p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.451D4, C42.342C23, (C4×Q16)⋊20C2, (C4×SD16)⋊3C2, C4○3(D4⋊Q8), C4○3(Q8⋊Q8), D4⋊Q8⋊50C2, Q8⋊Q8⋊50C2, C4○3(C4⋊2Q16), C4⋊2Q16⋊49C2, D4.4(C4○D4), C4⋊C4.60C23, Q8.3(C4○D4), C4○3(D4.D4), C4○3(D4.7D4), D4.D4⋊50C2, C4.113(C4○D8), C4⋊C8.335C22, (C2×C4).305C24, (C4×C8).108C22, (C2×C8).317C23, D4.7D4.4C2, (C22×C4).446D4, C23.251(C2×D4), C4⋊Q8.266C22, (C2×Q8).75C23, (C4×Q8).71C22, (C2×D4).402C23, (C4×D4).320C22, C42.12C4⋊31C2, C4.Q8.153C22, C2.D8.172C22, C4○3(C23.20D4), C23.20D4⋊54C2, C4.142(C8.C22), C22⋊C8.190C22, (C2×C42).832C22, (C2×Q16).121C22, C22.565(C22×D4), C22⋊Q8.167C22, D4⋊C4.161C22, (C22×C4).1021C23, C23.37C23⋊5C2, Q8⋊C4.154C22, (C2×SD16).141C22, C42⋊C2.321C22, C2.106(C22.19C24), C2.26(C2×C4○D8), (C4×C4○D4).26C2, C4.190(C2×C4○D4), (C2×C4).1584(C2×D4), C2.30(C2×C8.C22), (C2×C4○D4).312C22, SmallGroup(128,1839)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.451D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 332 in 195 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, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, 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×Q8, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C42.12C4, C4×SD16, C4×Q16, D4.7D4, D4.D4, C4⋊2Q16, D4⋊Q8, Q8⋊Q8, C23.20D4, C4×C4○D4, C23.37C23, C42.451D4
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, C42.451D4
►On 64 points
(1 24 5 20)(2 64 6 60)(3 18 7 22)(4 58 8 62)(9 41 13 45)(10 31 14 27)(11 43 15 47)(12 25 16 29)(17 56 21 52)(19 50 23 54)(26 34 30 38)(28 36 32 40)(33 48 37 44)(35 42 39 46)(49 61 53 57)(51 63 55 59)
(1 30 55 45)(2 31 56 46)(3 32 49 47)(4 25 50 48)(5 26 51 41)(6 27 52 42)(7 28 53 43)(8 29 54 44)(9 24 38 59)(10 17 39 60)(11 18 40 61)(12 19 33 62)(13 20 34 63)(14 21 35 64)(15 22 36 57)(16 23 37 58)
(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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 22 21 18)(19 20 23 24)(25 26 29 30)(27 32 31 28)(33 34 37 38)(35 40 39 36)(41 44 45 48)(42 47 46 43)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)
G:=sub<Sym(64)| (1,24,5,20)(2,64,6,60)(3,18,7,22)(4,58,8,62)(9,41,13,45)(10,31,14,27)(11,43,15,47)(12,25,16,29)(17,56,21,52)(19,50,23,54)(26,34,30,38)(28,36,32,40)(33,48,37,44)(35,42,39,46)(49,61,53,57)(51,63,55,59), (1,30,55,45)(2,31,56,46)(3,32,49,47)(4,25,50,48)(5,26,51,41)(6,27,52,42)(7,28,53,43)(8,29,54,44)(9,24,38,59)(10,17,39,60)(11,18,40,61)(12,19,33,62)(13,20,34,63)(14,21,35,64)(15,22,36,57)(16,23,37,58), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,22,21,18)(19,20,23,24)(25,26,29,30)(27,32,31,28)(33,34,37,38)(35,40,39,36)(41,44,45,48)(42,47,46,43)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63)>;
G:=Group( (1,24,5,20)(2,64,6,60)(3,18,7,22)(4,58,8,62)(9,41,13,45)(10,31,14,27)(11,43,15,47)(12,25,16,29)(17,56,21,52)(19,50,23,54)(26,34,30,38)(28,36,32,40)(33,48,37,44)(35,42,39,46)(49,61,53,57)(51,63,55,59), (1,30,55,45)(2,31,56,46)(3,32,49,47)(4,25,50,48)(5,26,51,41)(6,27,52,42)(7,28,53,43)(8,29,54,44)(9,24,38,59)(10,17,39,60)(11,18,40,61)(12,19,33,62)(13,20,34,63)(14,21,35,64)(15,22,36,57)(16,23,37,58), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,22,21,18)(19,20,23,24)(25,26,29,30)(27,32,31,28)(33,34,37,38)(35,40,39,36)(41,44,45,48)(42,47,46,43)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63) );
G=PermutationGroup([[(1,24,5,20),(2,64,6,60),(3,18,7,22),(4,58,8,62),(9,41,13,45),(10,31,14,27),(11,43,15,47),(12,25,16,29),(17,56,21,52),(19,50,23,54),(26,34,30,38),(28,36,32,40),(33,48,37,44),(35,42,39,46),(49,61,53,57),(51,63,55,59)], [(1,30,55,45),(2,31,56,46),(3,32,49,47),(4,25,50,48),(5,26,51,41),(6,27,52,42),(7,28,53,43),(8,29,54,44),(9,24,38,59),(10,17,39,60),(11,18,40,61),(12,19,33,62),(13,20,34,63),(14,21,35,64),(15,22,36,57),(16,23,37,58)], [(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)], [(1,4,5,8),(2,7,6,3),(9,12,13,16),(10,15,14,11),(17,22,21,18),(19,20,23,24),(25,26,29,30),(27,32,31,28),(33,34,37,38),(35,40,39,36),(41,44,45,48),(42,47,46,43),(49,56,53,52),(50,51,54,55),(57,64,61,60),(58,59,62,63)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4S | 4T | 4U | 4V | 4W | 8A | ··· | 8H |
| order | 1 | 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 | 4 | 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 | C4○D4 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C42.451D4 | C42.12C4 | C4×SD16 | C4×Q16 | D4.7D4 | D4.D4 | C4⋊2Q16 | D4⋊Q8 | Q8⋊Q8 | C23.20D4 | C4×C4○D4 | C23.37C23 | 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 C42.451D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 12 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[12,12,0,0,5,12,0,0,0,0,0,16,0,0,1,0],[12,5,0,0,5,5,0,0,0,0,0,1,0,0,1,0] >;
C42.451D4 in GAP, Magma, Sage, TeX
C_4^2._{451}D_4 % in TeX
G:=Group("C4^2.451D4"); // GroupNames label
G:=SmallGroup(128,1839);
// by ID
G=gap.SmallGroup(128,1839);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations