direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C42⋊3C4, C42⋊5C12, (C4×C12)⋊6C4, (C6×Q8)⋊4C4, (C2×Q8)⋊3C12, C23⋊C4.2C6, (C22×C6).4D4, C23.4(C3×D4), C4.4D4.2C6, C6.35(C23⋊C4), (C6×D4).177C22, (C2×C4).2(C2×C12), (C2×D4).4(C2×C6), C2.9(C3×C23⋊C4), (C2×C12).13(C2×C4), (C3×C23⋊C4).4C2, (C2×C6).76(C22⋊C4), (C3×C4.4D4).11C2, C22.13(C3×C22⋊C4), SmallGroup(192,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊3C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >
Subgroups: 178 in 70 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C23⋊C4, C4.4D4, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C42⋊3C4, C3×C23⋊C4, C3×C4.4D4, C3×C42⋊3C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C23⋊C4, C3×C22⋊C4, C42⋊3C4, C3×C23⋊C4, C3×C42⋊3C4
►On 48 points
(1 15 11)(2 16 12)(3 13 9)(4 14 10)(5 34 26)(6 35 27)(7 36 28)(8 33 25)(17 21 32)(18 22 29)(19 23 30)(20 24 31)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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)
(1 39 23 6)(2 40 24 7)(3 37 21 8)(4 38 22 5)(9 41 17 25)(10 42 18 26)(11 43 19 27)(12 44 20 28)(13 45 32 33)(14 46 29 34)(15 47 30 35)(16 48 31 36)
(1 5 6 24)(2 23 38 39)(3 40 8 4)(7 37 22 21)(9 44 25 10)(11 26 27 20)(12 19 42 43)(13 48 33 14)(15 34 35 31)(16 30 46 47)(17 28 41 18)(29 32 36 45)
G:=sub<Sym(48)| (1,15,11)(2,16,12)(3,13,9)(4,14,10)(5,34,26)(6,35,27)(7,36,28)(8,33,25)(17,21,32)(18,22,29)(19,23,30)(20,24,31)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (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), (1,39,23,6)(2,40,24,7)(3,37,21,8)(4,38,22,5)(9,41,17,25)(10,42,18,26)(11,43,19,27)(12,44,20,28)(13,45,32,33)(14,46,29,34)(15,47,30,35)(16,48,31,36), (1,5,6,24)(2,23,38,39)(3,40,8,4)(7,37,22,21)(9,44,25,10)(11,26,27,20)(12,19,42,43)(13,48,33,14)(15,34,35,31)(16,30,46,47)(17,28,41,18)(29,32,36,45)>;
G:=Group( (1,15,11)(2,16,12)(3,13,9)(4,14,10)(5,34,26)(6,35,27)(7,36,28)(8,33,25)(17,21,32)(18,22,29)(19,23,30)(20,24,31)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (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), (1,39,23,6)(2,40,24,7)(3,37,21,8)(4,38,22,5)(9,41,17,25)(10,42,18,26)(11,43,19,27)(12,44,20,28)(13,45,32,33)(14,46,29,34)(15,47,30,35)(16,48,31,36), (1,5,6,24)(2,23,38,39)(3,40,8,4)(7,37,22,21)(9,44,25,10)(11,26,27,20)(12,19,42,43)(13,48,33,14)(15,34,35,31)(16,30,46,47)(17,28,41,18)(29,32,36,45) );
G=PermutationGroup([[(1,15,11),(2,16,12),(3,13,9),(4,14,10),(5,34,26),(6,35,27),(7,36,28),(8,33,25),(17,21,32),(18,22,29),(19,23,30),(20,24,31),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(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)], [(1,39,23,6),(2,40,24,7),(3,37,21,8),(4,38,22,5),(9,41,17,25),(10,42,18,26),(11,43,19,27),(12,44,20,28),(13,45,32,33),(14,46,29,34),(15,47,30,35),(16,48,31,36)], [(1,5,6,24),(2,23,38,39),(3,40,8,4),(7,37,22,21),(9,44,25,10),(11,26,27,20),(12,19,42,43),(13,48,33,14),(15,34,35,31),(16,30,46,47),(17,28,41,18),(29,32,36,45)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | ··· | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | ··· | 12F | 12G | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 8 | ··· | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C23⋊C4 | C42⋊3C4 | C3×C23⋊C4 | C3×C42⋊3C4 |
| kernel | C3×C42⋊3C4 | C3×C23⋊C4 | C3×C4.4D4 | C42⋊3C4 | C4×C12 | C6×Q8 | C23⋊C4 | C4.4D4 | C42 | C2×Q8 | C22×C6 | C23 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C42⋊3C4 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 4 | 4 |
| 0 | 0 | 9 | 4 | 4 | 4 |
| 0 | 0 | 9 | 9 | 4 | 9 |
| 0 | 0 | 9 | 9 | 9 | 4 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 9 | 9 |
| 0 | 0 | 4 | 9 | 4 | 4 |
| 0 | 0 | 9 | 9 | 4 | 9 |
| 0 | 0 | 4 | 4 | 4 | 9 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,4,9,9,9,0,0,9,4,9,9,0,0,4,4,4,9,0,0,4,4,9,4],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,4,4,9,4,0,0,9,9,9,4,0,0,9,4,4,4,0,0,9,4,9,9] >;
C3×C42⋊3C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_3C_4
% in TeX
G:=Group("C3xC4^2:3C4"); // GroupNames label
G:=SmallGroup(192,160);
// by ID
G=gap.SmallGroup(192,160);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1522,248,2951,375,6053]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=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^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations