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, (C2×C8).460C23, (C4×C8).352C22, (C2×C4).339C24, 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, C23.36D4⋊45C2, C4⋊1D4.148C22, (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
Subgroups: 420 in 212 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×12], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×8], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4 [×2], C4×M4(2), C23.36D4 [×4], C4.4D8 [×2], C4.SD16 [×2], C42.28C22 [×4], C2×C4⋊Q8, C22.26C24, C42.243D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C42.243D4
Generators and relations
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 >
►On 64 points
(1 60 55 20)(2 61 56 21)(3 62 49 22)(4 63 50 23)(5 64 51 24)(6 57 52 17)(7 58 53 18)(8 59 54 19)(9 44 33 32)(10 45 34 25)(11 46 35 26)(12 47 36 27)(13 48 37 28)(14 41 38 29)(15 42 39 30)(16 43 40 31)
(1 45 5 41)(2 42 6 46)(3 47 7 43)(4 44 8 48)(9 19 13 23)(10 24 14 20)(11 21 15 17)(12 18 16 22)(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 38 33 14)(10 13 34 37)(11 36 35 12)(15 40 39 16)(17 58 57 18)(19 64 59 24)(20 23 60 63)(21 62 61 22)(25 48 45 28)(26 27 46 47)(29 44 41 32)(30 31 42 43)
G:=sub<Sym(64)| (1,60,55,20)(2,61,56,21)(3,62,49,22)(4,63,50,23)(5,64,51,24)(6,57,52,17)(7,58,53,18)(8,59,54,19)(9,44,33,32)(10,45,34,25)(11,46,35,26)(12,47,36,27)(13,48,37,28)(14,41,38,29)(15,42,39,30)(16,43,40,31), (1,45,5,41)(2,42,6,46)(3,47,7,43)(4,44,8,48)(9,19,13,23)(10,24,14,20)(11,21,15,17)(12,18,16,22)(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)(9,38,33,14)(10,13,34,37)(11,36,35,12)(15,40,39,16)(17,58,57,18)(19,64,59,24)(20,23,60,63)(21,62,61,22)(25,48,45,28)(26,27,46,47)(29,44,41,32)(30,31,42,43)>;
G:=Group( (1,60,55,20)(2,61,56,21)(3,62,49,22)(4,63,50,23)(5,64,51,24)(6,57,52,17)(7,58,53,18)(8,59,54,19)(9,44,33,32)(10,45,34,25)(11,46,35,26)(12,47,36,27)(13,48,37,28)(14,41,38,29)(15,42,39,30)(16,43,40,31), (1,45,5,41)(2,42,6,46)(3,47,7,43)(4,44,8,48)(9,19,13,23)(10,24,14,20)(11,21,15,17)(12,18,16,22)(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)(9,38,33,14)(10,13,34,37)(11,36,35,12)(15,40,39,16)(17,58,57,18)(19,64,59,24)(20,23,60,63)(21,62,61,22)(25,48,45,28)(26,27,46,47)(29,44,41,32)(30,31,42,43) );
G=PermutationGroup([(1,60,55,20),(2,61,56,21),(3,62,49,22),(4,63,50,23),(5,64,51,24),(6,57,52,17),(7,58,53,18),(8,59,54,19),(9,44,33,32),(10,45,34,25),(11,46,35,26),(12,47,36,27),(13,48,37,28),(14,41,38,29),(15,42,39,30),(16,43,40,31)], [(1,45,5,41),(2,42,6,46),(3,47,7,43),(4,44,8,48),(9,19,13,23),(10,24,14,20),(11,21,15,17),(12,18,16,22),(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),(9,38,33,14),(10,13,34,37),(11,36,35,12),(15,40,39,16),(17,58,57,18),(19,64,59,24),(20,23,60,63),(21,62,61,22),(25,48,45,28),(26,27,46,47),(29,44,41,32),(30,31,42,43)])
Matrix representation ►G ⊆ 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] >;
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 |
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