p-group, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)⋊22D4, C42.267C23, (C8×D4)⋊38C2, C8.86(C2×D4), C4.76(C4×D4), C8⋊6D4⋊33C2, C8⋊9D4⋊32C2, C4⋊C8⋊88C22, (C4×C8)⋊57C22, C22≀C2.4C4, C4⋊D4.19C4, C22⋊2(C8○D4), C24.81(C2×C4), C22.15(C4×D4), C8⋊C4⋊60C22, C22⋊Q8.18C4, C22⋊C8⋊77C22, (C2×C8).405C23, (C2×C4).652C24, (C22×C8)⋊52C22, (C4×D4).55C22, M4(2)○(C22⋊C8), C4.198(C22×D4), C8○2M4(2)⋊32C2, C23.36(C22×C4), C22.D4.4C4, C2.16(Q8○M4(2)), (C22×M4(2))⋊26C2, (C2×M4(2))⋊78C22, C22.179(C23×C4), (C23×C4).527C22, (C22×C4).919C23, C42.6C22⋊31C2, C22.19C24.11C2, C42⋊C2.294C22, C2.50(C2×C4×D4), (C2×C8○D4)⋊23C2, C2.18(C2×C8○D4), C4⋊C4.116(C2×C4), (C2×C22⋊C8)⋊44C2, C4.303(C2×C4○D4), C22⋊C8○(C2×M4(2)), (C2×D4).173(C2×C4), (C2×C4).1084(C2×D4), C22⋊C4.36(C2×C4), (C2×C4).67(C22×C4), (C2×Q8).156(C2×C4), (C22×C8)⋊C2⋊30C2, (C2×C4).830(C4○D4), (C22×C4).342(C2×C4), (C2×C4○D4).287C22, SmallGroup(128,1665)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 380 in 251 conjugacy classes, 142 normal (52 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×17], C8 [×4], C8 [×6], C2×C4 [×4], C2×C4 [×10], C2×C4 [×14], D4 [×12], Q8 [×2], C23 [×3], C23 [×2], C23 [×5], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], C2×C8 [×12], M4(2) [×4], M4(2) [×10], C22×C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C22⋊C8 [×4], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C22×C8 [×4], C2×M4(2) [×2], C2×M4(2) [×4], C2×M4(2) [×4], C8○D4 [×4], C23×C4, C2×C4○D4, C8○2M4(2), C2×C22⋊C8, (C22×C8)⋊C2, C42.6C22, C8×D4 [×2], C8⋊9D4 [×4], C8⋊6D4 [×2], C22.19C24, C22×M4(2), C2×C8○D4, M4(2)⋊22D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4, Q8○M4(2), M4(2)⋊22D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=dad=a5, 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)
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)
(1 11 23 27)(2 16 24 32)(3 13 17 29)(4 10 18 26)(5 15 19 31)(6 12 20 28)(7 9 21 25)(8 14 22 30)
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 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), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,29)(10,26)(11,31)(12,28)(13,25)(14,30)(15,27)(16,32), (1,11,23,27)(2,16,24,32)(3,13,17,29)(4,10,18,26)(5,15,19,31)(6,12,20,28)(7,9,21,25)(8,14,22,30), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)>;
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), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,29)(10,26)(11,31)(12,28)(13,25)(14,30)(15,27)(16,32), (1,11,23,27)(2,16,24,32)(3,13,17,29)(4,10,18,26)(5,15,19,31)(6,12,20,28)(7,9,21,25)(8,14,22,30), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24) );
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)], [(1,23),(2,20),(3,17),(4,22),(5,19),(6,24),(7,21),(8,18),(9,29),(10,26),(11,31),(12,28),(13,25),(14,30),(15,27),(16,32)], [(1,11,23,27),(2,16,24,32),(3,13,17,29),(4,10,18,26),(5,15,19,31),(6,12,20,28),(7,9,21,25),(8,14,22,30)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 2 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 11 | 2 |
| 0 | 2 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 10 | 2 |
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,9,0,0,2,0,0,0,0,0,15,11,0,0,15,2],[0,9,0,0,2,0,0,0,0,0,15,10,0,0,15,2] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4O | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | M4(2)⋊22D4 | C8○2M4(2) | C2×C22⋊C8 | (C22×C8)⋊C2 | C42.6C22 | C8×D4 | C8⋊9D4 | C8⋊6D4 | C22.19C24 | C22×M4(2) | C2×C8○D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | M4(2) | C2×C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 2 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_{22}D_4 % in TeX
G:=Group("M4(2):22D4"); // GroupNames label
G:=SmallGroup(128,1665);
// by ID
G=gap.SmallGroup(128,1665);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^5,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations