p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.243D4, C42.709C23, C4.4D8⋊41C2, C4⋊C4.94C23, C4.21(C8⋊C22), (C4×M4(2))⋊42C2, (C4×C8).352C22, (C2×C4).339C24, (C2×C8).460C23, C4.SD16⋊42C2, C23.681(C2×D4), (C22×C4).463D4, C4⋊Q8.276C22, (C2×Q8).94C23, C4.61(C4.4D4), (C2×D4).106C23, C4.21(C8.C22), C8⋊C4.170C22, C4⋊1D4.148C22, C23.36D4⋊45C2, (C2×C42).850C22, C22.599(C22×D4), D4⋊C4.134C22, (C22×C4).1037C23, Q8⋊C4.126C22, C4.4D4.137C22, C22.41(C4.4D4), C42.28C22⋊32C2, (C2×M4(2)).376C22, C22.26C24.35C2, (C2×C4⋊Q8)⋊36C2, C4.48(C2×C4○D4), (C2×C4).517(C2×D4), C2.40(C2×C8⋊C22), C2.50(C2×C4.4D4), C2.40(C2×C8.C22), (C2×C4).303(C4○D4), (C2×C4⋊C4).626C22, (C2×C4○D4).151C22, SmallGroup(128,1873)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.243D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >
Subgroups: 420 in 212 conjugacy classes, 96 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C4×M4(2), C23.36D4, C4.4D8, C4.SD16, C42.28C22, C2×C4⋊Q8, C22.26C24, C42.243D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C8⋊C22, C8.C22, C22×D4, C2×C4○D4, C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C42.243D4
►On 64 points
(1 60 55 18)(2 61 56 19)(3 62 49 20)(4 63 50 21)(5 64 51 22)(6 57 52 23)(7 58 53 24)(8 59 54 17)(9 48 39 30)(10 41 40 31)(11 42 33 32)(12 43 34 25)(13 44 35 26)(14 45 36 27)(15 46 37 28)(16 47 38 29)
(1 43 5 47)(2 48 6 44)(3 45 7 41)(4 42 8 46)(9 23 13 19)(10 20 14 24)(11 17 15 21)(12 22 16 18)(25 51 29 55)(26 56 30 52)(27 53 31 49)(28 50 32 54)(33 59 37 63)(34 64 38 60)(35 61 39 57)(36 58 40 62)
(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 50 55 4)(2 3 56 49)(5 54 51 8)(6 7 52 53)(9 40 39 10)(11 38 33 16)(12 15 34 37)(13 36 35 14)(17 64 59 22)(18 21 60 63)(19 62 61 20)(23 58 57 24)(25 46 43 28)(26 27 44 45)(29 42 47 32)(30 31 48 41)
G:=sub<Sym(64)| (1,60,55,18)(2,61,56,19)(3,62,49,20)(4,63,50,21)(5,64,51,22)(6,57,52,23)(7,58,53,24)(8,59,54,17)(9,48,39,30)(10,41,40,31)(11,42,33,32)(12,43,34,25)(13,44,35,26)(14,45,36,27)(15,46,37,28)(16,47,38,29), (1,43,5,47)(2,48,6,44)(3,45,7,41)(4,42,8,46)(9,23,13,19)(10,20,14,24)(11,17,15,21)(12,22,16,18)(25,51,29,55)(26,56,30,52)(27,53,31,49)(28,50,32,54)(33,59,37,63)(34,64,38,60)(35,61,39,57)(36,58,40,62), (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), (1,50,55,4)(2,3,56,49)(5,54,51,8)(6,7,52,53)(9,40,39,10)(11,38,33,16)(12,15,34,37)(13,36,35,14)(17,64,59,22)(18,21,60,63)(19,62,61,20)(23,58,57,24)(25,46,43,28)(26,27,44,45)(29,42,47,32)(30,31,48,41)>;
G:=Group( (1,60,55,18)(2,61,56,19)(3,62,49,20)(4,63,50,21)(5,64,51,22)(6,57,52,23)(7,58,53,24)(8,59,54,17)(9,48,39,30)(10,41,40,31)(11,42,33,32)(12,43,34,25)(13,44,35,26)(14,45,36,27)(15,46,37,28)(16,47,38,29), (1,43,5,47)(2,48,6,44)(3,45,7,41)(4,42,8,46)(9,23,13,19)(10,20,14,24)(11,17,15,21)(12,22,16,18)(25,51,29,55)(26,56,30,52)(27,53,31,49)(28,50,32,54)(33,59,37,63)(34,64,38,60)(35,61,39,57)(36,58,40,62), (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), (1,50,55,4)(2,3,56,49)(5,54,51,8)(6,7,52,53)(9,40,39,10)(11,38,33,16)(12,15,34,37)(13,36,35,14)(17,64,59,22)(18,21,60,63)(19,62,61,20)(23,58,57,24)(25,46,43,28)(26,27,44,45)(29,42,47,32)(30,31,48,41) );
G=PermutationGroup([[(1,60,55,18),(2,61,56,19),(3,62,49,20),(4,63,50,21),(5,64,51,22),(6,57,52,23),(7,58,53,24),(8,59,54,17),(9,48,39,30),(10,41,40,31),(11,42,33,32),(12,43,34,25),(13,44,35,26),(14,45,36,27),(15,46,37,28),(16,47,38,29)], [(1,43,5,47),(2,48,6,44),(3,45,7,41),(4,42,8,46),(9,23,13,19),(10,20,14,24),(11,17,15,21),(12,22,16,18),(25,51,29,55),(26,56,30,52),(27,53,31,49),(28,50,32,54),(33,59,37,63),(34,64,38,60),(35,61,39,57),(36,58,40,62)], [(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)], [(1,50,55,4),(2,3,56,49),(5,54,51,8),(6,7,52,53),(9,40,39,10),(11,38,33,16),(12,15,34,37),(13,36,35,14),(17,64,59,22),(18,21,60,63),(19,62,61,20),(23,58,57,24),(25,46,43,28),(26,27,44,45),(29,42,47,32),(30,31,48,41)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.243D4 | C4×M4(2) | C23.36D4 | C4.4D8 | C4.SD16 | C42.28C22 | C2×C4⋊Q8 | C22.26C24 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.243D4 ►in GL6(𝔽17)
| 1 | 8 | 0 | 0 | 0 | 0 |
| 4 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,4,0,0,0,0,8,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,16,0,0,0,1,0,0,0,0,1,16,16,0],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,0,1,16,16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,16,1,0] >;
C42.243D4 in GAP, Magma, Sage, TeX
C_4^2._{243}D_4 % in TeX
G:=Group("C4^2.243D4"); // GroupNames label
G:=SmallGroup(128,1873);
// by ID
G=gap.SmallGroup(128,1873);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,521,248,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations