direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.23D6, C62.65D4, C62.200C23, C6.47(C6×D4), (C6×D4).22C6, (C6×D4).27S3, Dic3⋊C4⋊14C6, C6.D4⋊8C6, (C2×C12).241D6, C23.28(S3×C6), (C22×C6).30D6, (C22×Dic3)⋊8C6, (C6×C12).262C22, (C2×C62).55C22, C6.123(D4⋊2S3), (C6×Dic3).137C22, C32⋊23(C22.D4), (Dic3×C2×C6)⋊9C2, (D4×C3×C6).16C2, (C2×C6).8(C3×D4), (C2×C4).16(S3×C6), (C2×D4).5(C3×S3), C6.29(C3×C4○D4), C2.11(C6×C3⋊D4), C22.57(S3×C2×C6), (C2×C12).71(C2×C6), (C3×C6).257(C2×D4), C6.148(C2×C3⋊D4), C22.4(C3×C3⋊D4), (C3×Dic3⋊C4)⋊36C2, C2.15(C3×D4⋊2S3), (C2×C6).45(C3⋊D4), (C22×C6).29(C2×C6), (C2×C6).55(C22×C6), C3⋊5(C3×C22.D4), (C3×C6).137(C4○D4), (C3×C6.D4)⋊24C2, (C2×C6).333(C22×S3), (C2×Dic3).36(C2×C6), SmallGroup(288,706)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.23D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=e6=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 394 in 183 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22.D4, C3×Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×D4, C6×D4, C6×Dic3, C6×Dic3, C6×C12, D4×C32, C2×C62, C23.23D6, C3×C22.D4, C3×Dic3⋊C4, C3×C6.D4, C3×C6.D4, Dic3×C2×C6, D4×C3×C6, C3×C23.23D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C22.D4, S3×C6, D4⋊2S3, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, C23.23D6, C3×C22.D4, C3×D4⋊2S3, C6×C3⋊D4, C3×C23.23D6
►On 48 points
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 13)(2 17)(3 15)(4 14)(5 18)(6 16)(7 20)(8 24)(9 22)(10 23)(11 21)(12 19)(25 38)(26 47)(27 40)(28 43)(29 42)(30 45)(31 41)(32 44)(33 37)(34 46)(35 39)(36 48)
(1 8)(2 9)(3 7)(4 12)(5 10)(6 11)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)
(1 6)(2 4)(3 5)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)
(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 36 11 30)(2 32 12 26)(3 34 10 28)(4 29 9 35)(5 25 7 31)(6 27 8 33)(13 40 21 37)(14 44 22 47)(15 38 23 41)(16 48 24 45)(17 42 19 39)(18 46 20 43)
G:=sub<Sym(48)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (1,13)(2,17)(3,15)(4,14)(5,18)(6,16)(7,20)(8,24)(9,22)(10,23)(11,21)(12,19)(25,38)(26,47)(27,40)(28,43)(29,42)(30,45)(31,41)(32,44)(33,37)(34,46)(35,39)(36,48), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47), (1,6)(2,4)(3,5)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (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), (1,36,11,30)(2,32,12,26)(3,34,10,28)(4,29,9,35)(5,25,7,31)(6,27,8,33)(13,40,21,37)(14,44,22,47)(15,38,23,41)(16,48,24,45)(17,42,19,39)(18,46,20,43)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (1,13)(2,17)(3,15)(4,14)(5,18)(6,16)(7,20)(8,24)(9,22)(10,23)(11,21)(12,19)(25,38)(26,47)(27,40)(28,43)(29,42)(30,45)(31,41)(32,44)(33,37)(34,46)(35,39)(36,48), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47), (1,6)(2,4)(3,5)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (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), (1,36,11,30)(2,32,12,26)(3,34,10,28)(4,29,9,35)(5,25,7,31)(6,27,8,33)(13,40,21,37)(14,44,22,47)(15,38,23,41)(16,48,24,45)(17,42,19,39)(18,46,20,43) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48)], [(1,13),(2,17),(3,15),(4,14),(5,18),(6,16),(7,20),(8,24),(9,22),(10,23),(11,21),(12,19),(25,38),(26,47),(27,40),(28,43),(29,42),(30,45),(31,41),(32,44),(33,37),(34,46),(35,39),(36,48)], [(1,8),(2,9),(3,7),(4,12),(5,10),(6,11),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,48),(38,43),(39,44),(40,45),(41,46),(42,47)], [(1,6),(2,4),(3,5),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44)], [(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)], [(1,36,11,30),(2,32,12,26),(3,34,10,28),(4,29,9,35),(5,25,7,31),(6,27,8,33),(13,40,21,37),(14,44,22,47),(15,38,23,41),(16,48,24,45),(17,42,19,39),(18,46,20,43)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6AG | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | 12R | 12S | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C4○D4 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C3×C4○D4 | C3×C3⋊D4 | D4⋊2S3 | C3×D4⋊2S3 |
| kernel | C3×C23.23D6 | C3×Dic3⋊C4 | C3×C6.D4 | Dic3×C2×C6 | D4×C3×C6 | C23.23D6 | Dic3⋊C4 | C6.D4 | C22×Dic3 | C6×D4 | C6×D4 | C62 | C2×C12 | C22×C6 | C3×C6 | C2×D4 | C2×C6 | C2×C6 | C2×C4 | C23 | C6 | C22 | C6 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 4 | 6 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 8 | 8 | 2 | 4 |
Matrix representation of C3×C23.23D6 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 10 | 12 |
| 0 | 3 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 11 | 8 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,10,0,0,0,0,1,10,0,0,0,12],[0,4,0,0,3,0,0,0,0,0,5,11,0,0,0,8] >;
C3×C23.23D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{23}D_6 % in TeX
G:=Group("C3xC2^3.23D6"); // GroupNames label
G:=SmallGroup(288,706);
// by ID
G=gap.SmallGroup(288,706);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,555,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^6=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations