p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.10Q8, C42.390D4, C4⋊C8⋊4C4, C4.32C4≀C2, (C2×C4).118D8, (C2×C4).52Q16, C4.2(C4.Q8), C4.2(C2.D8), C42.41(C2×C4), (C2×C4).15C42, (C2×C4).92SD16, (C22×C4).724D4, C4.44(D4⋊C4), C4.29(Q8⋊C4), C4⋊M4(2).8C2, C2.14(C42⋊6C4), C2.C42.13C4, C2.9(C22.4Q16), (C2×C42).138C22, C22.12(D4⋊C4), C23.143(C22⋊C4), C42.12C4.12C2, C22.15(Q8⋊C4), C2.11(M4(2)⋊4C4), C22.53(C2.C42), (C4×C4⋊C4).3C2, (C2×C4).98(C4⋊C4), (C22×C4).158(C2×C4), (C2×C4).340(C22⋊C4), SmallGroup(128,35)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.10Q8
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a2b-1c2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=ab2c3 >
Subgroups: 168 in 88 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×M4(2), C4×C4⋊C4, C4⋊M4(2), C42.12C4, C42.10Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4≀C2, C4.Q8, C2.D8, C42⋊6C4, C22.4Q16, M4(2)⋊4C4, C42.10Q8
►On 32 points
(1 31 15 23)(2 24 16 32)(3 25 9 17)(4 18 10 26)(5 27 11 19)(6 20 12 28)(7 29 13 21)(8 22 14 30)
(1 25 11 21)(2 26 12 22)(3 27 13 23)(4 28 14 24)(5 29 15 17)(6 30 16 18)(7 31 9 19)(8 32 10 20)
(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 6 31 28 15 12 23 20)(2 27 32 11 16 19 24 5)(3 4 25 26 9 10 17 18)(7 8 29 30 13 14 21 22)
G:=sub<Sym(32)| (1,31,15,23)(2,24,16,32)(3,25,9,17)(4,18,10,26)(5,27,11,19)(6,20,12,28)(7,29,13,21)(8,22,14,30), (1,25,11,21)(2,26,12,22)(3,27,13,23)(4,28,14,24)(5,29,15,17)(6,30,16,18)(7,31,9,19)(8,32,10,20), (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), (1,6,31,28,15,12,23,20)(2,27,32,11,16,19,24,5)(3,4,25,26,9,10,17,18)(7,8,29,30,13,14,21,22)>;
G:=Group( (1,31,15,23)(2,24,16,32)(3,25,9,17)(4,18,10,26)(5,27,11,19)(6,20,12,28)(7,29,13,21)(8,22,14,30), (1,25,11,21)(2,26,12,22)(3,27,13,23)(4,28,14,24)(5,29,15,17)(6,30,16,18)(7,31,9,19)(8,32,10,20), (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), (1,6,31,28,15,12,23,20)(2,27,32,11,16,19,24,5)(3,4,25,26,9,10,17,18)(7,8,29,30,13,14,21,22) );
G=PermutationGroup([[(1,31,15,23),(2,24,16,32),(3,25,9,17),(4,18,10,26),(5,27,11,19),(6,20,12,28),(7,29,13,21),(8,22,14,30)], [(1,25,11,21),(2,26,12,22),(3,27,13,23),(4,28,14,24),(5,29,15,17),(6,30,16,18),(7,31,9,19),(8,32,10,20)], [(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)], [(1,6,31,28,15,12,23,20),(2,27,32,11,16,19,24,5),(3,4,25,26,9,10,17,18),(7,8,29,30,13,14,21,22)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| 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 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C4≀C2 | M4(2)⋊4C4 |
| kernel | C42.10Q8 | C4×C4⋊C4 | C4⋊M4(2) | C42.12C4 | C2.C42 | C4⋊C8 | C42 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 2 |
Matrix representation of C42.10Q8 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 16 | 1 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 5 | 7 |
| 0 | 4 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 5 | 7 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,16,16,0,0,2,1],[0,16,0,0,1,0,0,0,0,0,0,5,0,0,7,7],[0,16,0,0,4,0,0,0,0,0,10,5,0,0,7,7] >;
C42.10Q8 in GAP, Magma, Sage, TeX
C_4^2._{10}Q_8 % in TeX
G:=Group("C4^2.10Q8"); // GroupNames label
G:=SmallGroup(128,35);
// by ID
G=gap.SmallGroup(128,35);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=a^2*b^-1*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*b^2*c^3>;
// generators/relations