p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.450D4, C42.341C23, (C4×D8)⋊21C2, D4⋊2(C4○D4), C4○3(C4⋊D8), C4⋊D8⋊47C2, Q8⋊2(C4○D4), (C4×SD16)⋊2C2, C4○3(D4⋊D4), D4⋊D4⋊52C2, C4○3(D4⋊2Q8), C4○3(C4.Q16), C4○3(C4⋊SD16), C4⋊SD16⋊49C2, C4.Q16⋊50C2, D4⋊2Q8⋊48C2, C4⋊C4.59C23, C4.112(C4○D8), C4⋊C8.285C22, (C2×C4).304C24, (C4×C8).107C22, (C2×C8).149C23, (C4×D4).74C22, (C2×D4).88C23, (C22×C4).445D4, C23.250(C2×D4), C4⋊Q8.265C22, C4.149(C8⋊C22), (C2×D8).124C22, (C2×Q8).374C23, (C4×Q8).301C22, C2.D8.171C22, C42.12C4⋊30C2, C4.Q8.152C22, C4○3(C23.19D4), C4⋊1D4.140C22, C23.19D4⋊53C2, C4⋊D4.162C22, C22⋊C8.189C22, C22.26C24⋊5C2, (C2×C42).831C22, C22.564(C22×D4), D4⋊C4.184C22, (C22×C4).1020C23, Q8⋊C4.153C22, (C2×SD16).140C22, C42⋊C2.320C22, C2.105(C22.19C24), (C4×C4○D4)⋊10C2, C2.25(C2×C4○D8), C4.189(C2×C4○D4), C2.31(C2×C8⋊C22), (C2×C4).1583(C2×D4), (C2×C4○D4).311C22, SmallGroup(128,1838)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.450D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, bd=db, dcd=a2c3 >
Subgroups: 428 in 217 conjugacy classes, 92 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C2×C4○D4, C2×C4○D4, C42.12C4, C4×D8, C4×SD16, D4⋊D4, C4⋊D8, C4⋊SD16, D4⋊2Q8, C4.Q16, C23.19D4, C4×C4○D4, C22.26C24, C42.450D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C22.19C24, C2×C4○D8, C2×C8⋊C22, C42.450D4
►On 64 points
(1 57 5 61)(2 55 6 51)(3 59 7 63)(4 49 8 53)(9 18 13 22)(10 41 14 45)(11 20 15 24)(12 43 16 47)(17 34 21 38)(19 36 23 40)(25 50 29 54)(26 62 30 58)(27 52 31 56)(28 64 32 60)(33 42 37 46)(35 44 39 48)
(1 35 25 9)(2 36 26 10)(3 37 27 11)(4 38 28 12)(5 39 29 13)(6 40 30 14)(7 33 31 15)(8 34 32 16)(17 64 43 49)(18 57 44 50)(19 58 45 51)(20 59 46 52)(21 60 47 53)(22 61 48 54)(23 62 41 55)(24 63 42 56)
(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)(10 16)(11 15)(12 14)(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)| (1,57,5,61)(2,55,6,51)(3,59,7,63)(4,49,8,53)(9,18,13,22)(10,41,14,45)(11,20,15,24)(12,43,16,47)(17,34,21,38)(19,36,23,40)(25,50,29,54)(26,62,30,58)(27,52,31,56)(28,64,32,60)(33,42,37,46)(35,44,39,48), (1,35,25,9)(2,36,26,10)(3,37,27,11)(4,38,28,12)(5,39,29,13)(6,40,30,14)(7,33,31,15)(8,34,32,16)(17,64,43,49)(18,57,44,50)(19,58,45,51)(20,59,46,52)(21,60,47,53)(22,61,48,54)(23,62,41,55)(24,63,42,56), (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)(10,16)(11,15)(12,14)(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( (1,57,5,61)(2,55,6,51)(3,59,7,63)(4,49,8,53)(9,18,13,22)(10,41,14,45)(11,20,15,24)(12,43,16,47)(17,34,21,38)(19,36,23,40)(25,50,29,54)(26,62,30,58)(27,52,31,56)(28,64,32,60)(33,42,37,46)(35,44,39,48), (1,35,25,9)(2,36,26,10)(3,37,27,11)(4,38,28,12)(5,39,29,13)(6,40,30,14)(7,33,31,15)(8,34,32,16)(17,64,43,49)(18,57,44,50)(19,58,45,51)(20,59,46,52)(21,60,47,53)(22,61,48,54)(23,62,41,55)(24,63,42,56), (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)(10,16)(11,15)(12,14)(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([[(1,57,5,61),(2,55,6,51),(3,59,7,63),(4,49,8,53),(9,18,13,22),(10,41,14,45),(11,20,15,24),(12,43,16,47),(17,34,21,38),(19,36,23,40),(25,50,29,54),(26,62,30,58),(27,52,31,56),(28,64,32,60),(33,42,37,46),(35,44,39,48)], [(1,35,25,9),(2,36,26,10),(3,37,27,11),(4,38,28,12),(5,39,29,13),(6,40,30,14),(7,33,31,15),(8,34,32,16),(17,64,43,49),(18,57,44,50),(19,58,45,51),(20,59,46,52),(21,60,47,53),(22,61,48,54),(23,62,41,55),(24,63,42,56)], [(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),(10,16),(11,15),(12,14),(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)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4S | 4T | 4U | 8A | ··· | 8H |
| order | 1 | 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 | 4 | 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 | C4○D4 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C42.450D4 | C42.12C4 | C4×D8 | C4×SD16 | D4⋊D4 | C4⋊D8 | C4⋊SD16 | D4⋊2Q8 | C4.Q16 | C23.19D4 | C4×C4○D4 | C22.26C24 | C42 | C22×C4 | D4 | Q8 | C4 | C4 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.450D4 ►in GL4(𝔽17) generated by
| 16 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 2 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,2,1,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[16,16,0,0,2,1,0,0,0,0,3,3,0,0,14,3],[1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C42.450D4 in GAP, Magma, Sage, TeX
C_4^2._{450}D_4 % in TeX
G:=Group("C4^2.450D4"); // GroupNames label
G:=SmallGroup(128,1838);
// by ID
G=gap.SmallGroup(128,1838);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d=a^2*c^3>;
// generators/relations