metabelian, supersoluble, monomial
Aliases: C62⋊6D4, C62.120C23, (C2×C6)⋊4D12, C23.28S32, D6⋊Dic3⋊8C2, C6.172(S3×D4), C6.87(C2×D12), C3⋊4(C12⋊7D4), (C3×Dic3)⋊12D4, (C22×C6).78D6, C6.70(C4○D12), Dic3⋊5(C3⋊D4), C32⋊14(C4⋊D4), (C22×S3).26D6, Dic3⋊Dic3⋊37C2, C3⋊4(C23.14D6), C6.D12⋊22C2, C6.57(D4⋊2S3), C22⋊3(C3⋊D12), (C2×Dic3).105D6, (C22×Dic3)⋊10S3, (C2×C62).39C22, C2.30(D6.3D6), (C6×Dic3).147C22, (C2×C3⋊D4)⋊5S3, (C6×C3⋊D4)⋊11C2, (Dic3×C2×C6)⋊11C2, C2.44(S3×C3⋊D4), C6.24(C2×C3⋊D4), (C2×C6)⋊11(C3⋊D4), C22.143(C2×S32), (C3×C6).166(C2×D4), (C2×C32⋊7D4)⋊4C2, (S3×C2×C6).48C22, (C2×C3⋊D12)⋊12C2, (C3×C6).88(C4○D4), C2.25(C2×C3⋊D12), (C2×C6).139(C22×S3), (C22×C3⋊S3).34C22, (C2×C3⋊Dic3).72C22, SmallGroup(288,626)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊6D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 898 in 215 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C3⋊D12, C6×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C12⋊7D4, C23.14D6, D6⋊Dic3, C6.D12, Dic3⋊Dic3, C2×C3⋊D12, Dic3×C2×C6, C6×C3⋊D4, C2×C32⋊7D4, C62⋊6D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, S32, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C3⋊D12, C2×S32, C12⋊7D4, C23.14D6, D6.3D6, C2×C3⋊D12, S3×C3⋊D4, C62⋊6D4
►On 48 points
(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 19 5 23 3 21)(2 20 6 24 4 22)(7 29 11 27 9 25)(8 30 12 28 10 26)(13 31 15 33 17 35)(14 32 16 34 18 36)(37 46 39 48 41 44)(38 47 40 43 42 45)
(1 36 7 38)(2 35 8 37)(3 34 9 42)(4 33 10 41)(5 32 11 40)(6 31 12 39)(13 30 46 20)(14 29 47 19)(15 28 48 24)(16 27 43 23)(17 26 44 22)(18 25 45 21)
(1 7)(2 26)(3 11)(4 30)(5 9)(6 28)(8 22)(10 20)(12 24)(13 33)(14 18)(15 31)(17 35)(19 25)(21 29)(23 27)(32 34)(37 44)(39 48)(40 42)(41 46)(45 47)
G:=sub<Sym(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), (1,19,5,23,3,21)(2,20,6,24,4,22)(7,29,11,27,9,25)(8,30,12,28,10,26)(13,31,15,33,17,35)(14,32,16,34,18,36)(37,46,39,48,41,44)(38,47,40,43,42,45), (1,36,7,38)(2,35,8,37)(3,34,9,42)(4,33,10,41)(5,32,11,40)(6,31,12,39)(13,30,46,20)(14,29,47,19)(15,28,48,24)(16,27,43,23)(17,26,44,22)(18,25,45,21), (1,7)(2,26)(3,11)(4,30)(5,9)(6,28)(8,22)(10,20)(12,24)(13,33)(14,18)(15,31)(17,35)(19,25)(21,29)(23,27)(32,34)(37,44)(39,48)(40,42)(41,46)(45,47)>;
G:=Group( (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), (1,19,5,23,3,21)(2,20,6,24,4,22)(7,29,11,27,9,25)(8,30,12,28,10,26)(13,31,15,33,17,35)(14,32,16,34,18,36)(37,46,39,48,41,44)(38,47,40,43,42,45), (1,36,7,38)(2,35,8,37)(3,34,9,42)(4,33,10,41)(5,32,11,40)(6,31,12,39)(13,30,46,20)(14,29,47,19)(15,28,48,24)(16,27,43,23)(17,26,44,22)(18,25,45,21), (1,7)(2,26)(3,11)(4,30)(5,9)(6,28)(8,22)(10,20)(12,24)(13,33)(14,18)(15,31)(17,35)(19,25)(21,29)(23,27)(32,34)(37,44)(39,48)(40,42)(41,46)(45,47) );
G=PermutationGroup([[(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)], [(1,19,5,23,3,21),(2,20,6,24,4,22),(7,29,11,27,9,25),(8,30,12,28,10,26),(13,31,15,33,17,35),(14,32,16,34,18,36),(37,46,39,48,41,44),(38,47,40,43,42,45)], [(1,36,7,38),(2,35,8,37),(3,34,9,42),(4,33,10,41),(5,32,11,40),(6,31,12,39),(13,30,46,20),(14,29,47,19),(15,28,48,24),(16,27,43,23),(17,26,44,22),(18,25,45,21)], [(1,7),(2,26),(3,11),(4,30),(5,9),(6,28),(8,22),(10,20),(12,24),(13,33),(14,18),(15,31),(17,35),(19,25),(21,29),(23,27),(32,34),(37,44),(39,48),(40,42),(41,46),(45,47)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6J | 6K | ··· | 6S | 6T | 6U | 12A | ··· | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 36 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 6 | ··· | 6 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C3⋊D4 | C4○D12 | S32 | S3×D4 | D4⋊2S3 | C3⋊D12 | C2×S32 | D6.3D6 | S3×C3⋊D4 |
| kernel | C62⋊6D4 | D6⋊Dic3 | C6.D12 | Dic3⋊Dic3 | C2×C3⋊D12 | Dic3×C2×C6 | C6×C3⋊D4 | C2×C32⋊7D4 | C22×Dic3 | C2×C3⋊D4 | C3×Dic3 | C62 | C2×Dic3 | C22×S3 | C22×C6 | C3×C6 | Dic3 | C2×C6 | C2×C6 | C6 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C62⋊6D4 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,0,0,0,3,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,9,0,0,0,0,0,1] >;
C62⋊6D4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_6D_4
% in TeX
G:=Group("C6^2:6D4"); // GroupNames label
G:=SmallGroup(288,626);
// by ID
G=gap.SmallGroup(288,626);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations