p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.62Q8, C2.D8⋊8C4, C4.Q8⋊13C4, C8.21(C4⋊C4), (C2×C8).50Q8, C4.10(C4×Q8), (C2×C8).228D4, C4.50(C4⋊Q8), C2.18(C8○D8), C42⋊6C4.8C2, C22.178(C4×D4), C4.199(C4⋊D4), C23.209(C4○D4), (C22×C8).488C22, C23.25D4.3C2, C22.25(C22⋊Q8), C22.3(C42.C2), C42⋊C2.40C22, (C2×C42).1069C22, (C22×C4).1394C23, (C2×M4(2)).202C22, C42.6C22.11C2, C2.12(C23.65C23), (C2×C4×C8).46C2, C4.43(C2×C4⋊C4), C4⋊C4.91(C2×C4), (C2×C8).184(C2×C4), (C2×C4).206(C2×Q8), (C2×C4).1540(C2×D4), (C2×C8.C4).13C2, (C2×C4).589(C4○D4), (C2×C4).412(C22×C4), SmallGroup(128,677)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.62Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b2c2, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=bc3 >
Subgroups: 164 in 104 conjugacy classes, 56 normal (26 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), C22×C4, C22×C4, C4×C8, C4⋊C8, C4.Q8, C2.D8, C8.C4, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C42⋊6C4, C2×C4×C8, C42.6C22, C23.25D4, C2×C8.C4, C42.62Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, C8○D8, C42.62Q8
►On 32 points
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 11 13 15)(10 12 14 16)(25 27 29 31)(26 28 30 32)
(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)
(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 30 7 28 5 26 3 32)(2 14 8 12 6 10 4 16)(9 24 15 22 13 20 11 18)(17 25 23 31 21 29 19 27)
G:=sub<Sym(32)| (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,11,13,15)(10,12,14,16)(25,27,29,31)(26,28,30,32), (1,20,5,24)(2,21,6,17)(3,22,7,18)(4,23,8,19)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31), (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), (1,30,7,28,5,26,3,32)(2,14,8,12,6,10,4,16)(9,24,15,22,13,20,11,18)(17,25,23,31,21,29,19,27)>;
G:=Group( (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,11,13,15)(10,12,14,16)(25,27,29,31)(26,28,30,32), (1,20,5,24)(2,21,6,17)(3,22,7,18)(4,23,8,19)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31), (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), (1,30,7,28,5,26,3,32)(2,14,8,12,6,10,4,16)(9,24,15,22,13,20,11,18)(17,25,23,31,21,29,19,27) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,11,13,15),(10,12,14,16),(25,27,29,31),(26,28,30,32)], [(1,20,5,24),(2,21,6,17),(3,22,7,18),(4,23,8,19),(9,32,13,28),(10,25,14,29),(11,26,15,30),(12,27,16,31)], [(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)], [(1,30,7,28,5,26,3,32),(2,14,8,12,6,10,4,16),(9,24,15,22,13,20,11,18),(17,25,23,31,21,29,19,27)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | Q8 | D4 | Q8 | C4○D4 | C4○D4 | C8○D8 |
| kernel | C42.62Q8 | C42⋊6C4 | C2×C4×C8 | C42.6C22 | C23.25D4 | C2×C8.C4 | C4.Q8 | C2.D8 | C42 | C2×C8 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 2 | 2 | 2 | 16 |
Matrix representation of C42.62Q8 ►in GL4(𝔽17) generated by
| 16 | 6 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 2 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 13 | 6 | 0 | 0 |
| 8 | 4 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 7 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,6,4,0,0,0,0,1,11,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,2,15,0,0,0,0,8,0,0,0,0,8],[13,8,0,0,6,4,0,0,0,0,1,7,0,0,15,16] >;
C42.62Q8 in GAP, Magma, Sage, TeX
C_4^2._{62}Q_8 % in TeX
G:=Group("C4^2.62Q8"); // GroupNames label
G:=SmallGroup(128,677);
// by ID
G=gap.SmallGroup(128,677);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^2*c^2,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=b*c^3>;
// generators/relations