direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4.9C42, C42⋊3C12, C12.32C42, (C4×C12)⋊4C4, (C2×C8)⋊1C12, (C2×C24)⋊3C4, C4.9(C4×C12), C12.50(C4⋊C4), (C2×C12).36Q8, C23.8(C3×D4), (C2×C12).277D4, (C22×C6).27D4, C42⋊C2.1C6, (C2×M4(2)).4C6, (C6×M4(2)).16C2, C12.101(C22⋊C4), C6.21(C2.C42), (C22×C12).386C22, C4.1(C3×C4⋊C4), (C2×C4).8(C3×D4), (C2×C4).1(C3×Q8), C22.1(C3×C4⋊C4), (C2×C6).18(C4⋊C4), (C2×C4).63(C2×C12), C4.17(C3×C22⋊C4), (C2×C12).324(C2×C4), (C22×C4).21(C2×C6), C22.7(C3×C22⋊C4), (C2×C6).70(C22⋊C4), C2.2(C3×C2.C42), (C3×C42⋊C2).15C2, SmallGroup(192,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.9C42
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 154 in 94 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22×C12, C4.9C42, C3×C42⋊C2, C6×M4(2), C3×C4.9C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C4.9C42, C3×C2.C42, C3×C4.9C42
►On 48 points
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 43 35)(6 44 36)(7 41 33)(8 42 34)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 35 11 32)(2 36 12 29)(3 33 9 30)(4 34 10 31)(5 27 48 24)(6 28 45 21)(7 25 46 22)(8 26 47 23)(13 44 20 37)(14 41 17 38)(15 42 18 39)(16 43 19 40)
(5 8 7 6)(9 11)(10 12)(17 19)(18 20)(25 27)(26 28)(29 30 31 32)(33 36 35 34)(37 38 39 40)(41 44 43 42)(45 46 47 48)
G:=sub<Sym(48)| (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,43,35)(6,44,36)(7,41,33)(8,42,34)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,35,11,32)(2,36,12,29)(3,33,9,30)(4,34,10,31)(5,27,48,24)(6,28,45,21)(7,25,46,22)(8,26,47,23)(13,44,20,37)(14,41,17,38)(15,42,18,39)(16,43,19,40), (5,8,7,6)(9,11)(10,12)(17,19)(18,20)(25,27)(26,28)(29,30,31,32)(33,36,35,34)(37,38,39,40)(41,44,43,42)(45,46,47,48)>;
G:=Group( (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,43,35)(6,44,36)(7,41,33)(8,42,34)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,35,11,32)(2,36,12,29)(3,33,9,30)(4,34,10,31)(5,27,48,24)(6,28,45,21)(7,25,46,22)(8,26,47,23)(13,44,20,37)(14,41,17,38)(15,42,18,39)(16,43,19,40), (5,8,7,6)(9,11)(10,12)(17,19)(18,20)(25,27)(26,28)(29,30,31,32)(33,36,35,34)(37,38,39,40)(41,44,43,42)(45,46,47,48) );
G=PermutationGroup([[(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,43,35),(6,44,36),(7,41,33),(8,42,34),(9,25,17),(10,26,18),(11,27,19),(12,28,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(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)], [(1,35,11,32),(2,36,12,29),(3,33,9,30),(4,34,10,31),(5,27,48,24),(6,28,45,21),(7,25,46,22),(8,26,47,23),(13,44,20,37),(14,41,17,38),(15,42,18,39),(16,43,19,40)], [(5,8,7,6),(9,11),(10,12),(17,19),(18,20),(25,27),(26,28),(29,30,31,32),(33,36,35,34),(37,38,39,40),(41,44,43,42),(45,46,47,48)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12Z | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | Q8 | D4 | C3×D4 | C3×Q8 | C3×D4 | C4.9C42 | C3×C4.9C42 |
| kernel | C3×C4.9C42 | C3×C42⋊C2 | C6×M4(2) | C4.9C42 | C4×C12 | C2×C24 | C42⋊C2 | C2×M4(2) | C42 | C2×C8 | C2×C12 | C2×C12 | C22×C6 | C2×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C3×C4.9C42 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 27 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 71 | 0 | 0 | 46 |
| 1 | 0 | 0 | 14 |
| 0 | 72 | 0 | 13 |
| 0 | 0 | 46 | 46 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[27,46,1,71,0,0,1,0,1,0,0,0,0,0,0,46],[1,0,0,0,0,72,0,0,0,0,46,0,14,13,46,27] >;
C3×C4.9C42 in GAP, Magma, Sage, TeX
C_3\times C_4._9C_4^2
% in TeX
G:=Group("C3xC4.9C4^2"); // GroupNames label
G:=SmallGroup(192,143);
// by ID
G=gap.SmallGroup(192,143);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,248,2111,6053]);
// 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,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations