direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8.C22, Q16⋊2C6, SD16⋊2C6, C12.64D4, M4(2)⋊2C6, C12.49C23, C24.12C22, C8.(C2×C6), (C2×Q8)⋊6C6, (C3×Q16)⋊6C2, (C6×Q8)⋊11C2, C4○D4.4C6, D4.3(C2×C6), (C2×C6).25D4, C4.15(C3×D4), C2.16(C6×D4), C6.79(C2×D4), Q8.6(C2×C6), (C3×SD16)⋊6C2, C4.6(C22×C6), C22.6(C3×D4), (C3×M4(2))⋊4C2, (C2×C12).70C22, (C3×D4).13C22, (C3×Q8).14C22, (C2×C4).11(C2×C6), (C3×C4○D4).5C2, SmallGroup(96,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.C22
G = < a,b,c,d | a3=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8.C22, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×C8.C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C8.C22, C6×D4, C3×C8.C22
►On 48 points
(1 14 18)(2 15 19)(3 16 20)(4 9 21)(5 10 22)(6 11 23)(7 12 24)(8 13 17)(25 34 47)(26 35 48)(27 36 41)(28 37 42)(29 38 43)(30 39 44)(31 40 45)(32 33 46)
(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)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 39)(35 37)(36 40)(41 45)(42 48)(44 46)
(1 41)(2 46)(3 43)(4 48)(5 45)(6 42)(7 47)(8 44)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)
G:=sub<Sym(48)| (1,14,18)(2,15,19)(3,16,20)(4,9,21)(5,10,22)(6,11,23)(7,12,24)(8,13,17)(25,34,47)(26,35,48)(27,36,41)(28,37,42)(29,38,43)(30,39,44)(31,40,45)(32,33,46), (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)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46), (1,41)(2,46)(3,43)(4,48)(5,45)(6,42)(7,47)(8,44)(9,26)(10,31)(11,28)(12,25)(13,30)(14,27)(15,32)(16,29)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34)>;
G:=Group( (1,14,18)(2,15,19)(3,16,20)(4,9,21)(5,10,22)(6,11,23)(7,12,24)(8,13,17)(25,34,47)(26,35,48)(27,36,41)(28,37,42)(29,38,43)(30,39,44)(31,40,45)(32,33,46), (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)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46), (1,41)(2,46)(3,43)(4,48)(5,45)(6,42)(7,47)(8,44)(9,26)(10,31)(11,28)(12,25)(13,30)(14,27)(15,32)(16,29)(17,39)(18,36)(19,33)(20,38)(21,35)(22,40)(23,37)(24,34) );
G=PermutationGroup([[(1,14,18),(2,15,19),(3,16,20),(4,9,21),(5,10,22),(6,11,23),(7,12,24),(8,13,17),(25,34,47),(26,35,48),(27,36,41),(28,37,42),(29,38,43),(30,39,44),(31,40,45),(32,33,46)], [(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),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(33,39),(35,37),(36,40),(41,45),(42,48),(44,46)], [(1,41),(2,46),(3,43),(4,48),(5,45),(6,42),(7,47),(8,44),(9,26),(10,31),(11,28),(12,25),(13,30),(14,27),(15,32),(16,29),(17,39),(18,36),(19,33),(20,38),(21,35),(22,40),(23,37),(24,34)]])
C3×C8.C22 is a maximal subgroup of
D12.39D4 M4(2).15D6 M4(2).16D6 D12.40D4 D24⋊C22 C24.C23 SD16.D6
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C8.C22 | C3×C8.C22 |
| kernel | C3×C8.C22 | C3×M4(2) | C3×SD16 | C3×Q16 | C6×Q8 | C3×C4○D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C12 | C2×C6 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 |
Matrix representation of C3×C8.C22 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 6 | 2 | 2 |
| 3 | 6 | 3 | 2 |
| 5 | 3 | 4 | 0 |
| 2 | 1 | 5 | 4 |
| 1 | 0 | 6 | 0 |
| 0 | 1 | 2 | 1 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 6 | 3 | 2 |
| 1 | 3 | 3 | 4 |
| 2 | 5 | 4 | 6 |
| 1 | 5 | 1 | 6 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,5,2,6,6,3,1,2,3,4,5,2,2,0,4],[1,0,0,0,0,1,0,0,6,2,6,0,0,1,0,6],[1,1,2,1,6,3,5,5,3,3,4,1,2,4,6,6] >;
C3×C8.C22 in GAP, Magma, Sage, TeX
C_3\times C_8.C_2^2
% in TeX
G:=Group("C3xC8.C2^2"); // GroupNames label
G:=SmallGroup(96,184);
// by ID
G=gap.SmallGroup(96,184);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,295,938,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations