p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.366D4, C42.723C23, (C2×C4)⋊8D8, (C4×D8)⋊8C2, C4.94(C2×D8), C4○3(C8⋊7D4), C8⋊11(C4○D4), C4○2(C8⋊4D4), C8⋊7D4⋊42C2, C8⋊4D4⋊25C2, C4○2(C8⋊2Q8), C8⋊2Q8⋊36C2, C22.1(C2×D8), C4.26(C4○D8), C4○3(C4.4D8), C4.4D8⋊47C2, C2.13(C22×D8), C4⋊C4.108C23, (C4×C8).411C22, (C2×C4).367C24, (C2×C8).564C23, (C4×D4).89C22, C23.395(C2×D4), (C22×C4).619D4, C4⋊Q8.290C22, (C2×D4).123C23, (C2×D8).133C22, C2.D8.182C22, C4⋊D4.172C22, C4⋊1D4.155C22, (C22×C8).539C22, C22.627(C22×D4), D4⋊C4.147C22, (C22×C4).1572C23, C22.26C24⋊12C2, (C2×C42).1136C22, C2.64(C22.26C24), (C2×C4×C8)⋊26C2, C4.52(C2×C4○D4), C2.36(C2×C4○D8), (C2×C4).700(C2×D4), SmallGroup(128,1901)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 484 in 228 conjugacy classes, 100 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], D4 [×24], Q8 [×4], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C4×C8 [×2], D4⋊C4 [×8], C2.D8 [×4], C2×C42, C4×D4 [×4], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8 [×4], C2×C4○D4 [×2], C2×C4×C8, C4×D8 [×4], C8⋊7D4 [×4], C4.4D8 [×2], C8⋊4D4, C8⋊2Q8, C22.26C24 [×2], C42.366D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×D8 [×6], C4○D8 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C22×D8, C2×C4○D8, C42.366D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=a2c3 >
►On 64 points
(1 51 5 55)(2 52 6 56)(3 53 7 49)(4 54 8 50)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(17 36 21 40)(18 37 22 33)(19 38 23 34)(20 39 24 35)(25 63 29 59)(26 64 30 60)(27 57 31 61)(28 58 32 62)
(1 33 31 10)(2 34 32 11)(3 35 25 12)(4 36 26 13)(5 37 27 14)(6 38 28 15)(7 39 29 16)(8 40 30 9)(17 60 47 50)(18 61 48 51)(19 62 41 52)(20 63 42 53)(21 64 43 54)(22 57 44 55)(23 58 45 56)(24 59 46 49)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 48)(9 57)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 56)(40 55)
G:=sub<Sym(64)| (1,51,5,55)(2,52,6,56)(3,53,7,49)(4,54,8,50)(9,47,13,43)(10,48,14,44)(11,41,15,45)(12,42,16,46)(17,36,21,40)(18,37,22,33)(19,38,23,34)(20,39,24,35)(25,63,29,59)(26,64,30,60)(27,57,31,61)(28,58,32,62), (1,33,31,10)(2,34,32,11)(3,35,25,12)(4,36,26,13)(5,37,27,14)(6,38,28,15)(7,39,29,16)(8,40,30,9)(17,60,47,50)(18,61,48,51)(19,62,41,52)(20,63,42,53)(21,64,43,54)(22,57,44,55)(23,58,45,56)(24,59,46,49), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,48)(9,57)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,56)(40,55)>;
G:=Group( (1,51,5,55)(2,52,6,56)(3,53,7,49)(4,54,8,50)(9,47,13,43)(10,48,14,44)(11,41,15,45)(12,42,16,46)(17,36,21,40)(18,37,22,33)(19,38,23,34)(20,39,24,35)(25,63,29,59)(26,64,30,60)(27,57,31,61)(28,58,32,62), (1,33,31,10)(2,34,32,11)(3,35,25,12)(4,36,26,13)(5,37,27,14)(6,38,28,15)(7,39,29,16)(8,40,30,9)(17,60,47,50)(18,61,48,51)(19,62,41,52)(20,63,42,53)(21,64,43,54)(22,57,44,55)(23,58,45,56)(24,59,46,49), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,48)(9,57)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,56)(40,55) );
G=PermutationGroup([(1,51,5,55),(2,52,6,56),(3,53,7,49),(4,54,8,50),(9,47,13,43),(10,48,14,44),(11,41,15,45),(12,42,16,46),(17,36,21,40),(18,37,22,33),(19,38,23,34),(20,39,24,35),(25,63,29,59),(26,64,30,60),(27,57,31,61),(28,58,32,62)], [(1,33,31,10),(2,34,32,11),(3,35,25,12),(4,36,26,13),(5,37,27,14),(6,38,28,15),(7,39,29,16),(8,40,30,9),(17,60,47,50),(18,61,48,51),(19,62,41,52),(20,63,42,53),(21,64,43,54),(22,57,44,55),(23,58,45,56),(24,59,46,49)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,48),(9,57),(10,64),(11,63),(12,62),(13,61),(14,60),(15,59),(16,58),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,32),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,56),(40,55)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 14 | 14 |
| 0 | 0 | 14 | 3 |
G:=sub<GL(4,GF(17))| [0,13,0,0,4,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,13,0],[0,16,0,0,1,0,0,0,0,0,3,3,0,0,14,3],[1,0,0,0,0,16,0,0,0,0,14,14,0,0,14,3] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D8 | C4○D8 |
| kernel | C42.366D4 | C2×C4×C8 | C4×D8 | C8⋊7D4 | C4.4D8 | C8⋊4D4 | C8⋊2Q8 | C22.26C24 | C42 | C22×C4 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{366}D_4 % in TeX
G:=Group("C4^2.366D4"); // GroupNames label
G:=SmallGroup(128,1901);
// by ID
G=gap.SmallGroup(128,1901);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,520,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations