p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.20C23, (C2×D4)⋊9Q8, C4⋊Q8⋊7C22, D4.5(C2×Q8), C4⋊C8⋊10C22, C4⋊C4.341D4, D4⋊2Q8⋊3C2, D4.Q8⋊14C2, D4⋊Q8⋊20C2, C2.13(D4○D8), C4⋊C4.44C23, (C2×C8).28C23, C2.D8⋊50C22, C4.Q8⋊59C22, C4.32(C22×Q8), (C2×C4).279C24, C22⋊C4.142D4, (C4×D4).69C22, C23.448(C2×D4), C4.67(C22⋊Q8), C42.C2⋊1C22, C2.20(D4○SD16), (C2×D4).397C23, C23.25D4⋊5C2, M4(2)⋊C4⋊19C2, D4⋊C4.26C22, C42.6C22⋊9C2, C22.11C24.8C2, (C22×C4).998C23, (C22×C8).181C22, C23.37D4.3C2, C22.539(C22×D4), C22.10(C22⋊Q8), C23.41C23⋊3C2, (C22×D4).355C22, (C2×M4(2)).68C22, C42⋊C2.118C22, C4.89(C2×C4○D4), (C2×C4).481(C2×D4), (C2×C4).103(C2×Q8), C2.60(C2×C22⋊Q8), (C2×D4⋊C4).28C2, (C2×C4).481(C4○D4), (C2×C4⋊C4).605C22, SmallGroup(128,1813)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 199 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×14], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], D4 [×4], D4 [×6], Q8 [×2], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×2], C24, D4⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4 [×4], C4×D4 [×2], C22⋊Q8 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×2], C4⋊Q8, C22×C8, C2×M4(2), C22×D4, C2×D4⋊C4, C23.37D4, C42.6C22, C23.25D4, M4(2)⋊C4, D4⋊Q8 [×2], D4⋊2Q8 [×2], D4.Q8 [×4], C22.11C24, C23.41C23, C42.20C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○D8, D4○SD16, C42.20C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=1, c2=e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, 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 21 28 18)(2 22 25 19)(3 23 26 20)(4 24 27 17)(5 11 30 16)(6 12 31 13)(7 9 32 14)(8 10 29 15)
(1 30 26 7)(2 8 27 31)(3 32 28 5)(4 6 25 29)(9 18 16 23)(10 24 13 19)(11 20 14 21)(12 22 15 17)
(5 16)(6 13)(7 14)(8 15)(9 32)(10 29)(11 30)(12 31)(17 24)(18 21)(19 22)(20 23)
(1 4 26 25)(2 28 27 3)(5 8 32 31)(6 30 29 7)(9 12 16 15)(10 14 13 11)(17 23 22 18)(19 21 24 20)
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), (1,21,28,18)(2,22,25,19)(3,23,26,20)(4,24,27,17)(5,11,30,16)(6,12,31,13)(7,9,32,14)(8,10,29,15), (1,30,26,7)(2,8,27,31)(3,32,28,5)(4,6,25,29)(9,18,16,23)(10,24,13,19)(11,20,14,21)(12,22,15,17), (5,16)(6,13)(7,14)(8,15)(9,32)(10,29)(11,30)(12,31)(17,24)(18,21)(19,22)(20,23), (1,4,26,25)(2,28,27,3)(5,8,32,31)(6,30,29,7)(9,12,16,15)(10,14,13,11)(17,23,22,18)(19,21,24,20)>;
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), (1,21,28,18)(2,22,25,19)(3,23,26,20)(4,24,27,17)(5,11,30,16)(6,12,31,13)(7,9,32,14)(8,10,29,15), (1,30,26,7)(2,8,27,31)(3,32,28,5)(4,6,25,29)(9,18,16,23)(10,24,13,19)(11,20,14,21)(12,22,15,17), (5,16)(6,13)(7,14)(8,15)(9,32)(10,29)(11,30)(12,31)(17,24)(18,21)(19,22)(20,23), (1,4,26,25)(2,28,27,3)(5,8,32,31)(6,30,29,7)(9,12,16,15)(10,14,13,11)(17,23,22,18)(19,21,24,20) );
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)], [(1,21,28,18),(2,22,25,19),(3,23,26,20),(4,24,27,17),(5,11,30,16),(6,12,31,13),(7,9,32,14),(8,10,29,15)], [(1,30,26,7),(2,8,27,31),(3,32,28,5),(4,6,25,29),(9,18,16,23),(10,24,13,19),(11,20,14,21),(12,22,15,17)], [(5,16),(6,13),(7,14),(8,15),(9,32),(10,29),(11,30),(12,31),(17,24),(18,21),(19,22),(20,23)], [(1,4,26,25),(2,28,27,3),(5,8,32,31),(6,30,29,7),(9,12,16,15),(10,14,13,11),(17,23,22,18),(19,21,24,20)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 16 |
| 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 | 16 | 16 | 1 | 2 |
| 0 | 0 | 1 | 0 | 16 | 16 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 14 | 6 | 6 |
| 0 | 0 | 3 | 0 | 14 | 11 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,0,0,0,0,15,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,1,0,0,16,0,16,0,0,0,0,0,1,16,0,0,0,0,2,16],[2,14,0,0,0,0,13,15,0,0,0,0,0,0,3,14,14,3,0,0,14,14,14,0,0,0,0,0,6,14,0,0,0,0,6,11],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,16,0,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,0,0,15,0,16] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 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 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 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 | D4 | D4 | Q8 | C4○D4 | D4○D8 | D4○SD16 |
| kernel | C42.20C23 | C2×D4⋊C4 | C23.37D4 | C42.6C22 | C23.25D4 | M4(2)⋊C4 | D4⋊Q8 | D4⋊2Q8 | D4.Q8 | C22.11C24 | C23.41C23 | C22⋊C4 | C4⋊C4 | C2×D4 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{20}C_2^3 % in TeX
G:=Group("C4^2.20C2^3"); // GroupNames label
G:=SmallGroup(128,1813);
// by ID
G=gap.SmallGroup(128,1813);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations