direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8.26D4, C42.282C23, M4(2).31C23, C4○D8⋊8C4, (C2×D8)⋊19C4, D8⋊13(C2×C4), C4.49(C4×D4), C4○(C8.26D4), Q16⋊13(C2×C4), (C2×Q16)⋊19C4, SD16⋊9(C2×C4), C8.129(C2×D4), (C2×C8).394D4, C4≀C2⋊17C22, C22.7(C4×D4), (C2×SD16)⋊11C4, C8○D4⋊18C22, C4.31(C23×C4), C8.25(C22×C4), C8⋊C4⋊40C22, (C2×C8).246C23, (C2×C4).211C24, C4○D8.23C22, C4○D4.23C23, D4.13(C22×C4), C4.202(C22×D4), Q8.13(C22×C4), C8.C4⋊11C22, C23.257(C4○D4), (C2×C42).768C22, (C22×C8).253C22, (C22×C4).1518C23, (C2×M4(2)).358C22, C2.71(C2×C4×D4), (C2×C4≀C2)⋊31C2, (C2×C8⋊C4)⋊7C2, (C2×C8○D4)⋊26C2, (C2×C8).98(C2×C4), (C2×C4○D8).17C2, C4○D4.22(C2×C4), (C2×C8.C4)⋊23C2, C22.2(C2×C4○D4), (C2×D4).178(C2×C4), (C2×C4).1578(C2×D4), (C2×Q8).161(C2×C4), (C2×C4).694(C4○D4), (C2×C4).473(C22×C4), (C2×C4○D4).294C22, SmallGroup(128,1686)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 348 in 232 conjugacy classes, 140 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×6], C22 [×3], C22 [×10], C8 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C42, C2×C8 [×2], C2×C8 [×10], C2×C8 [×10], M4(2) [×4], M4(2) [×10], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C8⋊C4 [×4], C4≀C2 [×8], C8.C4 [×4], C2×C42, C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×8], C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C2×C8⋊C4, C2×C4≀C2 [×2], C2×C8.C4, C8.26D4 [×8], C2×C8○D4 [×2], C2×C4○D8, C2×C8.26D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C8.26D4 [×2], C2×C4×D4, C2×C8.26D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >
►On 32 points
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(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)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)(18 22)(20 24)(25 31 29 27)(26 28 30 32)
(1 30 3 32 5 26 7 28)(2 27 4 29 6 31 8 25)(9 23 11 17 13 19 15 21)(10 20 12 22 14 24 16 18)
G:=sub<Sym(32)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (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), (2,6)(4,8)(9,11,13,15)(10,16,14,12)(18,22)(20,24)(25,31,29,27)(26,28,30,32), (1,30,3,32,5,26,7,28)(2,27,4,29,6,31,8,25)(9,23,11,17,13,19,15,21)(10,20,12,22,14,24,16,18)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (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), (2,6)(4,8)(9,11,13,15)(10,16,14,12)(18,22)(20,24)(25,31,29,27)(26,28,30,32), (1,30,3,32,5,26,7,28)(2,27,4,29,6,31,8,25)(9,23,11,17,13,19,15,21)(10,20,12,22,14,24,16,18) );
G=PermutationGroup([(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(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)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12),(18,22),(20,24),(25,31,29,27),(26,28,30,32)], [(1,30,3,32,5,26,7,28),(2,27,4,29,6,31,8,25),(9,23,11,17,13,19,15,21),(10,20,12,22,14,24,16,18)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 4 | 2 |
| 0 | 0 | 8 | 4 | 11 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 10 | 0 | 1 | 13 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 9 | 0 | 8 | 4 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 9 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,13,8,0,0,2,0,0,4,0,0,0,0,4,11,0,0,0,0,2,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,10,0,0,0,16,0,0,0,0,0,0,4,1,0,0,0,0,0,13],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,9,16,2,0,0,0,0,0,1,0,0,13,8,0,9,0,0,0,4,0,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C4○D4 | C8.26D4 |
| kernel | C2×C8.26D4 | C2×C8⋊C4 | C2×C4≀C2 | C2×C8.C4 | C8.26D4 | C2×C8○D4 | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C4○D8 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_8._{26}D_4 % in TeX
G:=Group("C2xC8.26D4"); // GroupNames label
G:=SmallGroup(128,1686);
// by ID
G=gap.SmallGroup(128,1686);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
// generators/relations