p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.14C42, (C4×C8)⋊19C4, C4.1(C4×Q8), C8○2(C4.Q8), C8○3(C2.D8), C4.Q8⋊14C4, C2.D8⋊14C4, C8.44(C4⋊C4), (C2×C8).60Q8, C2.5(C8○D8), C4.171(C4×D4), (C2×C8).401D4, C8○2(C8.C4), C8.C4⋊10C4, C22.1(C4×Q8), C4.21(C2×C42), C22.93(C4×D4), C8○2(C42⋊6C4), C42.315(C2×C4), C42⋊6C4.15C2, C8○(C23.25D4), M4(2).19(C2×C4), C23.201(C4○D4), C8○2M4(2).15C2, (C22×C8).547C22, (C22×C4).1313C23, (C2×C42).1050C22, C23.25D4.21C2, C42⋊C2.265C22, C22.23(C42⋊C2), (C2×M4(2)).310C22, (C2×C4×C8).49C2, C4.76(C2×C4⋊C4), C2.11(C4×C4⋊C4), C8○(C2×C8.C4), (C2×C8)○(C2.D8), (C2×C8)○(C4.Q8), C4⋊C4.144(C2×C4), (C2×C8)○(C8.C4), (C2×C8).244(C2×C4), (C2×C4).182(C2×Q8), (C2×C8)○(C42⋊6C4), (C2×C4).1507(C2×D4), (C2×C8.C4).26C2, (C2×C4).544(C4○D4), (C2×C4).522(C22×C4), (C2×C8)○(C23.25D4), SmallGroup(128,504)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.14C42
G = < a,b,c | a8=b4=c4=1, bab-1=a3, ac=ca, cbc-1=a2b >
Subgroups: 164 in 116 conjugacy classes, 76 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4×C8, C4×C8, C8⋊C4, C4.Q8, C2.D8, C8.C4, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C42⋊6C4, C2×C4×C8, C8○2M4(2), C23.25D4, C2×C8.C4, C8.14C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×C4⋊C4, C8○D8, C8.14C42
►On 32 points
(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)
(1 11 19 32)(2 14 20 27)(3 9 21 30)(4 12 22 25)(5 15 23 28)(6 10 24 31)(7 13 17 26)(8 16 18 29)
(9 11 13 15)(10 12 14 16)(25 27 29 31)(26 28 30 32)
G:=sub<Sym(32)| (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), (1,11,19,32)(2,14,20,27)(3,9,21,30)(4,12,22,25)(5,15,23,28)(6,10,24,31)(7,13,17,26)(8,16,18,29), (9,11,13,15)(10,12,14,16)(25,27,29,31)(26,28,30,32)>;
G:=Group( (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), (1,11,19,32)(2,14,20,27)(3,9,21,30)(4,12,22,25)(5,15,23,28)(6,10,24,31)(7,13,17,26)(8,16,18,29), (9,11,13,15)(10,12,14,16)(25,27,29,31)(26,28,30,32) );
G=PermutationGroup([[(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)], [(1,11,19,32),(2,14,20,27),(3,9,21,30),(4,12,22,25),(5,15,23,28),(6,10,24,31),(7,13,17,26),(8,16,18,29)], [(9,11,13,15),(10,12,14,16),(25,27,29,31),(26,28,30,32)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | C4○D4 | C4○D4 | C8○D8 |
| kernel | C8.14C42 | C42⋊6C4 | C2×C4×C8 | C8○2M4(2) | C23.25D4 | C2×C8.C4 | C4×C8 | C4.Q8 | C2.D8 | C8.C4 | C2×C8 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 16 |
Matrix representation of C8.14C42 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 5 | 8 |
| 4 | 0 | 0 |
| 0 | 4 | 15 |
| 0 | 0 | 13 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 10 | 13 |
G:=sub<GL(3,GF(17))| [16,0,0,0,2,5,0,0,8],[4,0,0,0,4,0,0,15,13],[16,0,0,0,1,10,0,0,13] >;
C8.14C42 in GAP, Magma, Sage, TeX
C_8._{14}C_4^2 % in TeX
G:=Group("C8.14C4^2"); // GroupNames label
G:=SmallGroup(128,504);
// by ID
G=gap.SmallGroup(128,504);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^2*b>;
// generators/relations