direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C42⋊6C4, C42⋊9C12, M4(2)⋊2C12, C12.27C42, C4⋊C4⋊3C12, C6.26C4≀C2, (C4×C12)⋊16C4, C4.1(C4×C12), C12.51(C4⋊C4), (C2×C12).69Q8, (C2×C12).506D4, (C3×M4(2))⋊8C4, (C2×C42).10C6, C23.36(C3×D4), C42⋊C2.2C6, (C22×C6).151D4, (C2×M4(2)).6C6, (C6×M4(2)).18C2, C12.110(C22⋊C4), C6.23(C2.C42), (C22×C12).570C22, C4.2(C3×C4⋊C4), (C3×C4⋊C4)⋊10C4, C2.3(C3×C4≀C2), (C2×C4×C12).30C2, C22.3(C3×C4⋊C4), (C2×C4).12(C3×Q8), (C2×C6).20(C4⋊C4), (C2×C4).65(C2×C12), (C2×C4).142(C3×D4), C4.25(C3×C22⋊C4), (C2×C12).260(C2×C4), (C2×C6).71(C22⋊C4), (C22×C4).110(C2×C6), C22.28(C3×C22⋊C4), C2.4(C3×C2.C42), (C3×C42⋊C2).16C2, SmallGroup(192,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊6C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >
Subgroups: 170 in 110 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C42⋊6C4, C2×C4×C12, C3×C42⋊C2, C6×M4(2), C3×C42⋊6C4
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≀C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C42⋊6C4, C3×C2.C42, C3×C4≀C2, C3×C42⋊6C4
►On 48 points
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 44 40)(26 41 37)(27 42 38)(28 43 39)(29 35 48)(30 36 45)(31 33 46)(32 34 47)
(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 6 2 5)(3 8 4 7)(9 14 10 13)(11 16 12 15)(17 22 18 21)(19 24 20 23)(25 28 27 26)(29 32 31 30)(33 36 35 34)(37 40 39 38)(41 44 43 42)(45 48 47 46)
(1 35 3 25)(2 33 4 27)(5 34 7 28)(6 36 8 26)(9 29 11 40)(10 31 12 38)(13 32 15 39)(14 30 16 37)(17 48 19 44)(18 46 20 42)(21 47 23 43)(22 45 24 41)
G:=sub<Sym(48)| (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,44,40)(26,41,37)(27,42,38)(28,43,39)(29,35,48)(30,36,45)(31,33,46)(32,34,47), (25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,6,2,5)(3,8,4,7)(9,14,10,13)(11,16,12,15)(17,22,18,21)(19,24,20,23)(25,28,27,26)(29,32,31,30)(33,36,35,34)(37,40,39,38)(41,44,43,42)(45,48,47,46), (1,35,3,25)(2,33,4,27)(5,34,7,28)(6,36,8,26)(9,29,11,40)(10,31,12,38)(13,32,15,39)(14,30,16,37)(17,48,19,44)(18,46,20,42)(21,47,23,43)(22,45,24,41)>;
G:=Group( (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,44,40)(26,41,37)(27,42,38)(28,43,39)(29,35,48)(30,36,45)(31,33,46)(32,34,47), (25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,6,2,5)(3,8,4,7)(9,14,10,13)(11,16,12,15)(17,22,18,21)(19,24,20,23)(25,28,27,26)(29,32,31,30)(33,36,35,34)(37,40,39,38)(41,44,43,42)(45,48,47,46), (1,35,3,25)(2,33,4,27)(5,34,7,28)(6,36,8,26)(9,29,11,40)(10,31,12,38)(13,32,15,39)(14,30,16,37)(17,48,19,44)(18,46,20,42)(21,47,23,43)(22,45,24,41) );
G=PermutationGroup([[(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,44,40),(26,41,37),(27,42,38),(28,43,39),(29,35,48),(30,36,45),(31,33,46),(32,34,47)], [(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,6,2,5),(3,8,4,7),(9,14,10,13),(11,16,12,15),(17,22,18,21),(19,24,20,23),(25,28,27,26),(29,32,31,30),(33,36,35,34),(37,40,39,38),(41,44,43,42),(45,48,47,46)], [(1,35,3,25),(2,33,4,27),(5,34,7,28),(6,36,8,26),(9,29,11,40),(10,31,12,38),(13,32,15,39),(14,30,16,37),(17,48,19,44),(18,46,20,42),(21,47,23,43),(22,45,24,41)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12AB | 12AC | ··· | 12AJ | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | Q8 | D4 | C3×D4 | C3×Q8 | C3×D4 | C4≀C2 | C3×C4≀C2 |
| kernel | C3×C42⋊6C4 | C2×C4×C12 | C3×C42⋊C2 | C6×M4(2) | C42⋊6C4 | C4×C12 | C3×C4⋊C4 | C3×M4(2) | C2×C42 | C42⋊C2 | C2×M4(2) | C42 | C4⋊C4 | M4(2) | C2×C12 | C2×C12 | C22×C6 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 8 | 8 | 2 | 1 | 1 | 4 | 2 | 2 | 8 | 16 |
Matrix representation of C3×C42⋊6C4 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 13 | 27 |
| 1 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 46 | 46 |
| 27 | 0 | 0 |
| 0 | 72 | 71 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[72,0,0,0,1,13,0,0,27],[1,0,0,0,27,46,0,0,46],[27,0,0,0,72,0,0,71,1] >;
C3×C42⋊6C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_6C_4
% in TeX
G:=Group("C3xC4^2:6C4"); // GroupNames label
G:=SmallGroup(192,145);
// by ID
G=gap.SmallGroup(192,145);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,248,6053]);
// 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*b*d^-1=b*c=c*b,d*c*d^-1=c^-1>;
// generators/relations