p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.366C23, C4⋊C4.238D4, (C4×C8)⋊63C22, C4⋊Q8⋊10C22, C2.21(D4○D8), C4.4D8⋊39C2, C4⋊C4.89C23, C22⋊C4.78D4, C8⋊C4⋊64C22, (C2×C4).334C24, (C2×C8).455C23, C23.452(C2×D4), (C2×Q8).90C23, C42.C2⋊4C22, D4⋊C4⋊95C22, C8○2M4(2)⋊37C2, C2.33(D4○SD16), Q8⋊C4⋊54C22, C4.45(C4.4D4), (C2×D4).102C23, C4⋊1D4.62C22, C23.24D4⋊43C2, C23.36D4⋊44C2, C23.37D4⋊36C2, (C22×C8).461C22, C4.4D4.31C22, C22.594(C22×D4), C23.41C23⋊6C2, (C22×C4).1032C23, C22.29C24.13C2, C22.15(C4.4D4), (C22×D4).368C22, C42⋊C2.139C22, C42.78C22⋊25C2, C42.28C22⋊29C2, C42.29C22⋊17C2, (C2×M4(2)).371C22, C4.43(C2×C4○D4), (C2×C4).136(C2×D4), (C2×D4⋊C4)⋊57C2, C2.45(C2×C4.4D4), (C2×C4).489(C4○D4), (C2×C4⋊C4).624C22, (C2×C4○D4).149C22, SmallGroup(128,1868)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 452 in 202 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×8], D4 [×14], Q8 [×4], C23, C23 [×7], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], D4⋊C4 [×10], Q8⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42.C2 [×2], C42.C2, C4⋊1D4 [×2], C4⋊Q8 [×2], C4⋊Q8, C22×C8, C2×M4(2), C22×D4, C2×C4○D4, C8○2M4(2), C2×D4⋊C4, C23.24D4, C23.36D4, C23.37D4, C4.4D8 [×2], C42.78C22 [×2], C42.28C22 [×2], C42.29C22 [×2], C22.29C24, C23.41C23, C42.366C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D4○D8, D4○SD16, C42.366C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b-1, e2=a2b2, ab=ba, cac=a-1b2, ad=da, eae-1=ab2, cbc=b-1, bd=db, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
►On 32 points
(1 25 23 11)(2 26 24 12)(3 27 17 13)(4 28 18 14)(5 29 19 15)(6 30 20 16)(7 31 21 9)(8 32 22 10)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)
(2 18)(3 7)(4 24)(6 22)(8 20)(9 31)(10 12)(11 29)(13 27)(14 16)(15 25)(17 21)(26 32)(28 30)
(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 11 19 29)(2 12 20 30)(3 13 21 31)(4 14 22 32)(5 15 23 25)(6 16 24 26)(7 9 17 27)(8 10 18 28)
G:=sub<Sym(32)| (1,25,23,11)(2,26,24,12)(3,27,17,13)(4,28,18,14)(5,29,19,15)(6,30,20,16)(7,31,21,9)(8,32,22,10), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (2,18)(3,7)(4,24)(6,22)(8,20)(9,31)(10,12)(11,29)(13,27)(14,16)(15,25)(17,21)(26,32)(28,30), (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), (1,11,19,29)(2,12,20,30)(3,13,21,31)(4,14,22,32)(5,15,23,25)(6,16,24,26)(7,9,17,27)(8,10,18,28)>;
G:=Group( (1,25,23,11)(2,26,24,12)(3,27,17,13)(4,28,18,14)(5,29,19,15)(6,30,20,16)(7,31,21,9)(8,32,22,10), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (2,18)(3,7)(4,24)(6,22)(8,20)(9,31)(10,12)(11,29)(13,27)(14,16)(15,25)(17,21)(26,32)(28,30), (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), (1,11,19,29)(2,12,20,30)(3,13,21,31)(4,14,22,32)(5,15,23,25)(6,16,24,26)(7,9,17,27)(8,10,18,28) );
G=PermutationGroup([(1,25,23,11),(2,26,24,12),(3,27,17,13),(4,28,18,14),(5,29,19,15),(6,30,20,16),(7,31,21,9),(8,32,22,10)], [(1,21,5,17),(2,22,6,18),(3,23,7,19),(4,24,8,20),(9,29,13,25),(10,30,14,26),(11,31,15,27),(12,32,16,28)], [(2,18),(3,7),(4,24),(6,22),(8,20),(9,31),(10,12),(11,29),(13,27),(14,16),(15,25),(17,21),(26,32),(28,30)], [(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)], [(1,11,19,29),(2,12,20,30),(3,13,21,31),(4,14,22,32),(5,15,23,25),(6,16,24,26),(7,9,17,27),(8,10,18,28)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 7 | 9 | 0 | 0 | 0 | 0 |
| 2 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 7 | 9 | 0 | 0 | 0 | 0 |
| 2 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
G:=sub<GL(6,GF(17))| [7,2,0,0,0,0,9,10,0,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,14,14,0,0,0,0,3,14,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,6,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,8,0,0,0,0,2,6,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,14,14],[7,2,0,0,0,0,9,10,0,0,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,14,14,0,0,0,0,3,14,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○D8 | D4○SD16 |
| kernel | C42.366C23 | C8○2M4(2) | C2×D4⋊C4 | C23.24D4 | C23.36D4 | C23.37D4 | C4.4D8 | C42.78C22 | C42.28C22 | C42.29C22 | C22.29C24 | C23.41C23 | C22⋊C4 | C4⋊C4 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{366}C_2^3 % in TeX
G:=Group("C4^2.366C2^3"); // GroupNames label
G:=SmallGroup(128,1868);
// by ID
G=gap.SmallGroup(128,1868);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^-1,e^2=a^2*b^2,a*b=b*a,c*a*c=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations