direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.16D6, C62.168C23, Dic3⋊C4⋊7C6, (C4×Dic3)⋊9C6, (C6×Dic3)⋊8C4, (C2×Dic3)⋊3C12, (C2×C12).263D6, C62.58(C2×C4), C6.5(C22×C12), C22.6(S3×C12), C23.21(S3×C6), (Dic3×C12)⋊27C2, Dic3.8(C2×C12), C6.D4.1C6, (C22×C6).102D6, (C6×C12).241C22, (C2×C62).44C22, C6.109(D4⋊2S3), (C22×Dic3).3C6, C32⋊13(C42⋊C2), (C6×Dic3).166C22, C2.7(S3×C2×C12), C6.104(S3×C2×C4), (C2×C6).61(C4×S3), (C2×C4).24(S3×C6), (C2×C6).4(C2×C12), C6.19(C3×C4○D4), C22.12(S3×C2×C6), (C2×C12).50(C2×C6), C3⋊2(C3×C42⋊C2), C2.1(C3×D4⋊2S3), C22⋊C4.3(C3×S3), (C3×C22⋊C4).4C6, (Dic3×C2×C6).10C2, (C3×Dic3⋊C4)⋊26C2, (C3×C22⋊C4).17S3, (C2×C6).23(C22×C6), (C3×C6).76(C22×C4), (C22×C6).18(C2×C6), (C3×C6).125(C4○D4), (C2×C6).301(C22×S3), (C3×Dic3).29(C2×C4), (C2×Dic3).18(C2×C6), (C3×C6.D4).1C2, (C32×C22⋊C4).5C2, SmallGroup(288,648)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.16D6
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, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 322 in 169 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C42⋊C2, C3×Dic3, C3×Dic3, C3×C12, C62, C62, C62, C4×Dic3, Dic3⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Dic3, C6×Dic3, C6×C12, C2×C62, C23.16D6, C3×C42⋊C2, Dic3×C12, C3×Dic3⋊C4, C3×C6.D4, C32×C22⋊C4, Dic3×C2×C6, C3×C23.16D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C4○D4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C42⋊C2, S3×C6, S3×C2×C4, D4⋊2S3, C22×C12, C3×C4○D4, S3×C12, S3×C2×C6, C23.16D6, C3×C42⋊C2, S3×C2×C12, C3×D4⋊2S3, C3×C23.16D6
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 7)(2 27)(3 9)(4 29)(5 11)(6 31)(8 33)(10 35)(12 25)(13 19)(14 48)(15 21)(16 38)(17 23)(18 40)(20 42)(22 44)(24 46)(26 32)(28 34)(30 36)(37 43)(39 45)(41 47)
(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 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)
(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 40 26 18)(2 45 27 23)(3 38 28 16)(4 43 29 21)(5 48 30 14)(6 41 31 19)(7 46 32 24)(8 39 33 17)(9 44 34 22)(10 37 35 15)(11 42 36 20)(12 47 25 13)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,7)(2,27)(3,9)(4,29)(5,11)(6,31)(8,33)(10,35)(12,25)(13,19)(14,48)(15,21)(16,38)(17,23)(18,40)(20,42)(22,44)(24,46)(26,32)(28,34)(30,36)(37,43)(39,45)(41,47), (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), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40), (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), (1,40,26,18)(2,45,27,23)(3,38,28,16)(4,43,29,21)(5,48,30,14)(6,41,31,19)(7,46,32,24)(8,39,33,17)(9,44,34,22)(10,37,35,15)(11,42,36,20)(12,47,25,13)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,7)(2,27)(3,9)(4,29)(5,11)(6,31)(8,33)(10,35)(12,25)(13,19)(14,48)(15,21)(16,38)(17,23)(18,40)(20,42)(22,44)(24,46)(26,32)(28,34)(30,36)(37,43)(39,45)(41,47), (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), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40), (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), (1,40,26,18)(2,45,27,23)(3,38,28,16)(4,43,29,21)(5,48,30,14)(6,41,31,19)(7,46,32,24)(8,39,33,17)(9,44,34,22)(10,37,35,15)(11,42,36,20)(12,47,25,13) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,7),(2,27),(3,9),(4,29),(5,11),(6,31),(8,33),(10,35),(12,25),(13,19),(14,48),(15,21),(16,38),(17,23),(18,40),(20,42),(22,44),(24,46),(26,32),(28,34),(30,36),(37,43),(39,45),(41,47)], [(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)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40)], [(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)], [(1,40,26,18),(2,45,27,23),(3,38,28,16),(4,43,29,21),(5,48,30,14),(6,41,31,19),(7,46,32,24),(8,39,33,17),(9,44,34,22),(10,37,35,15),(11,42,36,20),(12,47,25,13)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6Y | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AB | 12AC | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 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 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | D6 | C4○D4 | C3×S3 | C4×S3 | S3×C6 | S3×C6 | C3×C4○D4 | S3×C12 | D4⋊2S3 | C3×D4⋊2S3 |
| kernel | C3×C23.16D6 | Dic3×C12 | C3×Dic3⋊C4 | C3×C6.D4 | C32×C22⋊C4 | Dic3×C2×C6 | C23.16D6 | C6×Dic3 | C4×Dic3 | Dic3⋊C4 | C6.D4 | C3×C22⋊C4 | C22×Dic3 | C2×Dic3 | C3×C22⋊C4 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C6 | C22 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 1 | 2 | 1 | 4 | 2 | 4 | 4 | 2 | 8 | 8 | 2 | 4 |
Matrix representation of C3×C23.16D6 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 2 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 2 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 8 |
| 0 | 6 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,2,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[2,0,0,0,0,6,0,0,0,0,5,0,0,0,5,8],[0,2,0,0,6,0,0,0,0,0,1,0,0,0,1,12] >;
C3×C23.16D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{16}D_6 % in TeX
G:=Group("C3xC2^3.16D6"); // GroupNames label
G:=SmallGroup(288,648);
// by ID
G=gap.SmallGroup(288,648);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,555,142,9414]);
// 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,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=e^5>;
// generators/relations