metabelian, supersoluble, monomial
Aliases: C6.17D24, C12.72D12, C62.19D4, C4⋊Dic3⋊1S3, (C3×C6).15D8, C12⋊S3⋊6C4, C12.15(C4×S3), C6.6(D4⋊S3), C6.4(D6⋊C4), (C2×C12).77D6, (C2×C6).50D12, (C3×C12).34D4, (C3×C6).9SD16, C6.8(C24⋊C2), C3⋊1(C6.D8), C3⋊1(C2.D24), C2.2(C3⋊D24), C12.30(C3⋊D4), C32⋊6(D4⋊C4), (C6×C12).26C22, C6.1(Q8⋊2S3), C4.1(C6.D6), C4.14(C3⋊D12), C2.1(C32⋊5SD16), C2.5(C6.D12), C22.13(C3⋊D12), (C6×C3⋊C8)⋊4C2, (C2×C3⋊C8)⋊2S3, (C2×C4).53S32, (C3×C4⋊Dic3)⋊2C2, (C3×C12).24(C2×C4), (C2×C12⋊S3).8C2, (C2×C6).31(C3⋊D4), (C3×C6).31(C22⋊C4), SmallGroup(288,212)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.17D24
G = < a,b,c | a6=b24=c2=1, bab-1=cac=a-1, cbc=a3b-1 >
Subgroups: 738 in 127 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3⋊S3, C3×C6, C3⋊C8, C24, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D4⋊C4, C3×Dic3, C3×C12, C2×C3⋊S3, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×D12, C3×C3⋊C8, C6×Dic3, C12⋊S3, C12⋊S3, C6×C12, C22×C3⋊S3, C6.D8, C2.D24, C6×C3⋊C8, C3×C4⋊Dic3, C2×C12⋊S3, C6.17D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, S32, C24⋊C2, D24, D6⋊C4, D4⋊S3, Q8⋊2S3, C6.D6, C3⋊D12, C6.D8, C2.D24, C3⋊D24, C32⋊5SD16, C6.D12, C6.17D24
►On 48 points
(1 46 17 38 9 30)(2 31 10 39 18 47)(3 48 19 40 11 32)(4 33 12 41 20 25)(5 26 21 42 13 34)(6 35 14 43 22 27)(7 28 23 44 15 36)(8 37 16 45 24 29)
(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 15)(2 27)(3 13)(4 25)(5 11)(6 47)(7 9)(8 45)(10 43)(12 41)(14 39)(16 37)(17 23)(18 35)(19 21)(20 33)(22 31)(24 29)(26 40)(28 38)(30 36)(32 34)(42 48)(44 46)
G:=sub<Sym(48)| (1,46,17,38,9,30)(2,31,10,39,18,47)(3,48,19,40,11,32)(4,33,12,41,20,25)(5,26,21,42,13,34)(6,35,14,43,22,27)(7,28,23,44,15,36)(8,37,16,45,24,29), (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), (1,15)(2,27)(3,13)(4,25)(5,11)(6,47)(7,9)(8,45)(10,43)(12,41)(14,39)(16,37)(17,23)(18,35)(19,21)(20,33)(22,31)(24,29)(26,40)(28,38)(30,36)(32,34)(42,48)(44,46)>;
G:=Group( (1,46,17,38,9,30)(2,31,10,39,18,47)(3,48,19,40,11,32)(4,33,12,41,20,25)(5,26,21,42,13,34)(6,35,14,43,22,27)(7,28,23,44,15,36)(8,37,16,45,24,29), (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), (1,15)(2,27)(3,13)(4,25)(5,11)(6,47)(7,9)(8,45)(10,43)(12,41)(14,39)(16,37)(17,23)(18,35)(19,21)(20,33)(22,31)(24,29)(26,40)(28,38)(30,36)(32,34)(42,48)(44,46) );
G=PermutationGroup([[(1,46,17,38,9,30),(2,31,10,39,18,47),(3,48,19,40,11,32),(4,33,12,41,20,25),(5,26,21,42,13,34),(6,35,14,43,22,27),(7,28,23,44,15,36),(8,37,16,45,24,29)], [(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)], [(1,15),(2,27),(3,13),(4,25),(5,11),(6,47),(7,9),(8,45),(10,43),(12,41),(14,39),(16,37),(17,23),(18,35),(19,21),(20,33),(22,31),(24,29),(26,40),(28,38),(30,36),(32,34),(42,48),(44,46)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | D12 | C3⋊D4 | C24⋊C2 | D24 | S32 | D4⋊S3 | Q8⋊2S3 | C6.D6 | C3⋊D12 | C3⋊D12 | C3⋊D24 | C32⋊5SD16 |
| kernel | C6.17D24 | C6×C3⋊C8 | C3×C4⋊Dic3 | C2×C12⋊S3 | C12⋊S3 | C2×C3⋊C8 | C4⋊Dic3 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C12 | C2×C6 | C2×C6 | C6 | C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C6.17D24 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 41 | 48 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 25 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[41,38,0,0,0,0,48,0,0,0,0,0,0,0,27,46,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,25,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C6.17D24 in GAP, Magma, Sage, TeX
C_6._{17}D_{24} % in TeX
G:=Group("C6.17D24"); // GroupNames label
G:=SmallGroup(288,212);
// by ID
G=gap.SmallGroup(288,212);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,204,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^-1>;
// generators/relations