direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.8D6, C62.170C23, C4⋊Dic3⋊3C6, Dic3⋊C4⋊8C6, C23.8(S3×C6), (C4×Dic3)⋊10C6, (C2×C12).264D6, (C22×C6).26D6, (Dic3×C12)⋊28C2, C6.D4.3C6, C6.116(C4○D12), (C6×C12).189C22, (C2×C62).46C22, C6.111(D4⋊2S3), C32⋊10(C42⋊2C2), (C6×Dic3).92C22, C6.7(C3×C4○D4), (C2×C4).25(S3×C6), (C2×C12).2(C2×C6), C2.9(C3×C4○D12), C22.40(S3×C2×C6), (C3×C4⋊Dic3)⋊27C2, C3⋊2(C3×C42⋊2C2), C2.7(C3×D4⋊2S3), (C3×C22⋊C4).3C6, C22⋊C4.2(C3×S3), (C3×Dic3⋊C4)⋊27C2, (C3×C22⋊C4).16S3, (C22×C6).20(C2×C6), (C2×C6).25(C22×C6), (C3×C6).127(C4○D4), (C2×C6).303(C22×S3), (C2×Dic3).19(C2×C6), (C3×C6.D4).8C2, (C32×C22⋊C4).3C2, SmallGroup(288,650)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.8D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 290 in 137 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C42⋊2C2, C3×Dic3, C3×C12, C62, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C6×Dic3, C6×C12, C2×C62, C23.8D6, C3×C42⋊2C2, Dic3×C12, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×C6.D4, C32×C22⋊C4, C3×C23.8D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, C42⋊2C2, S3×C6, C4○D12, D4⋊2S3, C3×C4○D4, S3×C2×C6, C23.8D6, C3×C42⋊2C2, C3×C4○D12, C3×D4⋊2S3, C3×C23.8D6
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(2 36)(4 26)(6 28)(8 30)(10 32)(12 34)(13 39)(14 20)(15 41)(16 22)(17 43)(18 24)(19 45)(21 47)(23 37)(38 44)(40 46)(42 48)
(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 35)(2 36)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 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 18 29 44)(2 23 30 37)(3 16 31 42)(4 21 32 47)(5 14 33 40)(6 19 34 45)(7 24 35 38)(8 17 36 43)(9 22 25 48)(10 15 26 41)(11 20 27 46)(12 13 28 39)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (2,36)(4,26)(6,28)(8,30)(10,32)(12,34)(13,39)(14,20)(15,41)(16,22)(17,43)(18,24)(19,45)(21,47)(23,37)(38,44)(40,46)(42,48), (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), (1,35)(2,36)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (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), (1,18,29,44)(2,23,30,37)(3,16,31,42)(4,21,32,47)(5,14,33,40)(6,19,34,45)(7,24,35,38)(8,17,36,43)(9,22,25,48)(10,15,26,41)(11,20,27,46)(12,13,28,39)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (2,36)(4,26)(6,28)(8,30)(10,32)(12,34)(13,39)(14,20)(15,41)(16,22)(17,43)(18,24)(19,45)(21,47)(23,37)(38,44)(40,46)(42,48), (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), (1,35)(2,36)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (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), (1,18,29,44)(2,23,30,37)(3,16,31,42)(4,21,32,47)(5,14,33,40)(6,19,34,45)(7,24,35,38)(8,17,36,43)(9,22,25,48)(10,15,26,41)(11,20,27,46)(12,13,28,39) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(2,36),(4,26),(6,28),(8,30),(10,32),(12,34),(13,39),(14,20),(15,41),(16,22),(17,43),(18,24),(19,45),(21,47),(23,37),(38,44),(40,46),(42,48)], [(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)], [(1,35),(2,36),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44)], [(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)], [(1,18,29,44),(2,23,30,37),(3,16,31,42),(4,21,32,47),(5,14,33,40),(6,19,34,45),(7,24,35,38),(8,17,36,43),(9,22,25,48),(10,15,26,41),(11,20,27,46),(12,13,28,39)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 12S | ··· | 12Z | 12AA | 12AB | 12AC | 12AD |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 | D4⋊2S3 | C3×D4⋊2S3 |
| kernel | C3×C23.8D6 | Dic3×C12 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C3×C6.D4 | C32×C22⋊C4 | C23.8D6 | C4×Dic3 | Dic3⋊C4 | C4⋊Dic3 | C6.D4 | C3×C22⋊C4 | C3×C22⋊C4 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | C2×C4 | C23 | C6 | C6 | C2 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 6 | 2 | 4 | 2 | 4 | 12 | 8 | 2 | 4 |
Matrix representation of C3×C23.8D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 7 | 11 |
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 11 | 10 |
| 0 | 0 | 6 | 2 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,1,3,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,7,7,0,0,0,11],[0,5,0,0,8,0,0,0,0,0,11,6,0,0,10,2] >;
C3×C23.8D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._8D_6
% in TeX
G:=Group("C3xC2^3.8D6"); // GroupNames label
G:=SmallGroup(288,650);
// by ID
G=gap.SmallGroup(288,650);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1598,555,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=c,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,f*b*f^-1=b*c=c*b,e*b*e^-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=e^5>;
// generators/relations