metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.3C42, C23.38D12, M4(2)⋊3Dic3, C6.12C4≀C2, C4⋊Dic3⋊11C4, (C4×Dic3)⋊6C4, C12.10(C4⋊C4), (C2×C12).10Q8, C4.3(C4×Dic3), C3⋊3(C42⋊6C4), (C2×C12).492D4, (C3×M4(2))⋊6C4, (C2×C4).25Dic6, (C22×C6).46D4, (C22×C4).341D6, (C2×M4(2)).6S3, C2.3(D12⋊C4), C12.94(C22⋊C4), C4.10(Dic3⋊C4), C22.42(D6⋊C4), (C6×M4(2)).10C2, C22.3(C4⋊Dic3), C4.27(C6.D4), C23.26D6.8C2, C2.14(C6.C42), C6.14(C2.C42), (C22×C12).124C22, (C2×C6).7(C4⋊C4), (C2×C4).69(C4×S3), (C2×C4×Dic3).2C2, (C2×C12).62(C2×C4), (C2×C4).39(C2×Dic3), (C2×C4).179(C3⋊D4), (C2×C6).54(C22⋊C4), SmallGroup(192,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.3C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b >
Subgroups: 264 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C4×Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C3×M4(2), C22×Dic3, C22×C12, C42⋊6C4, C2×C4×Dic3, C23.26D6, C6×M4(2), C12.3C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4≀C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C42⋊6C4, D12⋊C4, C6.C42, C12.3C42
►On 48 points
(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 43 18 33)(2 42 19 32)(3 41 20 31)(4 40 21 30)(5 39 22 29)(6 38 23 28)(7 37 24 27)(8 48 13 26)(9 47 14 25)(10 46 15 36)(11 45 16 35)(12 44 17 34)
(1 37 21 36 7 43 15 30)(2 44 22 31 8 38 16 25)(3 39 23 26 9 45 17 32)(4 46 24 33 10 40 18 27)(5 41 13 28 11 47 19 34)(6 48 14 35 12 42 20 29)
G:=sub<Sym(48)| (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), (1,43,18,33)(2,42,19,32)(3,41,20,31)(4,40,21,30)(5,39,22,29)(6,38,23,28)(7,37,24,27)(8,48,13,26)(9,47,14,25)(10,46,15,36)(11,45,16,35)(12,44,17,34), (1,37,21,36,7,43,15,30)(2,44,22,31,8,38,16,25)(3,39,23,26,9,45,17,32)(4,46,24,33,10,40,18,27)(5,41,13,28,11,47,19,34)(6,48,14,35,12,42,20,29)>;
G:=Group( (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), (1,43,18,33)(2,42,19,32)(3,41,20,31)(4,40,21,30)(5,39,22,29)(6,38,23,28)(7,37,24,27)(8,48,13,26)(9,47,14,25)(10,46,15,36)(11,45,16,35)(12,44,17,34), (1,37,21,36,7,43,15,30)(2,44,22,31,8,38,16,25)(3,39,23,26,9,45,17,32)(4,46,24,33,10,40,18,27)(5,41,13,28,11,47,19,34)(6,48,14,35,12,42,20,29) );
G=PermutationGroup([[(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)], [(1,43,18,33),(2,42,19,32),(3,41,20,31),(4,40,21,30),(5,39,22,29),(6,38,23,28),(7,37,24,27),(8,48,13,26),(9,47,14,25),(10,46,15,36),(11,45,16,35),(12,44,17,34)], [(1,37,21,36,7,43,15,30),(2,44,22,31,8,38,16,25),(3,39,23,26,9,45,17,32),(4,46,24,33,10,40,18,27),(5,41,13,28,11,47,19,34),(6,48,14,35,12,42,20,29)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | Dic6 | C4×S3 | C3⋊D4 | D12 | C4≀C2 | D12⋊C4 |
| kernel | C12.3C42 | C2×C4×Dic3 | C23.26D6 | C6×M4(2) | C4×Dic3 | C4⋊Dic3 | C3×M4(2) | C2×M4(2) | C2×C12 | C2×C12 | C22×C6 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 |
Matrix representation of C12.3C42 ►in GL4(𝔽73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 1 | 27 |
| 30 | 60 | 0 | 0 |
| 30 | 43 | 0 | 0 |
| 0 | 0 | 72 | 19 |
| 0 | 0 | 46 | 1 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 27 | 71 |
| 0 | 0 | 13 | 46 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,1,0,0,0,0,46,1,0,0,0,27],[30,30,0,0,60,43,0,0,0,0,72,46,0,0,19,1],[66,14,0,0,59,7,0,0,0,0,27,13,0,0,71,46] >;
C12.3C42 in GAP, Magma, Sage, TeX
C_{12}._3C_4^2 % in TeX
G:=Group("C12.3C4^2"); // GroupNames label
G:=SmallGroup(192,114);
// by ID
G=gap.SmallGroup(192,114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,136,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations