direct product, metabelian, supersoluble, monomial
Aliases: C3×D12.C4, D12.C12, C24.66D6, Dic6.C12, (S3×C8)⋊8C6, C3⋊D4.C12, C8⋊S3⋊6C6, C8.12(S3×C6), C4.5(S3×C12), (S3×C24)⋊17C2, C12.53(C4×S3), C24.15(C2×C6), C4○D12.3C6, (C3×D12).3C4, D6.2(C2×C12), C12.13(C2×C12), (C2×C12).322D6, C32⋊11(C8○D4), M4(2)⋊5(C3×S3), (C3×M4(2))⋊6C6, C62.60(C2×C4), C22.1(S3×C12), (C3×Dic6).3C4, (C3×M4(2))⋊11S3, C12.39(C22×C6), (C3×C24).48C22, C6.16(C22×C12), Dic3.4(C2×C12), (S3×C12).60C22, (C3×C12).171C23, (C6×C12).114C22, C12.227(C22×S3), (C32×M4(2))⋊8C2, (C2×C3⋊C8)⋊3C6, (C6×C3⋊C8)⋊24C2, C3⋊2(C3×C8○D4), C4.39(S3×C2×C6), C3⋊C8.12(C2×C6), C2.17(S3×C2×C12), C6.115(S3×C2×C4), (C2×C6).6(C2×C12), (C2×C6).24(C4×S3), (C2×C4).46(S3×C6), (C3×C3⋊D4).3C4, (C3×C8⋊S3)⋊14C2, (C4×S3).17(C2×C6), (S3×C6).14(C2×C4), (C3×C12).68(C2×C4), (C2×C12).25(C2×C6), (C3×C4○D12).9C2, (C3×C3⋊C8).43C22, (C3×C6).87(C22×C4), (C3×Dic3).20(C2×C4), SmallGroup(288,678)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D12.C4
G = < a,b,c,d | a3=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >
Subgroups: 250 in 134 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8○D4, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C24, C3×M4(2), C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D12.C4, C3×C8○D4, S3×C24, C3×C8⋊S3, C6×C3⋊C8, C32×M4(2), C3×C4○D12, C3×D12.C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C8○D4, S3×C6, S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, D12.C4, C3×C8○D4, S3×C2×C12, C3×D12.C4
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(40 44 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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)
(1 13 10 16 7 19 4 22)(2 20 11 23 8 14 5 17)(3 15 12 18 9 21 6 24)(25 37 28 46 31 43 34 40)(26 44 29 41 32 38 35 47)(27 39 30 48 33 45 36 42)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37), (1,13,10,16,7,19,4,22)(2,20,11,23,8,14,5,17)(3,15,12,18,9,21,6,24)(25,37,28,46,31,43,34,40)(26,44,29,41,32,38,35,47)(27,39,30,48,33,45,36,42)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37), (1,13,10,16,7,19,4,22)(2,20,11,23,8,14,5,17)(3,15,12,18,9,21,6,24)(25,37,28,46,31,43,34,40)(26,44,29,41,32,38,35,47)(27,39,30,48,33,45,36,42) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37)], [(1,13,10,16,7,19,4,22),(2,20,11,23,8,14,5,17),(3,15,12,18,9,21,6,24),(25,37,28,46,31,43,34,40),(26,44,29,41,32,38,35,47),(27,39,30,48,33,45,36,42)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | 12N | 12O | 12P | 12Q | 12R | 12S | 24A | ··· | 24H | 24I | ··· | 24P | 24Q | ··· | 24AB | 24AC | 24AD | 24AE | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | S3 | D6 | D6 | C3×S3 | C4×S3 | C4×S3 | C8○D4 | S3×C6 | S3×C6 | S3×C12 | S3×C12 | C3×C8○D4 | D12.C4 | C3×D12.C4 |
| kernel | C3×D12.C4 | S3×C24 | C3×C8⋊S3 | C6×C3⋊C8 | C32×M4(2) | C3×C4○D12 | D12.C4 | C3×Dic6 | C3×D12 | C3×C3⋊D4 | S3×C8 | C8⋊S3 | C2×C3⋊C8 | C3×M4(2) | C4○D12 | Dic6 | D12 | C3⋊D4 | C3×M4(2) | C24 | C2×C12 | M4(2) | C12 | C2×C6 | C32 | C8 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×D12.C4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 65 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 9 | 0 | 0 |
| 65 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 63 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[65,0,0,0,0,9,0,0,0,0,46,0,0,0,0,27],[0,65,0,0,9,0,0,0,0,0,0,63,0,0,51,0],[46,0,0,0,0,46,0,0,0,0,0,46,0,0,1,0] >;
C3×D12.C4 in GAP, Magma, Sage, TeX
C_3\times D_{12}.C_4 % in TeX
G:=Group("C3xD12.C4"); // GroupNames label
G:=SmallGroup(288,678);
// by ID
G=gap.SmallGroup(288,678);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,555,142,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations