p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊10D4, C42.376C23, (C2×C8)⋊17D4, C8.3(C2×D4), C8⋊5D4⋊5C2, C8⋊3D4⋊5C2, C4⋊C4.240D4, (C4×C8)⋊27C22, C8.2D4⋊5C2, C4⋊Q8⋊11C22, (C2×D8)⋊51C22, C22⋊C4.80D4, C4.12(C22×D4), C8⋊C4⋊48C22, C8.12D4⋊15C2, C4.41(C4⋊1D4), (C2×C8).595C23, (C2×C4).352C24, (C2×Q16)⋊21C22, (C22×SD16)⋊5C2, C4.4D4⋊8C22, C23.454(C2×D4), C8○2M4(2)⋊12C2, C2.35(D4○SD16), (C2×SD16)⋊79C22, (C2×D4).118C23, C4⋊1D4.63C22, (C2×Q8).106C23, C22.29C24⋊12C2, C22.18(C4⋊1D4), (C22×C8).271C22, C22.612(C22×D4), (C22×C4).1042C23, (C22×D4).378C22, (C22×Q8).311C22, C23.38C23⋊12C2, C42⋊C2.325C22, (C2×M4(2)).272C22, (C2×C4○D8)⋊21C2, (C2×C8⋊C22)⋊24C2, (C2×C4).138(C2×D4), C2.31(C2×C4⋊1D4), (C2×C8.C22)⋊24C2, (C2×C4○D4).158C22, SmallGroup(128,1886)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 604 in 282 conjugacy classes, 108 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], D4 [×18], Q8 [×10], C23, C23 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], D8 [×6], SD16 [×20], Q16 [×6], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], C24, C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×4], C4⋊1D4 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×8], C2×SD16 [×4], C2×Q16, C2×Q16 [×2], C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4 [×2], C8○2M4(2), C8⋊5D4 [×2], C8.12D4 [×2], C8⋊3D4 [×2], C8.2D4 [×2], C22.29C24, C23.38C23, C22×SD16, C2×C4○D8, C2×C8⋊C22, C2×C8.C22, M4(2)⋊10D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C22×D4 [×3], C2×C4⋊1D4, D4○SD16 [×2], M4(2)⋊10D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3, cbc-1=dbd=a4b, dcd=c-1 >
►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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)
(1 11 32 23)(2 16 25 20)(3 13 26 17)(4 10 27 22)(5 15 28 19)(6 12 29 24)(7 9 30 21)(8 14 31 18)
(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 26)(10 29)(11 32)(12 27)(13 30)(14 25)(15 28)(16 31)
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,11,32,23)(2,16,25,20)(3,13,26,17)(4,10,27,22)(5,15,28,19)(6,12,29,24)(7,9,30,21)(8,14,31,18), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,26)(10,29)(11,32)(12,27)(13,30)(14,25)(15,28)(16,31)>;
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,11,32,23)(2,16,25,20)(3,13,26,17)(4,10,27,22)(5,15,28,19)(6,12,29,24)(7,9,30,21)(8,14,31,18), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,26)(10,29)(11,32)(12,27)(13,30)(14,25)(15,28)(16,31) );
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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31)], [(1,11,32,23),(2,16,25,20),(3,13,26,17),(4,10,27,22),(5,15,28,19),(6,12,29,24),(7,9,30,21),(8,14,31,18)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,26),(10,29),(11,32),(12,27),(13,30),(14,25),(15,28),(16,31)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 12 | 0 | 0 | 7 |
| 0 | 0 | 0 | 12 | 12 | 5 |
| 0 | 0 | 12 | 5 | 12 | 5 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 16 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,7,7,5,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,16,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4○SD16 |
| kernel | M4(2)⋊10D4 | C8○2M4(2) | C8⋊5D4 | C8.12D4 | C8⋊3D4 | C8.2D4 | C22.29C24 | C23.38C23 | C22×SD16 | C2×C4○D8 | C2×C8⋊C22 | C2×C8.C22 | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_{10}D_4 % in TeX
G:=Group("M4(2):10D4"); // GroupNames label
G:=SmallGroup(128,1886);
// by ID
G=gap.SmallGroup(128,1886);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,521,2804,172]);
// 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,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations