p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).3Q8, C2.D8⋊6C4, C4.Q8⋊11C4, C4.27(C4×Q8), (C2×C8).326D4, C2.15(C8○D8), C42⋊6C4.7C2, C22.160(C4×D4), C2.15(C8.26D4), C4.C42.1C2, C4.116(C22⋊Q8), C4.26(C42⋊C2), C23.205(C4○D4), C8○2M4(2).17C2, (C22×C8).393C22, (C2×C42).295C22, C23.25D4.2C2, C22.7(C42.C2), (C22×C4).1378C23, C22.5(C42⋊2C2), C42⋊C2.271C22, C4.139(C22.D4), (C2×M4(2)).320C22, C22.7C42.30C2, C42.6C22.10C2, C2.13(C23.63C23), C4⋊C4.80(C2×C4), (C2×C8).40(C2×C4), (C2×C4).273(C2×Q8), (C2×C4).1534(C2×D4), (C2×C4).760(C4○D4), (C2×C4).396(C22×C4), SmallGroup(128,654)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).3Q8
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a4c2, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a6c3 >
Subgroups: 156 in 93 conjugacy classes, 48 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C22.7C42, C42⋊6C4, C4.C42, C8○2M4(2), C42.6C22, C23.25D4, M4(2).3Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C23.63C23, C8○D8, C8.26D4, M4(2).3Q8
►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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
(1 15 29 22 5 11 25 18)(2 16 30 23 6 12 26 19)(3 9 31 24 7 13 27 20)(4 10 32 17 8 14 28 21)
(1 12 25 23 5 16 29 19)(2 20 26 13 6 24 30 9)(3 14 27 17 7 10 31 21)(4 22 28 15 8 18 32 11)
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), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,15,29,22,5,11,25,18)(2,16,30,23,6,12,26,19)(3,9,31,24,7,13,27,20)(4,10,32,17,8,14,28,21), (1,12,25,23,5,16,29,19)(2,20,26,13,6,24,30,9)(3,14,27,17,7,10,31,21)(4,22,28,15,8,18,32,11)>;
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), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,15,29,22,5,11,25,18)(2,16,30,23,6,12,26,19)(3,9,31,24,7,13,27,20)(4,10,32,17,8,14,28,21), (1,12,25,23,5,16,29,19)(2,20,26,13,6,24,30,9)(3,14,27,17,7,10,31,21)(4,22,28,15,8,18,32,11) );
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)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)], [(1,15,29,22,5,11,25,18),(2,16,30,23,6,12,26,19),(3,9,31,24,7,13,27,20),(4,10,32,17,8,14,28,21)], [(1,12,25,23,5,16,29,19),(2,20,26,13,6,24,30,9),(3,14,27,17,7,10,31,21),(4,22,28,15,8,18,32,11)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8N | 8O | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 | C4○D4 | C8○D8 | C8.26D4 |
| kernel | M4(2).3Q8 | C22.7C42 | C42⋊6C4 | C4.C42 | C8○2M4(2) | C42.6C22 | C23.25D4 | C4.Q8 | C2.D8 | C2×C8 | M4(2) | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 6 | 2 | 8 | 2 |
Matrix representation of M4(2).3Q8 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 8 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,8,0,0,9,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[0,13,0,0,4,0,0,0,0,0,2,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,13,0] >;
M4(2).3Q8 in GAP, Magma, Sage, TeX
M_4(2)._3Q_8
% in TeX
G:=Group("M4(2).3Q8"); // GroupNames label
G:=SmallGroup(128,654);
// by ID
G=gap.SmallGroup(128,654);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,58,2804,718,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^4*c^2,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^3>;
// generators/relations