p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).30D4, (C2×D8)⋊12C4, C4.90(C4×D4), (C2×Q16)⋊12C4, (C2×SD16)⋊7C4, (C2×C8).113D4, C22⋊C4.70D4, C22.61(C4×D4), C23.33(C2×D4), C8.21(C22⋊C4), C8○2M4(2)⋊1C2, C4.207(C4⋊D4), C4.48(C4.4D4), C22.5(C4⋊1D4), C42⋊C22⋊17C2, C23.25D4⋊17C2, C22.21(C4⋊D4), C23.C23⋊8C2, (C22×C8).225C22, (C22×C4).702C23, C42⋊C2.42C22, (C2×M4(2)).325C22, C2.28(C24.3C22), (C2×C8).75(C2×C4), (C2×C4○D8).8C2, (C2×C4).250(C2×D4), C4.44(C2×C22⋊C4), (C2×D4).115(C2×C4), (C2×Q8).100(C2×C4), (C2×C4).764(C4○D4), (C2×C4).200(C22×C4), (C2×C4○D4).41C22, SmallGroup(128,708)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).30D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3b, cbc-1=dbd=a4b, dcd=bc-1 >
Subgroups: 300 in 142 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C23⋊C4, C4≀C2, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C8○2M4(2), C23.C23, C42⋊C22, C23.25D4, C2×C4○D8, M4(2).30D4
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, M4(2).30D4
►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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
(1 22 26 12)(2 19 27 9)(3 24 28 14)(4 21 29 11)(5 18 30 16)(6 23 31 13)(7 20 32 10)(8 17 25 15)
(1 29)(2 28)(3 31)(4 30)(5 25)(6 32)(7 27)(8 26)(9 14)(10 13)(11 16)(12 15)(17 18)(19 20)(21 22)(23 24)
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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,22,26,12)(2,19,27,9)(3,24,28,14)(4,21,29,11)(5,18,30,16)(6,23,31,13)(7,20,32,10)(8,17,25,15), (1,29)(2,28)(3,31)(4,30)(5,25)(6,32)(7,27)(8,26)(9,14)(10,13)(11,16)(12,15)(17,18)(19,20)(21,22)(23,24)>;
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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,22,26,12)(2,19,27,9)(3,24,28,14)(4,21,29,11)(5,18,30,16)(6,23,31,13)(7,20,32,10)(8,17,25,15), (1,29)(2,28)(3,31)(4,30)(5,25)(6,32)(7,27)(8,26)(9,14)(10,13)(11,16)(12,15)(17,18)(19,20)(21,22)(23,24) );
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)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)], [(1,22,26,12),(2,19,27,9),(3,24,28,14),(4,21,29,11),(5,18,30,16),(6,23,31,13),(7,20,32,10),(8,17,25,15)], [(1,29),(2,28),(3,31),(4,30),(5,25),(6,32),(7,27),(8,26),(9,14),(10,13),(11,16),(12,15),(17,18),(19,20),(21,22),(23,24)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | C4○D4 | M4(2).30D4 |
| kernel | M4(2).30D4 | C8○2M4(2) | C23.C23 | C42⋊C22 | C23.25D4 | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C22⋊C4 | C2×C8 | M4(2) | C2×C4 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of M4(2).30D4 ►in GL4(𝔽17) generated by
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 |
| 14 | 14 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 14 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,0,14,3,0,0,14,14,12,12,0,0,5,12,0,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,4,0,0,13,0,0,0],[0,0,3,14,0,0,14,14,3,14,0,0,14,14,0,0] >;
M4(2).30D4 in GAP, Magma, Sage, TeX
M_4(2)._{30}D_4 % in TeX
G:=Group("M4(2).30D4"); // GroupNames label
G:=SmallGroup(128,708);
// by ID
G=gap.SmallGroup(128,708);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2804,172,2028]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^3*b,c*b*c^-1=d*b*d=a^4*b,d*c*d=b*c^-1>;
// generators/relations