⋊10(C6.D4), (C2×C6).71(S3×C6), (C6×C3⋊Dic3)⋊6C2, (C2×C3⋊Dic3)⋊9C6, (C2×C6)⋊6(C3×Dic3), (C3×C6).73(C3×D4), C6.42(C3×C3⋊D4), C23.3(C3×C3⋊S3), C22.7(C6×C3⋊S3), C2.5(C6×C3⋊Dic3), (C3×C6).65(C2×C12), (C2×C6)⋊3(C3⋊Dic3), C3⋊2(C3×C6.D4), C6.24(C2×C3⋊Dic3), C22⋊3(C3×C3⋊Dic3), C2.3(C3×C32⋊7D4), C32⋊11(C3×C22⋊C4), (C22×C6).33(C3×S3), (C32×C6).71(C2×C4), (C3×C6).70(C2×Dic3), (C3×C6).112(C3⋊D4), (C22×C6).11(C3⋊S3), (C2×C6).65(C2×C3⋊S3), SmallGroup(432,495)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C62⋊5C4
G = < a,b,c,d | a3=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c3, dcd-1=c-1 >
Subgroups: 756 in 332 conjugacy classes, 102 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3⋊Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C32×C6, C32×C6, C32×C6, C6×Dic3, C2×C3⋊Dic3, C2×C62, C2×C62, C2×C62, C3×C3⋊Dic3, C3×C62, C3×C62, C3×C62, C3×C6.D4, C62⋊5C4, C6×C3⋊Dic3, C63, C3×C62⋊5C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C6.D4, C3×C22⋊C4, C3×C3⋊S3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C3×C3⋊Dic3, C6×C3⋊S3, C3×C6.D4, C62⋊5C4, C6×C3⋊Dic3, C3×C32⋊7D4, C3×C62⋊5C4
►On 72 points
(1 14 12)(2 15 10)(3 13 11)(4 22 20)(5 23 21)(6 24 19)(7 33 26)(8 31 27)(9 32 25)(16 30 34)(17 28 35)(18 29 36)(37 53 46)(38 54 47)(39 49 48)(40 50 43)(41 51 44)(42 52 45)(55 69 62)(56 70 63)(57 71 64)(58 72 65)(59 67 66)(60 68 61)
(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)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 36 13 17 10 30)(2 34 14 18 11 28)(3 35 15 16 12 29)(4 8 24 33 21 25)(5 9 22 31 19 26)(6 7 23 32 20 27)(37 54 48 40 51 45)(38 49 43 41 52 46)(39 50 44 42 53 47)(55 70 64 58 67 61)(56 71 65 59 68 62)(57 72 66 60 69 63)
(1 38 31 62)(2 40 32 64)(3 42 33 66)(4 63 16 39)(5 65 17 41)(6 61 18 37)(7 67 11 45)(8 69 12 47)(9 71 10 43)(13 52 26 59)(14 54 27 55)(15 50 25 57)(19 68 36 46)(20 70 34 48)(21 72 35 44)(22 56 30 49)(23 58 28 51)(24 60 29 53)
G:=sub<Sym(72)| (1,14,12)(2,15,10)(3,13,11)(4,22,20)(5,23,21)(6,24,19)(7,33,26)(8,31,27)(9,32,25)(16,30,34)(17,28,35)(18,29,36)(37,53,46)(38,54,47)(39,49,48)(40,50,43)(41,51,44)(42,52,45)(55,69,62)(56,70,63)(57,71,64)(58,72,65)(59,67,66)(60,68,61), (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)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,36,13,17,10,30)(2,34,14,18,11,28)(3,35,15,16,12,29)(4,8,24,33,21,25)(5,9,22,31,19,26)(6,7,23,32,20,27)(37,54,48,40,51,45)(38,49,43,41,52,46)(39,50,44,42,53,47)(55,70,64,58,67,61)(56,71,65,59,68,62)(57,72,66,60,69,63), (1,38,31,62)(2,40,32,64)(3,42,33,66)(4,63,16,39)(5,65,17,41)(6,61,18,37)(7,67,11,45)(8,69,12,47)(9,71,10,43)(13,52,26,59)(14,54,27,55)(15,50,25,57)(19,68,36,46)(20,70,34,48)(21,72,35,44)(22,56,30,49)(23,58,28,51)(24,60,29,53)>;
