direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×D8⋊2C4, D8⋊2C12, Q16⋊2C12, C24.102D4, M5(2)⋊5C6, C12.43SD16, (C3×D8)⋊8C4, C4.Q8⋊1C6, C8.1(C2×C12), (C3×Q16)⋊8C4, C4○D8.2C6, (C2×C6).25D8, C8.22(C3×D4), C24.36(C2×C4), C4.8(C3×SD16), C22.3(C3×D8), (C2×C12).279D4, (C3×M5(2))⋊13C2, C6.40(D4⋊C4), C12.72(C22⋊C4), (C2×C24).193C22, (C2×C8).12(C2×C6), (C3×C4○D8).7C2, (C3×C4.Q8)⋊10C2, (C2×C4).10(C3×D4), C4.4(C3×C22⋊C4), C2.9(C3×D4⋊C4), SmallGroup(192,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8⋊2C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >
Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C6, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C4.Q8, M5(2), C4○D8, C48, C3×C4⋊C4, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, D8⋊2C4, C3×C4.Q8, C3×M5(2), C3×C4○D8, C3×D8⋊2C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, C2×C12, C3×D4, D4⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, D8⋊2C4, C3×D4⋊C4, C3×D8⋊2C4
►On 48 points
(1 23 15)(2 24 16)(3 17 9)(4 18 10)(5 19 11)(6 20 12)(7 21 13)(8 22 14)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 48)(24 47)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 26 29 30)(27 32 31 28)(33 34 37 38)(35 40 39 36)(41 42 45 46)(43 48 47 44)
G:=sub<Sym(48)| (1,23,15)(2,24,16)(3,17,9)(4,18,10)(5,19,11)(6,20,12)(7,21,13)(8,22,14)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,48)(24,47), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,26,29,30)(27,32,31,28)(33,34,37,38)(35,40,39,36)(41,42,45,46)(43,48,47,44)>;
G:=Group( (1,23,15)(2,24,16)(3,17,9)(4,18,10)(5,19,11)(6,20,12)(7,21,13)(8,22,14)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,48)(24,47), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,26,29,30)(27,32,31,28)(33,34,37,38)(35,40,39,36)(41,42,45,46)(43,48,47,44) );
G=PermutationGroup([[(1,23,15),(2,24,16),(3,17,9),(4,18,10),(5,19,11),(6,20,12),(7,21,13),(8,22,14),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,48),(24,47)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22),(25,26,29,30),(27,32,31,28),(33,34,37,38),(35,40,39,36),(41,42,45,46),(43,48,47,44)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 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 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | SD16 | D8 | C3×D4 | C3×D4 | C3×SD16 | C3×D8 | D8⋊2C4 | C3×D8⋊2C4 |
| kernel | C3×D8⋊2C4 | C3×C4.Q8 | C3×M5(2) | C3×C4○D8 | D8⋊2C4 | C3×D8 | C3×Q16 | C4.Q8 | M5(2) | C4○D8 | D8 | Q16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×D8⋊2C4 ►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 | 40 |
| 0 | 0 | 57 | 40 |
| 0 | 0 | 40 | 40 |
| 0 | 0 | 57 | 40 |
| 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],[57,57,0,0,40,57,0,0,0,0,40,57,0,0,40,40],[0,0,57,57,0,0,40,57,40,57,0,0,40,40,0,0],[1,0,0,0,0,96,0,0,0,0,40,57,0,0,57,57] >;
C3×D8⋊2C4 in GAP, Magma, Sage, TeX
C_3\times D_8\rtimes_2C_4
% in TeX
G:=Group("C3xD8:2C4"); // GroupNames label
G:=SmallGroup(192,166);
// by ID
G=gap.SmallGroup(192,166);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,2194,136,2111,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations