p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.447D4, C42.332C23, C4○D4⋊3Q8, C4○(Q8.Q8), C4○(D4.Q8), Q8.2(C2×Q8), D4.2(C2×Q8), C4○(C4.Q16), C4○(D4⋊2Q8), C4○(D4⋊Q8), C4○(Q8⋊Q8), D4.Q8⋊53C2, Q8.Q8⋊53C2, D4⋊Q8⋊48C2, D4⋊2Q8⋊46C2, Q8⋊Q8⋊48C2, C4.Q16⋊48C2, C4⋊C4.39C23, C4.111(C4○D8), C4.27(C22×Q8), C4⋊C8.283C22, (C2×C4).274C24, (C2×C8).139C23, (C22×C4).431D4, C23.386(C2×D4), C4⋊Q8.260C22, (C2×D4).393C23, (C4×D4).315C22, C23.25D4⋊4C2, C4.124(C22⋊Q8), (C2×Q8).364C23, (C4×Q8).296C22, C4.Q8.147C22, C2.D8.166C22, (C22×C8).180C22, (C2×C42).820C22, C23.24D4.7C2, C22.534(C22×D4), C22.19(C22⋊Q8), D4⋊C4.158C22, C2.17(D8⋊C22), C23.37C23⋊2C2, (C22×C4).1544C23, Q8⋊C4.149C22, C42.C2.102C22, C42⋊C2.115C22, (C2×C4⋊C8)⋊29C2, C4.84(C2×C4○D4), C2.19(C2×C4○D8), (C2×C4)○(Q8.Q8), (C2×C4)○(Q8⋊Q8), (C2×C4)○(C4.Q16), (C4×C4○D4).23C2, (C2×C4).322(C2×Q8), C2.55(C2×C22⋊Q8), (C2×C4).1435(C2×D4), (C2×C4).840(C4○D4), (C2×C4○D4).307C22, SmallGroup(128,1808)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 194 conjugacy classes, 102 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×2], D4 [×5], Q8 [×2], Q8 [×5], C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C42⋊C2, C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×C4○D4, C23.24D4 [×2], C2×C4⋊C8, C23.25D4 [×2], D4⋊Q8, Q8⋊Q8, D4⋊2Q8, C4.Q16, D4.Q8 [×2], Q8.Q8 [×2], C4×C4○D4, C23.37C23, C42.447D4
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], C4○D8 [×2], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C4○D8, D8⋊C22, C42.447D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c3 >
►On 64 points
(1 55 10 63)(2 64 11 56)(3 49 12 57)(4 58 13 50)(5 51 14 59)(6 60 15 52)(7 53 16 61)(8 62 9 54)(17 43 39 30)(18 31 40 44)(19 45 33 32)(20 25 34 46)(21 47 35 26)(22 27 36 48)(23 41 37 28)(24 29 38 42)
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 9 10 8)(2 7 11 16)(3 15 12 6)(4 5 13 14)(17 38 39 24)(18 23 40 37)(19 36 33 22)(20 21 34 35)(25 26 46 47)(27 32 48 45)(28 44 41 31)(29 30 42 43)(49 60 57 52)(50 51 58 59)(53 64 61 56)(54 55 62 63)
G:=sub<Sym(64)| (1,55,10,63)(2,64,11,56)(3,49,12,57)(4,58,13,50)(5,51,14,59)(6,60,15,52)(7,53,16,61)(8,62,9,54)(17,43,39,30)(18,31,40,44)(19,45,33,32)(20,25,34,46)(21,47,35,26)(22,27,36,48)(23,41,37,28)(24,29,38,42), (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,55,21,51)(18,56,22,52)(19,49,23,53)(20,50,24,54)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,9,10,8)(2,7,11,16)(3,15,12,6)(4,5,13,14)(17,38,39,24)(18,23,40,37)(19,36,33,22)(20,21,34,35)(25,26,46,47)(27,32,48,45)(28,44,41,31)(29,30,42,43)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63)>;
G:=Group( (1,55,10,63)(2,64,11,56)(3,49,12,57)(4,58,13,50)(5,51,14,59)(6,60,15,52)(7,53,16,61)(8,62,9,54)(17,43,39,30)(18,31,40,44)(19,45,33,32)(20,25,34,46)(21,47,35,26)(22,27,36,48)(23,41,37,28)(24,29,38,42), (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,55,21,51)(18,56,22,52)(19,49,23,53)(20,50,24,54)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,9,10,8)(2,7,11,16)(3,15,12,6)(4,5,13,14)(17,38,39,24)(18,23,40,37)(19,36,33,22)(20,21,34,35)(25,26,46,47)(27,32,48,45)(28,44,41,31)(29,30,42,43)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63) );
G=PermutationGroup([(1,55,10,63),(2,64,11,56),(3,49,12,57),(4,58,13,50),(5,51,14,59),(6,60,15,52),(7,53,16,61),(8,62,9,54),(17,43,39,30),(18,31,40,44),(19,45,33,32),(20,25,34,46),(21,47,35,26),(22,27,36,48),(23,41,37,28),(24,29,38,42)], [(1,26,5,30),(2,27,6,31),(3,28,7,32),(4,29,8,25),(9,46,13,42),(10,47,14,43),(11,48,15,44),(12,41,16,45),(17,55,21,51),(18,56,22,52),(19,49,23,53),(20,50,24,54),(33,57,37,61),(34,58,38,62),(35,59,39,63),(36,60,40,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,9,10,8),(2,7,11,16),(3,15,12,6),(4,5,13,14),(17,38,39,24),(18,23,40,37),(19,36,33,22),(20,21,34,35),(25,26,46,47),(27,32,48,45),(28,44,41,31),(29,30,42,43),(49,60,57,52),(50,51,58,59),(53,64,61,56),(54,55,62,63)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 16 | 7 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 16 | 7 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,10,16,0,0,16,7],[3,14,0,0,14,14,0,0,0,0,10,16,0,0,16,7] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | D8⋊C22 |
| kernel | C42.447D4 | C23.24D4 | C2×C4⋊C8 | C23.25D4 | D4⋊Q8 | Q8⋊Q8 | D4⋊2Q8 | C4.Q16 | D4.Q8 | Q8.Q8 | C4×C4○D4 | C23.37C23 | C42 | C22×C4 | C4○D4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{447}D_4 % in TeX
G:=Group("C4^2.447D4"); // GroupNames label
G:=SmallGroup(128,1808);
// by ID
G=gap.SmallGroup(128,1808);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations