p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.278D4, C42.412C23, C4.502- 1+4, (C2×C4).53D8, C4.73(C2×D8), D4⋊Q8⋊8C2, C8⋊2Q8⋊12C2, C4.4D8⋊13C2, (C4×C8).76C22, C2.16(C22×D8), C22.25(C2×D8), C4⋊C4.165C23, C4⋊C8.288C22, (C2×C4).424C24, (C2×C8).165C23, C22.D8⋊6C2, C23.697(C2×D4), (C22×C4).507D4, C4⋊Q8.308C22, C2.D8.35C22, D4⋊C4.3C22, (C2×D4).173C23, (C4×D4).111C22, C4.28(C8.C22), C42.12C4⋊27C2, C4⋊1D4.169C22, C4⋊D4.196C22, C22⋊C8.180C22, (C2×C42).885C22, C22.684(C22×D4), (C22×C4).1089C23, C22.26C24.43C2, C2.72(C23.38C23), (C2×C4⋊Q8)⋊42C2, (C2×C4).868(C2×D4), C2.59(C2×C8.C22), (C2×C4⋊C4).645C22, SmallGroup(128,1958)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.278D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=a2c3 >
Subgroups: 412 in 202 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, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C42.12C4, D4⋊Q8, C22.D8, C4.4D8, C8⋊2Q8, C2×C4⋊Q8, C22.26C24, C42.278D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C8.C22, C22×D4, 2- 1+4, C23.38C23, C22×D8, C2×C8.C22, C42.278D4
►On 64 points
(1 47 25 63)(2 48 26 64)(3 41 27 57)(4 42 28 58)(5 43 29 59)(6 44 30 60)(7 45 31 61)(8 46 32 62)(9 54 33 22)(10 55 34 23)(11 56 35 24)(12 49 36 17)(13 50 37 18)(14 51 38 19)(15 52 39 20)(16 53 40 21)
(1 13 5 9)(2 34 6 38)(3 15 7 11)(4 36 8 40)(10 30 14 26)(12 32 16 28)(17 46 21 42)(18 59 22 63)(19 48 23 44)(20 61 24 57)(25 37 29 33)(27 39 31 35)(41 52 45 56)(43 54 47 50)(49 62 53 58)(51 64 55 60)
(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 28)(3 7)(4 26)(6 32)(8 30)(9 33)(10 12)(11 39)(13 37)(14 16)(15 35)(17 55)(19 53)(20 24)(21 51)(23 49)(27 31)(34 36)(38 40)(41 61)(42 48)(43 59)(44 46)(45 57)(47 63)(52 56)(58 64)(60 62)
G:=sub<Sym(64)| (1,47,25,63)(2,48,26,64)(3,41,27,57)(4,42,28,58)(5,43,29,59)(6,44,30,60)(7,45,31,61)(8,46,32,62)(9,54,33,22)(10,55,34,23)(11,56,35,24)(12,49,36,17)(13,50,37,18)(14,51,38,19)(15,52,39,20)(16,53,40,21), (1,13,5,9)(2,34,6,38)(3,15,7,11)(4,36,8,40)(10,30,14,26)(12,32,16,28)(17,46,21,42)(18,59,22,63)(19,48,23,44)(20,61,24,57)(25,37,29,33)(27,39,31,35)(41,52,45,56)(43,54,47,50)(49,62,53,58)(51,64,55,60), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,33)(10,12)(11,39)(13,37)(14,16)(15,35)(17,55)(19,53)(20,24)(21,51)(23,49)(27,31)(34,36)(38,40)(41,61)(42,48)(43,59)(44,46)(45,57)(47,63)(52,56)(58,64)(60,62)>;
G:=Group( (1,47,25,63)(2,48,26,64)(3,41,27,57)(4,42,28,58)(5,43,29,59)(6,44,30,60)(7,45,31,61)(8,46,32,62)(9,54,33,22)(10,55,34,23)(11,56,35,24)(12,49,36,17)(13,50,37,18)(14,51,38,19)(15,52,39,20)(16,53,40,21), (1,13,5,9)(2,34,6,38)(3,15,7,11)(4,36,8,40)(10,30,14,26)(12,32,16,28)(17,46,21,42)(18,59,22,63)(19,48,23,44)(20,61,24,57)(25,37,29,33)(27,39,31,35)(41,52,45,56)(43,54,47,50)(49,62,53,58)(51,64,55,60), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,33)(10,12)(11,39)(13,37)(14,16)(15,35)(17,55)(19,53)(20,24)(21,51)(23,49)(27,31)(34,36)(38,40)(41,61)(42,48)(43,59)(44,46)(45,57)(47,63)(52,56)(58,64)(60,62) );
G=PermutationGroup([[(1,47,25,63),(2,48,26,64),(3,41,27,57),(4,42,28,58),(5,43,29,59),(6,44,30,60),(7,45,31,61),(8,46,32,62),(9,54,33,22),(10,55,34,23),(11,56,35,24),(12,49,36,17),(13,50,37,18),(14,51,38,19),(15,52,39,20),(16,53,40,21)], [(1,13,5,9),(2,34,6,38),(3,15,7,11),(4,36,8,40),(10,30,14,26),(12,32,16,28),(17,46,21,42),(18,59,22,63),(19,48,23,44),(20,61,24,57),(25,37,29,33),(27,39,31,35),(41,52,45,56),(43,54,47,50),(49,62,53,58),(51,64,55,60)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,33),(10,12),(11,39),(13,37),(14,16),(15,35),(17,55),(19,53),(20,24),(21,51),(23,49),(27,31),(34,36),(38,40),(41,61),(42,48),(43,59),(44,46),(45,57),(47,63),(52,56),(58,64),(60,62)]])
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 | D8 | C8.C22 | 2- 1+4 |
| kernel | C42.278D4 | C42.12C4 | D4⋊Q8 | C22.D8 | C4.4D8 | C8⋊2Q8 | C2×C4⋊Q8 | C22.26C24 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.278D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 2 | 15 | 15 |
| 0 | 0 | 15 | 15 | 2 | 15 |
| 0 | 0 | 2 | 2 | 2 | 15 |
| 0 | 0 | 15 | 2 | 2 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,15,15,2,15,0,0,2,15,2,2,0,0,15,2,2,2,0,0,15,15,15,2],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
C42.278D4 in GAP, Magma, Sage, TeX
C_4^2._{278}D_4 % in TeX
G:=Group("C4^2.278D4"); // GroupNames label
G:=SmallGroup(128,1958);
// by ID
G=gap.SmallGroup(128,1958);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations