p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.18C23, C4○D4⋊7D4, C4⋊C8⋊8C22, D4.9(C2×D4), Q8.9(C2×D4), C4⋊D8⋊20C2, C4⋊C4.339D4, C4⋊SD16⋊4C2, C2.9(D4○D8), (C4×D4)⋊4C22, (C4×Q8)⋊4C22, (C2×D8)⋊41C22, (C2×C8).20C23, C4.76(C22×D4), D4.2D4⋊16C2, C4⋊C4.386C23, (C2×C4).249C24, Q8.D4⋊16C2, C22⋊C4.140D4, (C2×Q16)⋊41C22, (C2×D4).55C23, C23.446(C2×D4), (C2×Q8).42C23, C4.170(C4⋊D4), D4⋊C4⋊17C22, C22.29C24⋊9C2, C2.14(D4○SD16), Q8⋊C4⋊19C22, (C2×SD16)⋊74C22, C4⋊1D4.59C22, C23.36D4⋊7C2, C22.6(C4⋊D4), C42.6C22⋊7C2, (C22×C4).979C23, (C22×C8).178C22, C4.4D4.26C22, C22.509(C22×D4), C23.33C23⋊7C2, (C22×D4).343C22, (C2×M4(2)).56C22, C42⋊C2.104C22, (C2×C4○D8)⋊6C2, (C2×C8⋊C22)⋊17C2, C4.159(C2×C4○D4), (C2×C4).469(C2×D4), C2.67(C2×C4⋊D4), (C2×D4⋊C4)⋊29C2, (C2×C4).280(C4○D4), (C2×C4⋊C4).583C22, (C2×C4○D4).121C22, SmallGroup(128,1777)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 548 in 252 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×19], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×2], D4 [×19], Q8 [×2], Q8 [×3], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×6], SD16 [×8], Q16 [×2], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×6], C24, D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×4], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4 [×2], C22×C8, C2×M4(2), C2×D8, C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×2], C2×Q16, C4○D8 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C2×D4⋊C4, C23.36D4, C42.6C22, C4⋊D8 [×2], C4⋊SD16 [×2], D4.2D4 [×2], Q8.D4 [×2], C23.33C23, C22.29C24, C2×C4○D8, C2×C8⋊C22, C42.18C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○D8, D4○SD16, C42.18C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=a2, ab=ba, cac=a-1, ad=da, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >
►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 15 10 25)(2 16 11 26)(3 13 12 27)(4 14 9 28)(5 18 22 30)(6 19 23 31)(7 20 24 32)(8 17 21 29)
(1 26)(2 25)(3 28)(4 27)(5 8)(6 7)(9 13)(10 16)(11 15)(12 14)(17 30)(18 29)(19 32)(20 31)(21 22)(23 24)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 13)(7 14)(8 15)(9 32)(10 29)(11 30)(12 31)(21 25)(22 26)(23 27)(24 28)
(1 28 3 26)(2 15 4 13)(5 29 7 31)(6 18 8 20)(9 27 11 25)(10 14 12 16)(17 24 19 22)(21 32 23 30)
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), (1,15,10,25)(2,16,11,26)(3,13,12,27)(4,14,9,28)(5,18,22,30)(6,19,23,31)(7,20,24,32)(8,17,21,29), (1,26)(2,25)(3,28)(4,27)(5,8)(6,7)(9,13)(10,16)(11,15)(12,14)(17,30)(18,29)(19,32)(20,31)(21,22)(23,24), (1,17)(2,18)(3,19)(4,20)(5,16)(6,13)(7,14)(8,15)(9,32)(10,29)(11,30)(12,31)(21,25)(22,26)(23,27)(24,28), (1,28,3,26)(2,15,4,13)(5,29,7,31)(6,18,8,20)(9,27,11,25)(10,14,12,16)(17,24,19,22)(21,32,23,30)>;
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), (1,15,10,25)(2,16,11,26)(3,13,12,27)(4,14,9,28)(5,18,22,30)(6,19,23,31)(7,20,24,32)(8,17,21,29), (1,26)(2,25)(3,28)(4,27)(5,8)(6,7)(9,13)(10,16)(11,15)(12,14)(17,30)(18,29)(19,32)(20,31)(21,22)(23,24), (1,17)(2,18)(3,19)(4,20)(5,16)(6,13)(7,14)(8,15)(9,32)(10,29)(11,30)(12,31)(21,25)(22,26)(23,27)(24,28), (1,28,3,26)(2,15,4,13)(5,29,7,31)(6,18,8,20)(9,27,11,25)(10,14,12,16)(17,24,19,22)(21,32,23,30) );
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)], [(1,15,10,25),(2,16,11,26),(3,13,12,27),(4,14,9,28),(5,18,22,30),(6,19,23,31),(7,20,24,32),(8,17,21,29)], [(1,26),(2,25),(3,28),(4,27),(5,8),(6,7),(9,13),(10,16),(11,15),(12,14),(17,30),(18,29),(19,32),(20,31),(21,22),(23,24)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,13),(7,14),(8,15),(9,32),(10,29),(11,30),(12,31),(21,25),(22,26),(23,27),(24,28)], [(1,28,3,26),(2,15,4,13),(5,29,7,31),(6,18,8,20),(9,27,11,25),(10,14,12,16),(17,24,19,22),(21,32,23,30)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 10 | 10 |
| 0 | 0 | 5 | 0 | 0 | 10 |
| 0 | 0 | 0 | 12 | 12 | 12 |
| 0 | 0 | 12 | 5 | 12 | 12 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 16 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 5 | 0 | 0 | 10 |
| 0 | 0 | 5 | 12 | 5 | 12 |
| 0 | 0 | 0 | 12 | 5 | 12 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 12 | 0 | 7 | 0 |
| 0 | 0 | 0 | 12 | 5 | 12 |
| 0 | 0 | 5 | 12 | 5 | 12 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,10,5,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,10,10,12,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,0,0,16,0],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,5,5,0,0,0,0,0,12,12,0,0,7,0,5,5,0,0,10,10,12,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,12,0,5,0,0,0,0,12,12,0,0,7,7,5,5,0,0,10,0,12,12] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D4○D8 | D4○SD16 |
| kernel | C42.18C23 | C2×D4⋊C4 | C23.36D4 | C42.6C22 | C4⋊D8 | C4⋊SD16 | D4.2D4 | Q8.D4 | C23.33C23 | C22.29C24 | C2×C4○D8 | C2×C8⋊C22 | C22⋊C4 | C4⋊C4 | C4○D4 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{18}C_2^3 % in TeX
G:=Group("C4^2.18C2^3"); // GroupNames label
G:=SmallGroup(128,1777);
// by ID
G=gap.SmallGroup(128,1777);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=a^2,a*b=b*a,c*a*c=a^-1,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
// generators/relations