direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8.C8, C24.7C8, C8.1C24, C24.22Q8, C24.109D4, C42.8C12, M5(2).4C6, C8.6(C3×Q8), (C4×C8).10C6, C4.8(C2×C24), (C2×C8).9C12, C8.29(C3×D4), C6.15(C4⋊C8), (C4×C24).21C2, (C2×C24).20C4, C12.48(C2×C8), (C4×C12).23C4, C12.68(C4⋊C4), (C3×M5(2)).8C2, (C2×C6).17M4(2), (C2×C24).443C22, C22.5(C3×M4(2)), C2.5(C3×C4⋊C8), C4.19(C3×C4⋊C4), (C2×C8).97(C2×C6), (C2×C4).68(C2×C12), (C2×C12).329(C2×C4), SmallGroup(192,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.C8
G = < a,b,c | a3=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C3×C8.C8
►On 48 points
(1 29 48)(2 30 33)(3 31 34)(4 32 35)(5 17 36)(6 18 37)(7 19 38)(8 20 39)(9 21 40)(10 22 41)(11 23 42)(12 24 43)(13 25 44)(14 26 45)(15 27 46)(16 28 47)
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)(17 23 29 19 25 31 21 27)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 40 46 36 42 48 38 44)
(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)| (1,29,48)(2,30,33)(3,31,34)(4,32,35)(5,17,36)(6,18,37)(7,19,38)(8,20,39)(9,21,40)(10,22,41)(11,23,42)(12,24,43)(13,25,44)(14,26,45)(15,27,46)(16,28,47), (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,40,46,36,42,48,38,44), (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( (1,29,48)(2,30,33)(3,31,34)(4,32,35)(5,17,36)(6,18,37)(7,19,38)(8,20,39)(9,21,40)(10,22,41)(11,23,42)(12,24,43)(13,25,44)(14,26,45)(15,27,46)(16,28,47), (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,40,46,36,42,48,38,44), (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([[(1,29,48),(2,30,33),(3,31,34),(4,32,35),(5,17,36),(6,18,37),(7,19,38),(8,20,39),(9,21,40),(10,22,41),(11,23,42),(12,24,43),(13,25,44),(14,26,45),(15,27,46),(16,28,47)], [(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16),(17,23,29,19,25,31,21,27),(18,20,22,24,26,28,30,32),(33,35,37,39,41,43,45,47),(34,40,46,36,42,48,38,44)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24T | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | C3×M4(2) | C8.C8 | C3×C8.C8 |
| kernel | C3×C8.C8 | C4×C24 | C3×M5(2) | C8.C8 | C4×C12 | C2×C24 | C4×C8 | M5(2) | C24 | C42 | C2×C8 | C8 | C24 | C24 | C2×C6 | C8 | C8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 16 |
Matrix representation of C3×C8.C8 ►in GL3(𝔽97) generated by
| 61 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 50 | 0 |
| 0 | 0 | 64 |
| 47 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 64 | 0 |
G:=sub<GL(3,GF(97))| [61,0,0,0,1,0,0,0,1],[1,0,0,0,50,0,0,0,64],[47,0,0,0,0,64,0,1,0] >;
C3×C8.C8 in GAP, Magma, Sage, TeX
C_3\times C_8.C_8
% in TeX
G:=Group("C3xC8.C8"); // GroupNames label
G:=SmallGroup(192,170);
// by ID
G=gap.SmallGroup(192,170);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,3027,248,102,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=1,c^8=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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