metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.26D6, (C2×C12)⋊6C4, (C2×C4)⋊4Dic3, C12.37(C2×C4), C4○3(C4⋊Dic3), C4⋊Dic3⋊17C2, (C2×C4).102D6, (C22×C4).9S3, C3⋊4(C42⋊C2), (C4×Dic3)⋊15C2, C4○(C6.D4), C2.4(C4○D12), C6.16(C4○D4), (C2×C6).44C23, C6.24(C22×C4), C4.15(C2×Dic3), (C2×C12).93C22, (C22×C12).10C2, C6.D4.5C2, C22.5(C2×Dic3), C2.5(C22×Dic3), (C22×C6).36C22, C22.22(C22×S3), (C2×Dic3).35C22, (C2×C6).35(C2×C4), (C2×C4)○(C4⋊Dic3), SmallGroup(96,133)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.26D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 130 in 76 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Dic3, C2×C12, C2×C12, C22×C6, C42⋊C2, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C23.26D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C2×Dic3, C22×S3, C42⋊C2, C4○D12, C22×Dic3, C23.26D6
►On 48 points
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 31 19 38)(2 36 20 43)(3 29 21 48)(4 34 22 41)(5 27 23 46)(6 32 24 39)(7 25 13 44)(8 30 14 37)(9 35 15 42)(10 28 16 47)(11 33 17 40)(12 26 18 45)
G:=sub<Sym(48)| (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,31,19,38)(2,36,20,43)(3,29,21,48)(4,34,22,41)(5,27,23,46)(6,32,24,39)(7,25,13,44)(8,30,14,37)(9,35,15,42)(10,28,16,47)(11,33,17,40)(12,26,18,45)>;
G:=Group( (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,31,19,38)(2,36,20,43)(3,29,21,48)(4,34,22,41)(5,27,23,46)(6,32,24,39)(7,25,13,44)(8,30,14,37)(9,35,15,42)(10,28,16,47)(11,33,17,40)(12,26,18,45) );
G=PermutationGroup([[(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,31,19,38),(2,36,20,43),(3,29,21,48),(4,34,22,41),(5,27,23,46),(6,32,24,39),(7,25,13,44),(8,30,14,37),(9,35,15,42),(10,28,16,47),(11,33,17,40),(12,26,18,45)]])
C23.26D6 is a maximal subgroup of
C12.8C42 C42⋊3Dic3 C12.3C42 (C2×C24)⋊C4 C12.20C42 M4(2)⋊4Dic3 C24⋊C4⋊C2 C23.15D12 C23.18D12 C4⋊C4.225D6 C4⋊C4.234D6 (C2×D12)⋊13C4 C6.Q16⋊C2 (C2×Q8).51D6 C12.12C42 Dic3⋊C8⋊C2 C23.27D12 C23.28D12 C12.88(C2×Q8) C23.51D12 C23.52D12 C12.7C42 C23.53D12 M4(2)⋊24D6 (C6×D4)⋊6C4 (C6×Q8)⋊6C4 C4○D4⋊4Dic3 (C6×D4)⋊9C4 (C6×D4)⋊10C4 C42.274D6 C4×C4○D12 C24.42D6 C6.72+ 1+4 C6.82+ 1+4 C6.52- 1+4 C42.87D6 C42.88D6 C42.90D6 S3×C42⋊C2 C42⋊9D6 C42.102D6 C42.105D6 C42.106D6 C42.229D6 C42.117D6 C42.119D6 C6.712- 1+4 C6.432+ 1+4 C6.452+ 1+4 C6.472+ 1+4 C6.152- 1+4 C6.212- 1+4 C6.232- 1+4 C6.242- 1+4 C6.802- 1+4 C6.1222+ 1+4 C24.83D6 C24.49D6 C24.52D6 C6.422- 1+4 C6.442- 1+4 C6.1052- 1+4 Dic3×C4○D4 C6.1442+ 1+4 (C2×D4)⋊43D6 C6.1462+ 1+4 C23.26D18 C62.11C23 C62.25C23 C62.97C23 C62.247C23 (D5×C12)⋊C4 (C4×D5)⋊Dic3 (C6×Dic5)⋊7C4 C23.26D30 (C2×C12)⋊6F5
C23.26D6 is a maximal quotient of
C42.285D6 C42.270D6 C42⋊6Dic3 C4×C4⋊Dic3 C42⋊11Dic3 C42⋊7Dic3 C24.19D6 C4⋊C4⋊5Dic3 C42.187D6 C4×C6.D4 C24.74D6 C24.75D6 C23.26D18 C62.11C23 C62.25C23 C62.97C23 C62.247C23 (D5×C12)⋊C4 (C4×D5)⋊Dic3 (C6×Dic5)⋊7C4 C23.26D30 (C2×C12)⋊6F5
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Dic3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C23.26D6 | C4×Dic3 | C4⋊Dic3 | C6.D4 | C22×C12 | C2×C12 | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 4 | 2 | 1 | 4 | 8 |
Matrix representation of C23.26D6 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 11 |
| 8 | 0 | 0 |
| 0 | 0 | 11 |
| 0 | 6 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,12],[12,0,0,0,12,0,0,0,12],[1,0,0,0,12,0,0,0,12],[1,0,0,0,7,0,0,0,11],[8,0,0,0,0,6,0,11,0] >;
C23.26D6 in GAP, Magma, Sage, TeX
C_2^3._{26}D_6 % in TeX
G:=Group("C2^3.26D6"); // GroupNames label
G:=SmallGroup(96,133);
// by ID
G=gap.SmallGroup(96,133);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,362,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations