p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.384D4, C42.700C23, (C4×D8)⋊4C2, (C4×Q16)⋊3C2, C4○2(Q8.Q8), C4○2(D4.Q8), Q8.Q8⋊54C2, D4.Q8⋊54C2, C4○2(D4⋊D4), (C4×SD16)⋊28C2, D4.2(C4○D4), D4⋊D4.5C2, C4⋊C4.55C23, (C4×C8).69C22, Q8.1(C4○D4), C4○2(D4.7D4), C4○2(D4.2D4), D4.2D4⋊53C2, D4.7D4⋊53C2, C4.135(C4○D8), C4⋊C8.309C22, (C2×C8).315C23, (C2×C4).300C24, C4○2(Q8.D4), Q8.D4⋊53C2, (C2×D4).85C23, (C4×D4).72C22, C23.249(C2×D4), (C22×C4).169D4, (C4×Q8).69C22, (C2×D8).122C22, (C2×Q8).371C23, C2.D8.169C22, C4.Q8.150C22, C42.12C4⋊28C2, C4○2(C23.20D4), C4○2(C23.19D4), C23.19D4⋊52C2, C23.20D4⋊53C2, C4⋊D4.159C22, C22⋊C8.188C22, (C2×C42).827C22, (C2×Q16).119C22, C22.560(C22×D4), C22⋊Q8.164C22, D4⋊C4.160C22, C2.27(D8⋊C22), (C22×C4).1016C23, C23.36C23⋊1C2, Q8⋊C4.173C22, (C2×SD16).137C22, C4.4D4.128C22, C42.C2.105C22, C42⋊C2.319C22, C2.101(C22.19C24), (C4×C4○D4)⋊9C2, C2.23(C2×C4○D8), C4.185(C2×C4○D4), (C2×C4).493(C2×D4), (C2×C4○D4).310C22, SmallGroup(128,1834)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.384D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=ab2, bc=cb, bd=db, dcd=a2c3 >
Subgroups: 356 in 198 conjugacy classes, 90 normal (84 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, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, 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×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C42.12C4, C4×D8, C4×SD16, C4×Q16, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C4×C4○D4, C23.36C23, C42.384D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, C22.19C24, C2×C4○D8, D8⋊C22, C42.384D4
►On 64 points
(1 57 5 61)(2 55 6 51)(3 59 7 63)(4 49 8 53)(9 20 13 24)(10 43 14 47)(11 22 15 18)(12 45 16 41)(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 39 29 15)(2 40 30 16)(3 33 31 9)(4 34 32 10)(5 35 25 11)(6 36 26 12)(7 37 27 13)(8 38 28 14)(17 64 47 53)(18 57 48 54)(19 58 41 55)(20 59 42 56)(21 60 43 49)(22 61 44 50)(23 62 45 51)(24 63 46 52)
(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)| (1,57,5,61)(2,55,6,51)(3,59,7,63)(4,49,8,53)(9,20,13,24)(10,43,14,47)(11,22,15,18)(12,45,16,41)(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), (1,39,29,15)(2,40,30,16)(3,33,31,9)(4,34,32,10)(5,35,25,11)(6,36,26,12)(7,37,27,13)(8,38,28,14)(17,64,47,53)(18,57,48,54)(19,58,41,55)(20,59,42,56)(21,60,43,49)(22,61,44,50)(23,62,45,51)(24,63,46,52), (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( (1,57,5,61)(2,55,6,51)(3,59,7,63)(4,49,8,53)(9,20,13,24)(10,43,14,47)(11,22,15,18)(12,45,16,41)(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), (1,39,29,15)(2,40,30,16)(3,33,31,9)(4,34,32,10)(5,35,25,11)(6,36,26,12)(7,37,27,13)(8,38,28,14)(17,64,47,53)(18,57,48,54)(19,58,41,55)(20,59,42,56)(21,60,43,49)(22,61,44,50)(23,62,45,51)(24,63,46,52), (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([[(1,57,5,61),(2,55,6,51),(3,59,7,63),(4,49,8,53),(9,20,13,24),(10,43,14,47),(11,22,15,18),(12,45,16,41),(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)], [(1,39,29,15),(2,40,30,16),(3,33,31,9),(4,34,32,10),(5,35,25,11),(6,36,26,12),(7,37,27,13),(8,38,28,14),(17,64,47,53),(18,57,48,54),(19,58,41,55),(20,59,42,56),(21,60,43,49),(22,61,44,50),(23,62,45,51),(24,63,46,52)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 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 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | C4○D8 | D8⋊C22 |
| kernel | C42.384D4 | C42.12C4 | C4×D8 | C4×SD16 | C4×Q16 | D4⋊D4 | D4.7D4 | D4.2D4 | Q8.D4 | D4.Q8 | Q8.Q8 | C23.19D4 | C23.20D4 | C4×C4○D4 | C23.36C23 | C42 | C22×C4 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.384D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 7 | 13 |
| 0 | 0 | 12 | 10 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 10 | 4 |
| 0 | 0 | 13 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,7,12,0,0,13,10],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[14,14,0,0,3,14,0,0,0,0,10,13,0,0,4,7],[1,0,0,0,0,16,0,0,0,0,1,12,0,0,0,16] >;
C42.384D4 in GAP, Magma, Sage, TeX
C_4^2._{384}D_4 % in TeX
G:=Group("C4^2.384D4"); // GroupNames label
G:=SmallGroup(128,1834);
// by ID
G=gap.SmallGroup(128,1834);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,80,4037,1027,124]);
// 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*b^2,b*c=c*b,b*d=d*b,d*c*d=a^2*c^3>;
// generators/relations