direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C8.Q8, C48⋊3C4, C16⋊1C12, C24.9Q8, C12.44SD16, M5(2).1C6, C8.(C3×Q8), C4.Q8.1C6, C8.18(C2×C12), C24.86(C2×C4), C12.55(C4⋊C4), C4.9(C3×SD16), C8.C4.2C6, (C2×C12).282D4, (C2×C6).26SD16, C6.10(C4.Q8), (C3×M5(2)).3C2, C22.5(C3×SD16), (C2×C24).196C22, C4.6(C3×C4⋊C4), C2.3(C3×C4.Q8), (C2×C8).15(C2×C6), (C2×C4).13(C3×D4), (C3×C4.Q8).6C2, (C3×C8.C4).5C2, SmallGroup(192,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.Q8
G = < a,b,c,d | a3=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >
Smallest permutation representation of C3×C8.Q8
►On 48 points
Generators in S48
(1 30 36)(2 31 37)(3 32 38)(4 17 39)(5 18 40)(6 19 41)(7 20 42)(8 21 43)(9 22 44)(10 23 45)(11 24 46)(12 25 47)(13 26 48)(14 27 33)(15 28 34)(16 29 35)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 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)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 23 25 31)(18 26)(19 29 27 21)(20 32)(24 28)(33 43 41 35)(34 46)(37 39 45 47)(38 42)(40 48)
G:=sub<Sym(48)| (1,30,36)(2,31,37)(3,32,38)(4,17,39)(5,18,40)(6,19,41)(7,20,42)(8,21,43)(9,22,44)(10,23,45)(11,24,46)(12,25,47)(13,26,48)(14,27,33)(15,28,34)(16,29,35), (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48), (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), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,23,25,31)(18,26)(19,29,27,21)(20,32)(24,28)(33,43,41,35)(34,46)(37,39,45,47)(38,42)(40,48)>;
G:=Group( (1,30,36)(2,31,37)(3,32,38)(4,17,39)(5,18,40)(6,19,41)(7,20,42)(8,21,43)(9,22,44)(10,23,45)(11,24,46)(12,25,47)(13,26,48)(14,27,33)(15,28,34)(16,29,35), (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48), (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), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,23,25,31)(18,26)(19,29,27,21)(20,32)(24,28)(33,43,41,35)(34,46)(37,39,45,47)(38,42)(40,48) );
G=PermutationGroup([[(1,30,36),(2,31,37),(3,32,38),(4,17,39),(5,18,40),(6,19,41),(7,20,42),(8,21,43),(9,22,44),(10,23,45),(11,24,46),(12,25,47),(13,26,48),(14,27,33),(15,28,34),(16,29,35)], [(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8),(17,27,21,31,25,19,29,23),(18,20,22,24,26,28,30,32),(33,43,37,47,41,35,45,39),(34,36,38,40,42,44,46,48)], [(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)], [(2,4,10,12),(3,7),(5,13),(6,16,14,8),(11,15),(17,23,25,31),(18,26),(19,29,27,21),(20,32),(24,28),(33,43,41,35),(34,46),(37,39,45,47),(38,42),(40,48)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 24I | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | Q8 | D4 | SD16 | SD16 | C3×Q8 | C3×D4 | C3×SD16 | C3×SD16 | C8.Q8 | C3×C8.Q8 |
| kernel | C3×C8.Q8 | C3×C4.Q8 | C3×C8.C4 | C3×M5(2) | C8.Q8 | C48 | C4.Q8 | C8.C4 | M5(2) | C16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C8.Q8 ►in GL4(𝔽97) generated by
| 61 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 57 | 40 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 40 | 57 |
| 0 | 0 | 40 | 40 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 57 | 40 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 40 | 57 |
| 0 | 0 | 57 | 57 |
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[57,57,0,0,40,57,0,0,0,0,40,40,0,0,57,40],[0,0,57,57,0,0,40,57,1,0,0,0,0,1,0,0],[1,0,0,0,0,96,0,0,0,0,40,57,0,0,57,57] >;
C3×C8.Q8 in GAP, Magma, Sage, TeX
C_3\times C_8.Q_8
% in TeX
G:=Group("C3xC8.Q8"); // GroupNames label
G:=SmallGroup(192,171);
// by ID
G=gap.SmallGroup(192,171);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,1683,136,2111,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations
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