direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C42.C4, C42.2C12, (C4×C12).5C4, (C6×D4).6C4, (C2×D4).3C12, (C2×C12).19D4, C4.10D4⋊5C6, C4.4D4.3C6, C6.36(C23⋊C4), (C6×Q8).155C22, (C2×C4).3(C3×D4), (C2×C4).3(C2×C12), (C2×Q8).2(C2×C6), (C2×C12).14(C2×C4), C2.10(C3×C23⋊C4), (C3×C4.10D4)⋊12C2, (C2×C6).77(C22⋊C4), (C3×C4.4D4).12C2, C22.14(C3×C22⋊C4), SmallGroup(192,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.C4
G = < a,b,c,d | a3=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >
Subgroups: 146 in 64 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.10D4, C4.4D4, C4×C12, C3×C22⋊C4, C3×M4(2), C6×D4, C6×Q8, C42.C4, C3×C4.10D4, C3×C4.4D4, C3×C42.C4
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.C4, C3×C23⋊C4, C3×C42.C4
►On 48 points
(1 14 39)(2 15 40)(3 16 33)(4 9 34)(5 10 35)(6 11 36)(7 12 37)(8 13 38)(17 29 41)(18 30 42)(19 31 43)(20 32 44)(21 25 45)(22 26 46)(23 27 47)(24 28 48)
(1 3)(2 18 6 22)(4 20 8 24)(5 7)(9 32 13 28)(10 12)(11 26 15 30)(14 16)(17 19)(21 23)(25 27)(29 31)(33 39)(34 44 38 48)(35 37)(36 46 40 42)(41 43)(45 47)
(1 23 5 19)(2 20 6 24)(3 21 7 17)(4 18 8 22)(9 30 13 26)(10 31 14 27)(11 28 15 32)(12 29 16 25)(33 45 37 41)(34 42 38 46)(35 43 39 47)(36 48 40 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)
G:=sub<Sym(48)| (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (1,3)(2,18,6,22)(4,20,8,24)(5,7)(9,32,13,28)(10,12)(11,26,15,30)(14,16)(17,19)(21,23)(25,27)(29,31)(33,39)(34,44,38,48)(35,37)(36,46,40,42)(41,43)(45,47), (1,23,5,19)(2,20,6,24)(3,21,7,17)(4,18,8,22)(9,30,13,26)(10,31,14,27)(11,28,15,32)(12,29,16,25)(33,45,37,41)(34,42,38,46)(35,43,39,47)(36,48,40,44), (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)>;
G:=Group( (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (1,3)(2,18,6,22)(4,20,8,24)(5,7)(9,32,13,28)(10,12)(11,26,15,30)(14,16)(17,19)(21,23)(25,27)(29,31)(33,39)(34,44,38,48)(35,37)(36,46,40,42)(41,43)(45,47), (1,23,5,19)(2,20,6,24)(3,21,7,17)(4,18,8,22)(9,30,13,26)(10,31,14,27)(11,28,15,32)(12,29,16,25)(33,45,37,41)(34,42,38,46)(35,43,39,47)(36,48,40,44), (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) );
G=PermutationGroup([[(1,14,39),(2,15,40),(3,16,33),(4,9,34),(5,10,35),(6,11,36),(7,12,37),(8,13,38),(17,29,41),(18,30,42),(19,31,43),(20,32,44),(21,25,45),(22,26,46),(23,27,47),(24,28,48)], [(1,3),(2,18,6,22),(4,20,8,24),(5,7),(9,32,13,28),(10,12),(11,26,15,30),(14,16),(17,19),(21,23),(25,27),(29,31),(33,39),(34,44,38,48),(35,37),(36,46,40,42),(41,43),(45,47)], [(1,23,5,19),(2,20,6,24),(3,21,7,17),(4,18,8,22),(9,30,13,26),(10,31,14,27),(11,28,15,32),(12,29,16,25),(33,45,37,41),(34,42,38,46),(35,43,39,47),(36,48,40,44)], [(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)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 8 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 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.C4 | C3×C23⋊C4 | C3×C42.C4 |
| kernel | C3×C42.C4 | C3×C4.10D4 | C3×C4.4D4 | C42.C4 | C4×C12 | C6×D4 | C4.10D4 | C4.4D4 | C42 | C2×D4 | C2×C12 | C2×C4 | 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.C4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 46 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,72,0,0,0,0,0,46,0,0,0,0,46],[0,46,0,0,46,0,0,0,0,0,0,46,0,0,46,0],[0,0,0,72,0,0,1,0,1,0,0,0,0,1,0,0] >;
C3×C42.C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2.C_4
% in TeX
G:=Group("C3xC4^2.C4"); // GroupNames label
G:=SmallGroup(192,161);
// by ID
G=gap.SmallGroup(192,161);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951,375,172,6053]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^4=c^2,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