p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.211D4, C42.321C23, (C2×D4)⋊38D4, C4⋊C8⋊2C22, D4.4(C2×D4), C4⋊D8⋊17C2, C4⋊4(C8⋊C22), C4⋊Q8⋊53C22, D4.D4⋊1C2, (C2×C8).11C23, C4.67(C22×D4), D4.2D4⋊12C2, C4⋊1D4⋊33C22, C4⋊C4.377C23, C4⋊M4(2)⋊2C2, (C2×C4).240C24, (C2×D8).53C22, (C2×D4).49C23, (C22×C4).793D4, C23.652(C2×D4), (C2×Q8).36C23, C4.168(C4⋊D4), D4⋊C4⋊15C22, Q8⋊C4⋊17C22, (C4×D4).310C22, C23.36D4⋊5C2, C4.4D4⋊53C22, (C2×SD16).2C22, C22.32(C4⋊D4), C22.26C24⋊3C2, (C2×C42).809C22, (C22×C4).970C23, C22.500(C22×D4), C2.11(D8⋊C22), (C22×D4).567C22, (C2×M4(2)).47C22, (C2×C4×D4)⋊61C2, (C2×C8⋊C22)⋊14C2, C4.150(C2×C4○D4), C2.58(C2×C4⋊D4), C2.15(C2×C8⋊C22), (C2×C4).1419(C2×D4), (C2×C4).271(C4○D4), (C2×C4⋊C4).920C22, (C2×C4○D4).115C22, SmallGroup(128,1768)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 572 in 274 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×24], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×21], D4 [×4], D4 [×18], Q8 [×4], C23, C23 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4 [×3], C22×C4 [×11], C2×D4 [×8], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×4], C8⋊C22 [×8], C23×C4, C22×D4, C2×C4○D4 [×2], C23.36D4 [×2], C4⋊M4(2), C4⋊D8 [×2], D4.D4 [×2], D4.2D4 [×4], C2×C4×D4, C22.26C24, C2×C8⋊C22 [×2], C42.211D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8⋊C22, D8⋊C22, C42.211D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, cbc-1=b-1, bd=db, dcd=b2c3 >
►On 32 points
(1 13 25 23)(2 24 26 14)(3 15 27 17)(4 18 28 16)(5 9 29 19)(6 20 30 10)(7 11 31 21)(8 22 32 12)
(1 31 5 27)(2 28 6 32)(3 25 7 29)(4 30 8 26)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
G:=sub<Sym(32)| (1,13,25,23)(2,24,26,14)(3,15,27,17)(4,18,28,16)(5,9,29,19)(6,20,30,10)(7,11,31,21)(8,22,32,12), (1,31,5,27)(2,28,6,32)(3,25,7,29)(4,30,8,26)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)>;
G:=Group( (1,13,25,23)(2,24,26,14)(3,15,27,17)(4,18,28,16)(5,9,29,19)(6,20,30,10)(7,11,31,21)(8,22,32,12), (1,31,5,27)(2,28,6,32)(3,25,7,29)(4,30,8,26)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19) );
G=PermutationGroup([(1,13,25,23),(2,24,26,14),(3,15,27,17),(4,18,28,16),(5,9,29,19),(6,20,30,10),(7,11,31,21),(8,22,32,12)], [(1,31,5,27),(2,28,6,32),(3,25,7,29),(4,30,8,26),(9,17,13,21),(10,22,14,18),(11,19,15,23),(12,24,16,20)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 4 | 6 | 0 | 0 | 0 | 0 |
| 6 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 4 | 6 | 0 | 0 | 0 | 0 |
| 6 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,6,0,0,0,0,6,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0],[4,6,0,0,0,0,6,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | C42.211D4 | C23.36D4 | C4⋊M4(2) | C4⋊D8 | D4.D4 | D4.2D4 | C2×C4×D4 | C22.26C24 | C2×C8⋊C22 | C42 | C22×C4 | C2×D4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{211}D_4 % in TeX
G:=Group("C4^2.211D4"); // GroupNames label
G:=SmallGroup(128,1768);
// by ID
G=gap.SmallGroup(128,1768);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2019,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations