metabelian, supersoluble, monomial
Aliases: C12.58D6, C62.7C4, C12.3Dic3, C32⋊8M4(2), (C3×C12).4C4, (C6×C12).6C2, C4.(C3⋊Dic3), (C2×C12).12S3, C32⋊4C8⋊9C2, (C2×C6).8Dic3, C3⋊2(C4.Dic3), C6.13(C2×Dic3), C22.(C3⋊Dic3), (C3×C12).49C22, C4.15(C2×C3⋊S3), (C2×C4).2(C3⋊S3), (C3×C6).32(C2×C4), C2.3(C2×C3⋊Dic3), SmallGroup(144,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.58D6
G = < a,b,c | a12=b6=1, c2=a3, ab=ba, cac-1=a5, cbc-1=a6b-1 >
Subgroups: 114 in 60 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C3×C6, C3⋊C8, C2×C12, C3×C12, C62, C4.Dic3, C32⋊4C8, C6×C12, C12.58D6
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), C3⋊S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4.Dic3, C2×C3⋊Dic3, C12.58D6
►On 72 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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 60 47)(2 49 48)(3 50 37)(4 51 38)(5 52 39)(6 53 40)(7 54 41)(8 55 42)(9 56 43)(10 57 44)(11 58 45)(12 59 46)(13 67 32 19 61 26)(14 68 33 20 62 27)(15 69 34 21 63 28)(16 70 35 22 64 29)(17 71 36 23 65 30)(18 72 25 24 66 31)
(1 13 4 16 7 19 10 22)(2 18 5 21 8 24 11 15)(3 23 6 14 9 17 12 20)(25 52 28 55 31 58 34 49)(26 57 29 60 32 51 35 54)(27 50 30 53 33 56 36 59)(37 71 40 62 43 65 46 68)(38 64 41 67 44 70 47 61)(39 69 42 72 45 63 48 66)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,60,47)(2,49,48)(3,50,37)(4,51,38)(5,52,39)(6,53,40)(7,54,41)(8,55,42)(9,56,43)(10,57,44)(11,58,45)(12,59,46)(13,67,32,19,61,26)(14,68,33,20,62,27)(15,69,34,21,63,28)(16,70,35,22,64,29)(17,71,36,23,65,30)(18,72,25,24,66,31), (1,13,4,16,7,19,10,22)(2,18,5,21,8,24,11,15)(3,23,6,14,9,17,12,20)(25,52,28,55,31,58,34,49)(26,57,29,60,32,51,35,54)(27,50,30,53,33,56,36,59)(37,71,40,62,43,65,46,68)(38,64,41,67,44,70,47,61)(39,69,42,72,45,63,48,66)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,60,47)(2,49,48)(3,50,37)(4,51,38)(5,52,39)(6,53,40)(7,54,41)(8,55,42)(9,56,43)(10,57,44)(11,58,45)(12,59,46)(13,67,32,19,61,26)(14,68,33,20,62,27)(15,69,34,21,63,28)(16,70,35,22,64,29)(17,71,36,23,65,30)(18,72,25,24,66,31), (1,13,4,16,7,19,10,22)(2,18,5,21,8,24,11,15)(3,23,6,14,9,17,12,20)(25,52,28,55,31,58,34,49)(26,57,29,60,32,51,35,54)(27,50,30,53,33,56,36,59)(37,71,40,62,43,65,46,68)(38,64,41,67,44,70,47,61)(39,69,42,72,45,63,48,66) );
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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,60,47),(2,49,48),(3,50,37),(4,51,38),(5,52,39),(6,53,40),(7,54,41),(8,55,42),(9,56,43),(10,57,44),(11,58,45),(12,59,46),(13,67,32,19,61,26),(14,68,33,20,62,27),(15,69,34,21,63,28),(16,70,35,22,64,29),(17,71,36,23,65,30),(18,72,25,24,66,31)], [(1,13,4,16,7,19,10,22),(2,18,5,21,8,24,11,15),(3,23,6,14,9,17,12,20),(25,52,28,55,31,58,34,49),(26,57,29,60,32,51,35,54),(27,50,30,53,33,56,36,59),(37,71,40,62,43,65,46,68),(38,64,41,67,44,70,47,61),(39,69,42,72,45,63,48,66)]])
C12.58D6 is a maximal subgroup of
C12.D12 C12.14D12 D12⋊2Dic3 C12.82D12 C122⋊C2 C12.59D12 C62.8Q8 C12.19D12 C12.20D12 (C6×D4).S3 (C6×C12).C4 C62.39D4 S3×C4.Dic3 D12.2Dic3 D12⋊20D6 D12.32D6 C24.95D6 M4(2)×C3⋊S3 C62.131D4 C62.134D4 D4.(C3⋊Dic3) C62.73D4 C62.75D4 He3⋊7M4(2) C36.69D6 C33⋊7M4(2) C33⋊18M4(2)
C12.58D6 is a maximal quotient of
C122.C2 C12.57D12 C62⋊7C8 C36.69D6 He3⋊8M4(2) C33⋊7M4(2) C33⋊18M4(2)
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4.Dic3 |
| kernel | C12.58D6 | C32⋊4C8 | C6×C12 | C3×C12 | C62 | C2×C12 | C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 16 |
Matrix representation of C12.58D6 ►in GL4(𝔽73) generated by
| 49 | 0 | 0 | 0 |
| 0 | 70 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 24 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 65 |
| 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 49 | 0 |
G:=sub<GL(4,GF(73))| [49,0,0,0,0,70,0,0,0,0,3,0,0,0,0,24],[1,0,0,0,0,72,0,0,0,0,64,0,0,0,0,65],[0,46,0,0,1,0,0,0,0,0,0,49,0,0,8,0] >;
C12.58D6 in GAP, Magma, Sage, TeX
C_{12}._{58}D_6 % in TeX
G:=Group("C12.58D6"); // GroupNames label
G:=SmallGroup(144,91);
// by ID
G=gap.SmallGroup(144,91);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,50,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^12=b^6=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^6*b^-1>;
// generators/relations