G:=Group( (1,14,12)(2,15,10)(3,13,11)(4,22,20)(5,23,21)(6,24,19)(7,33,26)(8,31,27)(9,32,25)(16,30,34)(17,28,35)(18,29,36)(37,53,46)(38,54,47)(39,49,48)(40,50,43)(41,51,44)(42,52,45)(55,69,62)(56,70,63)(57,71,64)(58,72,65)(59,67,66)(60,68,61), (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)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,36,13,17,10,30)(2,34,14,18,11,28)(3,35,15,16,12,29)(4,8,24,33,21,25)(5,9,22,31,19,26)(6,7,23,32,20,27)(37,54,48,40,51,45)(38,49,43,41,52,46)(39,50,44,42,53,47)(55,70,64,58,67,61)(56,71,65,59,68,62)(57,72,66,60,69,63), (1,38,31,62)(2,40,32,64)(3,42,33,66)(4,63,16,39)(5,65,17,41)(6,61,18,37)(7,67,11,45)(8,69,12,47)(9,71,10,43)(13,52,26,59)(14,54,27,55)(15,50,25,57)(19,68,36,46)(20,70,34,48)(21,72,35,44)(22,56,30,49)(23,58,28,51)(24,60,29,53) );
G=PermutationGroup([[(1,14,12),(2,15,10),(3,13,11),(4,22,20),(5,23,21),(6,24,19),(7,33,26),(8,31,27),(9,32,25),(16,30,34),(17,28,35),(18,29,36),(37,53,46),(38,54,47),(39,49,48),(40,50,43),(41,51,44),(42,52,45),(55,69,62),(56,70,63),(57,71,64),(58,72,65),(59,67,66),(60,68,61)], [(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),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,36,13,17,10,30),(2,34,14,18,11,28),(3,35,15,16,12,29),(4,8,24,33,21,25),(5,9,22,31,19,26),(6,7,23,32,20,27),(37,54,48,40,51,45),(38,49,43,41,52,46),(39,50,44,42,53,47),(55,70,64,58,67,61),(56,71,65,59,68,62),(57,72,66,60,69,63)], [(1,38,31,62),(2,40,32,64),(3,42,33,66),(4,63,16,39),(5,65,17,41),(6,61,18,37),(7,67,11,45),(8,69,12,47),(9,71,10,43),(13,52,26,59),(14,54,27,55),(15,50,25,57),(19,68,36,46),(20,70,34,48),(21,72,35,44),(22,56,30,49),(23,58,28,51),(24,60,29,53)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6CP | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 18 | ··· | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Dic3 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3×Dic3 | S3×C6 | C3×C3⋊D4 |
| kernel | C3×C62⋊5C4 | C6×C3⋊Dic3 | C63 | C62⋊5C4 | C3×C62 | C2×C3⋊Dic3 | C2×C62 | C62 | C2×C62 | C32×C6 | C62 | C62 | C22×C6 | C3×C6 | C3×C6 | C2×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 2 | 8 | 4 | 8 | 16 | 4 | 16 | 8 | 32 |
Matrix representation of C3×C62⋊5C4 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C3×C62⋊5C4 in GAP, Magma, Sage, TeX
C_3\times C_6^2\rtimes_5C_4
% in TeX
G:=Group("C3xC6^2:5C4"); // GroupNames label
G:=SmallGroup(432,495);
// by ID
G=gap.SmallGroup(432,495);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^3,d*c*d^-1=c^-1>;
// generators/relations