metabelian, supersoluble, monomial
Aliases: C6.11D12, C62.24C22, (C6×C12)⋊2C2, (C2×C12)⋊2S3, C3⋊2(D6⋊C4), C6.15(C4×S3), (C2×C6).33D6, (C3×C6).26D4, C6.21(C3⋊D4), C32⋊6(C22⋊C4), C2.2(C12⋊S3), C2.2(C32⋊7D4), (C2×C3⋊S3)⋊2C4, C2.5(C4×C3⋊S3), (C2×C4)⋊1(C3⋊S3), (C3×C6).26(C2×C4), (C2×C3⋊Dic3)⋊2C2, C22.6(C2×C3⋊S3), (C22×C3⋊S3).2C2, SmallGroup(144,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.11D12
G = < a,b,c | a6=b12=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b-1 >
Subgroups: 378 in 102 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×C12, C22×S3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C6.11D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C2×C3⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6.11D12
►On 72 points
(1 19 67 53 40 25)(2 20 68 54 41 26)(3 21 69 55 42 27)(4 22 70 56 43 28)(5 23 71 57 44 29)(6 24 72 58 45 30)(7 13 61 59 46 31)(8 14 62 60 47 32)(9 15 63 49 48 33)(10 16 64 50 37 34)(11 17 65 51 38 35)(12 18 66 52 39 36)
(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 52 53 12)(2 11 54 51)(3 50 55 10)(4 9 56 49)(5 60 57 8)(6 7 58 59)(13 72 46 30)(14 29 47 71)(15 70 48 28)(16 27 37 69)(17 68 38 26)(18 25 39 67)(19 66 40 36)(20 35 41 65)(21 64 42 34)(22 33 43 63)(23 62 44 32)(24 31 45 61)
G:=sub<Sym(72)| (1,19,67,53,40,25)(2,20,68,54,41,26)(3,21,69,55,42,27)(4,22,70,56,43,28)(5,23,71,57,44,29)(6,24,72,58,45,30)(7,13,61,59,46,31)(8,14,62,60,47,32)(9,15,63,49,48,33)(10,16,64,50,37,34)(11,17,65,51,38,35)(12,18,66,52,39,36), (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), (1,52,53,12)(2,11,54,51)(3,50,55,10)(4,9,56,49)(5,60,57,8)(6,7,58,59)(13,72,46,30)(14,29,47,71)(15,70,48,28)(16,27,37,69)(17,68,38,26)(18,25,39,67)(19,66,40,36)(20,35,41,65)(21,64,42,34)(22,33,43,63)(23,62,44,32)(24,31,45,61)>;
G:=Group( (1,19,67,53,40,25)(2,20,68,54,41,26)(3,21,69,55,42,27)(4,22,70,56,43,28)(5,23,71,57,44,29)(6,24,72,58,45,30)(7,13,61,59,46,31)(8,14,62,60,47,32)(9,15,63,49,48,33)(10,16,64,50,37,34)(11,17,65,51,38,35)(12,18,66,52,39,36), (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), (1,52,53,12)(2,11,54,51)(3,50,55,10)(4,9,56,49)(5,60,57,8)(6,7,58,59)(13,72,46,30)(14,29,47,71)(15,70,48,28)(16,27,37,69)(17,68,38,26)(18,25,39,67)(19,66,40,36)(20,35,41,65)(21,64,42,34)(22,33,43,63)(23,62,44,32)(24,31,45,61) );
G=PermutationGroup([[(1,19,67,53,40,25),(2,20,68,54,41,26),(3,21,69,55,42,27),(4,22,70,56,43,28),(5,23,71,57,44,29),(6,24,72,58,45,30),(7,13,61,59,46,31),(8,14,62,60,47,32),(9,15,63,49,48,33),(10,16,64,50,37,34),(11,17,65,51,38,35),(12,18,66,52,39,36)], [(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)], [(1,52,53,12),(2,11,54,51),(3,50,55,10),(4,9,56,49),(5,60,57,8),(6,7,58,59),(13,72,46,30),(14,29,47,71),(15,70,48,28),(16,27,37,69),(17,68,38,26),(18,25,39,67),(19,66,40,36),(20,35,41,65),(21,64,42,34),(22,33,43,63),(23,62,44,32),(24,31,45,61)]])
C6.11D12 is a maximal subgroup of
C62.6C23 C62.18C23 C62.20C23 Dic3.D12 C62.24C23 C62.38C23 Dic3⋊4D12 Dic3⋊D12 C62.58C23 D6.D12 C62.65C23 C62.74C23 D6⋊D12 C62.77C23 S3×D6⋊C4 D6⋊4D12 C122⋊16C2 C4×C12⋊S3 C122⋊6C2 C122⋊2C2 C22⋊C4×C3⋊S3 C62.225C23 C62⋊12D4 C62.227C23 C62.228C23 C62.229C23 C62.69D4 C62.236C23 C62.237C23 C62.238C23 C12⋊3D12 C62.240C23 C12.31D12 C62.242C23 C4×C32⋊7D4 C62.129D4 C62⋊19D4 C62⋊13D4 C62⋊14D4 C62.261C23 C62.262C23 C62.21D6 C6.11D36 C62.78D6 C62.79D6 C62.148D6
C6.11D12 is a maximal quotient of
C122⋊C2 C62.110D4 C62.113D4 C62.114D4 C6.4Dic12 C12.60D12 C62.84D4 C12.19D12 C12.20D12 C62.37D4 C62.15Q8 C6.11D36 C62.31D6 C62.78D6 C62.79D6 C62.148D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 |
| kernel | C6.11D12 | C2×C3⋊Dic3 | C6×C12 | C22×C3⋊S3 | C2×C3⋊S3 | C2×C12 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 8 |
Matrix representation of C6.11D12 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 2 | 9 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 5 | 8 |
| 4 | 11 | 0 | 0 |
| 2 | 9 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,0,5,0,0,8,8],[4,2,0,0,11,9,0,0,0,0,5,0,0,0,8,8] >;
C6.11D12 in GAP, Magma, Sage, TeX
C_6._{11}D_{12} % in TeX
G:=Group("C6.11D12"); // GroupNames label
G:=SmallGroup(144,95);
// by ID
G=gap.SmallGroup(144,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations