direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3xC8oD8, D8:5C12, Q16:5C12, C24.77D4, SD16:3C12, C4wrC2:7C6, (C4xC8):10C6, (C4xC24):21C2, C8oD4:10C6, (C3xD8):11C4, C4oD8.5C6, C4.82(C6xD4), C8.30(C3xD4), C8.C4:8C6, C24.67(C2xC4), C8.11(C2xC12), (C3xQ16):11C4, (C3xSD16):7C4, D4.3(C2xC12), C6.120(C4xD4), C2.18(D4xC12), Q8.8(C2xC12), C12.487(C2xD4), C42.73(C2xC6), C4.15(C22xC12), (C4xC12).358C22, C12.160(C22xC4), (C2xC24).411C22, (C2xC12).910C23, M4(2).11(C2xC6), (C3xM4(2)).45C22, (C3xC4wrC2):15C2, (C3xC8oD4):15C2, (C2xC8).101(C2xC6), C4oD4.15(C2xC6), (C3xC4oD8).10C2, (C3xD4).20(C2xC4), (C3xQ8).21(C2xC4), (C3xC8.C4):17C2, C22.1(C3xC4oD4), (C2xC6).49(C4oD4), (C2xC4).85(C22xC6), (C3xC4oD4).53C22, SmallGroup(192,876)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC8oD8
G = < a,b,c,d | a3=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >
Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C12, C12, C2xC6, C2xC6, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C4oD4, C24, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, C4xC12, C2xC24, C2xC24, C3xM4(2), C3xM4(2), C3xD8, C3xSD16, C3xQ16, C3xC4oD4, C8oD8, C4xC24, C3xC4wrC2, C3xC8.C4, C3xC8oD4, C3xC4oD8, C3xC8oD8
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C23, C12, C2xC6, C22xC4, C2xD4, C4oD4, C2xC12, C3xD4, C22xC6, C4xD4, C22xC12, C6xD4, C3xC4oD4, C8oD8, D4xC12, C3xC8oD8
►On 48 points
(1 27 21)(2 28 22)(3 29 23)(4 30 24)(5 31 17)(6 32 18)(7 25 19)(8 26 20)(9 36 41)(10 37 42)(11 38 43)(12 39 44)(13 40 45)(14 33 46)(15 34 47)(16 35 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 6 3 8 5 2 7 4)(9 12 15 10 13 16 11 14)(17 22 19 24 21 18 23 20)(25 30 27 32 29 26 31 28)(33 36 39 34 37 40 35 38)(41 44 47 42 45 48 43 46)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)
G:=sub<Sym(48)| (1,27,21)(2,28,22)(3,29,23)(4,30,24)(5,31,17)(6,32,18)(7,25,19)(8,26,20)(9,36,41)(10,37,42)(11,38,43)(12,39,44)(13,40,45)(14,33,46)(15,34,47)(16,35,48), (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), (1,6,3,8,5,2,7,4)(9,12,15,10,13,16,11,14)(17,22,19,24,21,18,23,20)(25,30,27,32,29,26,31,28)(33,36,39,34,37,40,35,38)(41,44,47,42,45,48,43,46), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)>;
G:=Group( (1,27,21)(2,28,22)(3,29,23)(4,30,24)(5,31,17)(6,32,18)(7,25,19)(8,26,20)(9,36,41)(10,37,42)(11,38,43)(12,39,44)(13,40,45)(14,33,46)(15,34,47)(16,35,48), (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), (1,6,3,8,5,2,7,4)(9,12,15,10,13,16,11,14)(17,22,19,24,21,18,23,20)(25,30,27,32,29,26,31,28)(33,36,39,34,37,40,35,38)(41,44,47,42,45,48,43,46), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37) );
G=PermutationGroup([[(1,27,21),(2,28,22),(3,29,23),(4,30,24),(5,31,17),(6,32,18),(7,25,19),(8,26,20),(9,36,41),(10,37,42),(11,38,43),(12,39,44),(13,40,45),(14,33,46),(15,34,47),(16,35,48)], [(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)], [(1,6,3,8,5,2,7,4),(9,12,15,10,13,16,11,14),(17,22,19,24,21,18,23,20),(25,30,27,32,29,26,31,28),(33,36,39,34,37,40,35,38),(41,44,47,42,45,48,43,46)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 12Q | 12R | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | C4oD4 | C3xD4 | C3xC4oD4 | C8oD8 | C3xC8oD8 |
| kernel | C3xC8oD8 | C4xC24 | C3xC4wrC2 | C3xC8.C4 | C3xC8oD4 | C3xC4oD8 | C8oD8 | C3xD8 | C3xSD16 | C3xQ16 | C4xC8 | C4wrC2 | C8.C4 | C8oD4 | C4oD8 | D8 | SD16 | Q16 | C24 | C2xC6 | C8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C3xC8oD8 ►in GL3(F73) generated by
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 72 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 51 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[1,0,0,0,10,0,0,0,10],[72,0,0,0,63,0,0,0,51],[1,0,0,0,0,1,0,1,0] >;
C3xC8oD8 in GAP, Magma, Sage, TeX
C_3\times C_8\circ D_8
% in TeX
G:=Group("C3xC8oD8"); // GroupNames label
G:=SmallGroup(192,876);
// by ID
G=gap.SmallGroup(192,876);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,268,4204,2111,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations