metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.55D4, C6.6M4(2), C23.3Dic3, (C2xC6):2C8, C22:2(C3:C8), C3:2(C22:C8), (C2xC12).4C4, C6.10(C2xC8), (C2xC4).94D6, (C22xC6).5C4, (C22xC4).3S3, (C2xC4).4Dic3, C4.30(C3:D4), C6.12(C22:C4), (C22xC12).11C2, C2.3(C4.Dic3), C22.9(C2xDic3), (C2xC12).108C22, C2.1(C6.D4), C2.5(C2xC3:C8), (C2xC3:C8):10C2, (C2xC6).27(C2xC4), SmallGroup(96,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.55D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b3 >
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, Dic3, D6, C22:C4, C2xC8, M4(2), C3:C8, C2xDic3, C3:D4, C22:C8, C2xC3:C8, C4.Dic3, C6.D4, C12.55D4
Smallest permutation representation of C12.55D4
►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 33 42 22 7 27 48 16)(2 26 43 15 8 32 37 21)(3 31 44 20 9 25 38 14)(4 36 45 13 10 30 39 19)(5 29 46 18 11 35 40 24)(6 34 47 23 12 28 41 17)
(1 13 10 22 7 19 4 16)(2 18 11 15 8 24 5 21)(3 23 12 20 9 17 6 14)(25 38 34 47 31 44 28 41)(26 43 35 40 32 37 29 46)(27 48 36 45 33 42 30 39)
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), (1,33,42,22,7,27,48,16)(2,26,43,15,8,32,37,21)(3,31,44,20,9,25,38,14)(4,36,45,13,10,30,39,19)(5,29,46,18,11,35,40,24)(6,34,47,23,12,28,41,17), (1,13,10,22,7,19,4,16)(2,18,11,15,8,24,5,21)(3,23,12,20,9,17,6,14)(25,38,34,47,31,44,28,41)(26,43,35,40,32,37,29,46)(27,48,36,45,33,42,30,39)>;
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), (1,33,42,22,7,27,48,16)(2,26,43,15,8,32,37,21)(3,31,44,20,9,25,38,14)(4,36,45,13,10,30,39,19)(5,29,46,18,11,35,40,24)(6,34,47,23,12,28,41,17), (1,13,10,22,7,19,4,16)(2,18,11,15,8,24,5,21)(3,23,12,20,9,17,6,14)(25,38,34,47,31,44,28,41)(26,43,35,40,32,37,29,46)(27,48,36,45,33,42,30,39) );
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)], [(1,33,42,22,7,27,48,16),(2,26,43,15,8,32,37,21),(3,31,44,20,9,25,38,14),(4,36,45,13,10,30,39,19),(5,29,46,18,11,35,40,24),(6,34,47,23,12,28,41,17)], [(1,13,10,22,7,19,4,16),(2,18,11,15,8,24,5,21),(3,23,12,20,9,17,6,14),(25,38,34,47,31,44,28,41),(26,43,35,40,32,37,29,46),(27,48,36,45,33,42,30,39)]])
C12.55D4 is a maximal subgroup of
C6.C4wrC2 C4:Dic3:C4 (C22xS3):C8 (C2xDic3):C8 C24.3Dic3 (C2xC12):C8 (C6xD4):C4 (C6xQ8):C4 Dic3.5M4(2) Dic3.M4(2) C24:C4:C2 S3xC22:C8 D6:M4(2) D6:C8:C2 C42.285D6 C42.270D6 (C2xC6).40D8 C4:C4.228D6 C4:C4.230D6 C4:C4.231D6 C4:C4.233D6 C42.187D6 C4:C4.236D6 D4xC3:C8 C42.47D6 C12:3M4(2) (C2xC6).D8 C4:D4.S3 C6.Q16:C2 D12:16D4 D12:17D4 Dic6:17D4 (C2xQ8).49D6 (C2xC6).Q16 (C2xQ8).51D6 D12.36D4 D12.37D4 Dic6.37D4 C8xC3:D4 C24:33D4 C24:D4 C24:21D4 C24.6Dic3 (C2xC6):8D8 (C3xD4).31D4 (C3xQ8):13D4 (C2xC6):8Q16 (C6xD4).11C4 (C3xD4):14D4 (C3xD4).32D4 C36.55D4 C12.S4 C12.77D12 C62:7C8 C12.12S4 C60.93D4 C60.212D4 C30.7M4(2) C30.22M4(2)
C12.55D4 is a maximal quotient of
(C2xC12):3C8 C24.3Dic3 (C2xC12):C8 C12.57D8 C12.26Q16 C24.98D4 C24.D4 C24.99D4 C36.55D4 C12.77D12 C62:7C8 C60.93D4 C60.212D4 C30.7M4(2) C30.22M4(2)
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | C3:D4 | C3:C8 | C4.Dic3 |
| kernel | C12.55D4 | C2xC3:C8 | C22xC12 | C2xC12 | C22xC6 | C2xC6 | C22xC4 | C12 | C2xC4 | C2xC4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of C12.55D4 ►in GL3(F73) generated by
| 27 | 0 | 0 |
| 0 | 70 | 0 |
| 0 | 7 | 49 |
| 10 | 0 | 0 |
| 0 | 22 | 7 |
| 0 | 0 | 51 |
| 63 | 0 | 0 |
| 0 | 22 | 7 |
| 0 | 39 | 51 |
G:=sub<GL(3,GF(73))| [27,0,0,0,70,7,0,0,49],[10,0,0,0,22,0,0,7,51],[63,0,0,0,22,39,0,7,51] >;
C12.55D4 in GAP, Magma, Sage, TeX
C_{12}._{55}D_4 % in TeX
G:=Group("C12.55D4"); // GroupNames label
G:=SmallGroup(96,37);
// by ID
G=gap.SmallGroup(96,37);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^3>;
// generators/relations
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