p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.326D4, (C2×D8)⋊8C4, (C2×Q16)⋊8C4, C4.23(C4×D4), (C2×C8).268D4, (C2×SD16)⋊17C4, C2.19(C8○D8), C4.52(C4⋊1D4), C8.20(C22⋊C4), C22.187(C4×D4), C4.205(C4⋊D4), C23.211(C4○D4), C22.19(C4⋊D4), (C22×C8).495C22, C22.1(C4.4D4), (C22×C4).1414C23, (C2×C42).1078C22, (C2×M4(2)).211C22, C2.26(C24.3C22), (C2×C4×C8)⋊27C2, (C2×C4≀C2)⋊20C2, (C2×C4○D8).2C2, (C2×C8).186(C2×C4), (C2×C4).742(C2×D4), C4.42(C2×C22⋊C4), (C2×Q8).98(C2×C4), (C2×C8.C4)⋊10C2, (C2×D4).113(C2×C4), (C22×C8)⋊C2⋊24C2, (C2×C4).605(C4○D4), (C2×C4).428(C22×C4), (C2×C4○D4).39C22, SmallGroup(128,706)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.326D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 292 in 150 conjugacy classes, 56 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, C4≀C2, C8.C4, C2×C42, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4×C8, (C22×C8)⋊C2, C2×C4≀C2, C2×C8.C4, C2×C4○D8, C42.326D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, C8○D8, C42.326D4
►On 32 points
(1 3 5 7)(2 4 6 8)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 19 21 23)(18 20 22 24)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 29)
(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 10 19 27 5 14 23 31)(2 13 20 30 6 9 24 26)(3 16 21 25 7 12 17 29)(4 11 22 28 8 15 18 32)
G:=sub<Sym(32)| (1,3,5,7)(2,4,6,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,19,21,23)(18,20,22,24), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29), (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), (1,10,19,27,5,14,23,31)(2,13,20,30,6,9,24,26)(3,16,21,25,7,12,17,29)(4,11,22,28,8,15,18,32)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,19,21,23)(18,20,22,24), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29), (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), (1,10,19,27,5,14,23,31)(2,13,20,30,6,9,24,26)(3,16,21,25,7,12,17,29)(4,11,22,28,8,15,18,32) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,19,21,23),(18,20,22,24)], [(1,23,5,19),(2,24,6,20),(3,17,7,21),(4,18,8,22),(9,30,13,26),(10,31,14,27),(11,32,15,28),(12,25,16,29)], [(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)], [(1,10,19,27,5,14,23,31),(2,13,20,30,6,9,24,26),(3,16,21,25,7,12,17,29),(4,11,22,28,8,15,18,32)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 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 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 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 | D4 | D4 | C4○D4 | C4○D4 | C8○D8 |
| kernel | C42.326D4 | C2×C4×C8 | (C22×C8)⋊C2 | C2×C4≀C2 | C2×C8.C4 | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C42 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | 16 |
Matrix representation of C42.326D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 2 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 2 |
| 5 | 8 | 0 | 0 |
| 14 | 12 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 15 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,2,13,0,0,0,0,8,0,0,0,0,2],[5,14,0,0,8,12,0,0,0,0,0,15,0,0,2,0] >;
C42.326D4 in GAP, Magma, Sage, TeX
C_4^2._{326}D_4 % in TeX
G:=Group("C4^2.326D4"); // GroupNames label
G:=SmallGroup(128,706);
// by ID
G=gap.SmallGroup(128,706);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations