p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.443D4, C42.320C23, C4○D4⋊8D4, C4⋊5(C4○D8), C4○(C4⋊D8), D4.3(C2×D4), Q8.3(C2×D4), C4⋊D8⋊45C2, C4○(C4⋊SD16), C4○(C4⋊2Q16), C4⋊SD16⋊47C2, C4⋊2Q16⋊47C2, C4○(D4.D4), C4○(D4.2D4), C4○(Q8.D4), C4.66(C22×D4), D4.2D4⋊52C2, D4.D4⋊48C2, C4⋊C8.281C22, C4⋊C4.376C23, (C2×C8).135C23, (C2×C4).239C24, Q8.D4⋊52C2, (C2×D4).48C23, C23.382(C2×D4), (C22×C4).421D4, C4⋊Q8.255C22, (C2×Q8).35C23, C4.210(C4⋊D4), (C2×D8).117C22, (C4×D4).309C22, (C4×Q8).290C22, C23.24D4⋊15C2, C4⋊1D4.136C22, C22.13(C4⋊D4), C22.26C24⋊2C2, (C22×C8).177C22, (C2×C42).808C22, (C2×Q16).116C22, C22.499(C22×D4), D4⋊C4.155C22, C2.10(D8⋊C22), (C22×C4).1529C23, Q8⋊C4.145C22, (C2×SD16).133C22, C4.4D4.125C22, C42⋊C2.310C22, (C2×C4⋊C8)⋊27C2, (C2×C4○D8)⋊5C2, (C4×C4○D4)⋊6C2, (C2×C4)○(C4⋊D8), C2.12(C2×C4○D8), (C2×C4)○(C4⋊SD16), (C2×C4)○(C4⋊2Q16), C4.149(C2×C4○D4), C2.57(C2×C4⋊D4), (C2×C4)○(D4.2D4), (C2×C4).1418(C2×D4), (C2×C4)○(Q8.D4), (C2×C4).906(C4○D4), (C2×C4○D4).114C22, SmallGroup(128,1767)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 468 in 248 conjugacy classes, 102 normal (44 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×20], D4 [×2], D4 [×17], Q8 [×2], Q8 [×5], C23, C23 [×3], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×10], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C4○D8 [×8], C2×C4○D4, C2×C4○D4 [×2], C23.24D4 [×2], C2×C4⋊C8, C4⋊D8, C4⋊SD16, D4.D4, C4⋊2Q16, D4.2D4 [×2], Q8.D4 [×2], C4×C4○D4, C22.26C24, C2×C4○D8 [×2], C42.443D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C4○D8, D8⋊C22, C42.443D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, bc=cb, bd=db, dcd=b2c3 >
►On 64 points
(1 50 29 61)(2 62 30 51)(3 52 31 63)(4 64 32 53)(5 54 25 57)(6 58 26 55)(7 56 27 59)(8 60 28 49)(9 42 37 20)(10 21 38 43)(11 44 39 22)(12 23 40 45)(13 46 33 24)(14 17 34 47)(15 48 35 18)(16 19 36 41)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(25 39 29 35)(26 40 30 36)(27 33 31 37)(28 34 32 38)(41 55 45 51)(42 56 46 52)(43 49 47 53)(44 50 48 54)
(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)
(2 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 41)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)
G:=sub<Sym(64)| (1,50,29,61)(2,62,30,51)(3,52,31,63)(4,64,32,53)(5,54,25,57)(6,58,26,55)(7,56,27,59)(8,60,28,49)(9,42,37,20)(10,21,38,43)(11,44,39,22)(12,23,40,45)(13,46,33,24)(14,17,34,47)(15,48,35,18)(16,19,36,41), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(25,39,29,35)(26,40,30,36)(27,33,31,37)(28,34,32,38)(41,55,45,51)(42,56,46,52)(43,49,47,53)(44,50,48,54), (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), (2,8)(3,7)(4,6)(9,13)(10,12)(14,16)(17,41)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63)>;
G:=Group( (1,50,29,61)(2,62,30,51)(3,52,31,63)(4,64,32,53)(5,54,25,57)(6,58,26,55)(7,56,27,59)(8,60,28,49)(9,42,37,20)(10,21,38,43)(11,44,39,22)(12,23,40,45)(13,46,33,24)(14,17,34,47)(15,48,35,18)(16,19,36,41), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(25,39,29,35)(26,40,30,36)(27,33,31,37)(28,34,32,38)(41,55,45,51)(42,56,46,52)(43,49,47,53)(44,50,48,54), (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), (2,8)(3,7)(4,6)(9,13)(10,12)(14,16)(17,41)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63) );
G=PermutationGroup([(1,50,29,61),(2,62,30,51),(3,52,31,63),(4,64,32,53),(5,54,25,57),(6,58,26,55),(7,56,27,59),(8,60,28,49),(9,42,37,20),(10,21,38,43),(11,44,39,22),(12,23,40,45),(13,46,33,24),(14,17,34,47),(15,48,35,18),(16,19,36,41)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(25,39,29,35),(26,40,30,36),(27,33,31,37),(28,34,32,38),(41,55,45,51),(42,56,46,52),(43,49,47,53),(44,50,48,54)], [(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)], [(2,8),(3,7),(4,6),(9,13),(10,12),(14,16),(17,41),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(26,32),(27,31),(28,30),(33,37),(34,36),(38,40),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,64),(56,63)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 |
| 0 | 0 | 14 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,4,0,0,4,0,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[0,1,0,0,16,0,0,0,0,0,14,14,0,0,3,14],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 4S | 4T | 8A | ··· | 8H |
| 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 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | D4 | C4○D4 | C4○D8 | D8⋊C22 |
| kernel | C42.443D4 | C23.24D4 | C2×C4⋊C8 | C4⋊D8 | C4⋊SD16 | D4.D4 | C4⋊2Q16 | D4.2D4 | Q8.D4 | C4×C4○D4 | C22.26C24 | C2×C4○D8 | C42 | C22×C4 | C4○D4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{443}D_4 % in TeX
G:=Group("C4^2.443D4"); // GroupNames label
G:=SmallGroup(128,1767);
// by ID
G=gap.SmallGroup(128,1767);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations