Aliases: C62⋊3C8, (C2×C62).4C4, C3⋊Dic3.54D4, C32⋊6(C22⋊C8), C22⋊(C32⋊2C8), C62.11(C2×C4), (C3×C6).11M4(2), C23.2(C32⋊C4), C2.3(C62⋊C4), C2.3(C62.C4), (C3×C6).26(C2×C8), (C2×C32⋊2C8)⋊3C2, C2.5(C2×C32⋊2C8), (C2×C3⋊Dic3).17C4, C22.14(C2×C32⋊C4), (C3×C6).21(C22⋊C4), (C22×C3⋊Dic3).3C2, (C2×C3⋊Dic3).112C22, SmallGroup(288,435)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C62⋊3C8 |
Generators and relations for C62⋊3C8
G = < a,b,c | a6=b6=c8=1, ab=ba, cac-1=a-1b, cbc-1=a4b >
Subgroups: 408 in 104 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C8, C2×C4, C23, C32, Dic3, C2×C6, C2×C8, C22×C4, C3×C6, C3×C6, C2×Dic3, C22×C6, C22⋊C8, C3⋊Dic3, C3⋊Dic3, C62, C62, C62, C22×Dic3, C32⋊2C8, C2×C3⋊Dic3, C2×C3⋊Dic3, C2×C62, C2×C32⋊2C8, C22×C3⋊Dic3, C62⋊3C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C32⋊C4, C32⋊2C8, C2×C32⋊C4, C2×C32⋊2C8, C62.C4, C62⋊C4, C62⋊3C8
►On 48 points
(1 29 43)(2 40 30 24 44 16)(3 45 31)(4 10 46 18 32 34)(5 25 47)(6 36 26 20 48 12)(7 41 27)(8 14 42 22 28 38)(9 17 33)(11 35 19)(13 21 37)(15 39 23)
(1 23)(2 16 44 24 30 40)(3 17)(4 34 32 18 46 10)(5 19)(6 12 48 20 26 36)(7 21)(8 38 28 22 42 14)(9 31)(11 25)(13 27)(15 29)(33 45)(35 47)(37 41)(39 43)
(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)
G:=sub<Sym(48)| (1,29,43)(2,40,30,24,44,16)(3,45,31)(4,10,46,18,32,34)(5,25,47)(6,36,26,20,48,12)(7,41,27)(8,14,42,22,28,38)(9,17,33)(11,35,19)(13,21,37)(15,39,23), (1,23)(2,16,44,24,30,40)(3,17)(4,34,32,18,46,10)(5,19)(6,12,48,20,26,36)(7,21)(8,38,28,22,42,14)(9,31)(11,25)(13,27)(15,29)(33,45)(35,47)(37,41)(39,43), (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)>;
G:=Group( (1,29,43)(2,40,30,24,44,16)(3,45,31)(4,10,46,18,32,34)(5,25,47)(6,36,26,20,48,12)(7,41,27)(8,14,42,22,28,38)(9,17,33)(11,35,19)(13,21,37)(15,39,23), (1,23)(2,16,44,24,30,40)(3,17)(4,34,32,18,46,10)(5,19)(6,12,48,20,26,36)(7,21)(8,38,28,22,42,14)(9,31)(11,25)(13,27)(15,29)(33,45)(35,47)(37,41)(39,43), (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) );
G=PermutationGroup([[(1,29,43),(2,40,30,24,44,16),(3,45,31),(4,10,46,18,32,34),(5,25,47),(6,36,26,20,48,12),(7,41,27),(8,14,42,22,28,38),(9,17,33),(11,35,19),(13,21,37),(15,39,23)], [(1,23),(2,16,44,24,30,40),(3,17),(4,34,32,18,46,10),(5,19),(6,12,48,20,26,36),(7,21),(8,38,28,22,42,14),(9,31),(11,25),(13,27),(15,29),(33,45),(35,47),(37,41),(39,43)], [(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)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 9 | 9 | 9 | 9 | 18 | 18 | 4 | ··· | 4 | 18 | ··· | 18 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | C32⋊C4 | C32⋊2C8 | C2×C32⋊C4 | C62.C4 | C62⋊C4 |
| kernel | C62⋊3C8 | C2×C32⋊2C8 | C22×C3⋊Dic3 | C2×C3⋊Dic3 | C2×C62 | C62 | C3⋊Dic3 | C3×C6 | C23 | C22 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C62⋊3C8 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 71 | 20 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,71,18,0,0,0,0,20,2,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62⋊3C8 in GAP, Magma, Sage, TeX
C_6^2\rtimes_3C_8
% in TeX
G:=Group("C6^2:3C8"); // GroupNames label
G:=SmallGroup(288,435);
// by ID
G=gap.SmallGroup(288,435);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,100,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;
// generators/relations