metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48⋊1C4, C24.1Q8, C16⋊1Dic3, C8.3Dic6, C12.6SD16, M5(2).1S3, C3⋊3(C8.Q8), (C2×C4).8D12, (C2×C8).45D6, C24.72(C2×C4), (C2×C12).98D4, C12.26(C4⋊C4), C8⋊Dic3.1C2, (C2×C6).7SD16, C6.6(C4.Q8), C8.18(C2×Dic3), C2.3(C8⋊Dic3), C4.11(C24⋊C2), C24.C4.5C2, (C2×C24).49C22, C4.11(C4⋊Dic3), (C3×M5(2)).1C2, C22.5(C24⋊C2), SmallGroup(192,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.Q8
G = < a,b,c | a24=1, b4=a18, c2=a15b2, bab-1=a13, cac-1=a11, cbc-1=a12b3 >
Smallest permutation representation of C24.Q8
►On 48 points
Generators in S48
(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 31 10 28 19 25 4 46 13 43 22 40 7 37 16 34)(2 44 11 41 20 38 5 35 14 32 23 29 8 26 17 47)(3 33 12 30 21 27 6 48 15 45 24 42 9 39 18 36)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 34 37 46)(26 45 38 33)(27 32 39 44)(28 43 40 31)(29 30 41 42)(35 48 47 36)
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), (1,31,10,28,19,25,4,46,13,43,22,40,7,37,16,34)(2,44,11,41,20,38,5,35,14,32,23,29,8,26,17,47)(3,33,12,30,21,27,6,48,15,45,24,42,9,39,18,36), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,34,37,46)(26,45,38,33)(27,32,39,44)(28,43,40,31)(29,30,41,42)(35,48,47,36)>;
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), (1,31,10,28,19,25,4,46,13,43,22,40,7,37,16,34)(2,44,11,41,20,38,5,35,14,32,23,29,8,26,17,47)(3,33,12,30,21,27,6,48,15,45,24,42,9,39,18,36), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,34,37,46)(26,45,38,33)(27,32,39,44)(28,43,40,31)(29,30,41,42)(35,48,47,36) );
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)], [(1,31,10,28,19,25,4,46,13,43,22,40,7,37,16,34),(2,44,11,41,20,38,5,35,14,32,23,29,8,26,17,47),(3,33,12,30,21,27,6,48,15,45,24,42,9,39,18,36)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,34,37,46),(26,45,38,33),(27,32,39,44),(28,43,40,31),(29,30,41,42),(35,48,47,36)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 24 | 24 | 2 | 4 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | Dic3 | D6 | SD16 | SD16 | Dic6 | D12 | C24⋊C2 | C24⋊C2 | C8.Q8 | C24.Q8 |
| kernel | C24.Q8 | C8⋊Dic3 | C24.C4 | C3×M5(2) | C48 | M5(2) | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C24.Q8 ►in GL4(𝔽97) generated by
| 8 | 53 | 0 | 0 |
| 44 | 61 | 0 | 0 |
| 23 | 49 | 36 | 44 |
| 13 | 84 | 53 | 89 |
| 0 | 0 | 96 | 1 |
| 13 | 3 | 95 | 96 |
| 36 | 18 | 26 | 68 |
| 72 | 26 | 26 | 68 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 24 | 53 | 89 |
| 10 | 86 | 36 | 44 |
G:=sub<GL(4,GF(97))| [8,44,23,13,53,61,49,84,0,0,36,53,0,0,44,89],[0,13,36,72,0,3,18,26,96,95,26,26,1,96,68,68],[0,1,0,10,1,0,24,86,0,0,53,36,0,0,89,44] >;
C24.Q8 in GAP, Magma, Sage, TeX
C_{24}.Q_8 % in TeX
G:=Group("C24.Q8"); // GroupNames label
G:=SmallGroup(192,72);
// by ID
G=gap.SmallGroup(192,72);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,387,675,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=a^18,c^2=a^15*b^2,b*a*b^-1=a^13,c*a*c^-1=a^11,c*b*c^-1=a^12*b^3>;
// generators/relations
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