p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.383D4, C42.272C23, C4○D8⋊5C4, D8⋊11(C2×C4), C4○(D8⋊C4), Q16⋊11(C2×C4), SD16⋊7(C2×C4), C4.186(C4×D4), D8⋊C4⋊32C2, C22.5(C4×D4), C4○(Q16⋊C4), C4.20(C23×C4), C8.19(C22×C4), D4.4(C22×C4), Q16⋊C4⋊32C2, C4○(SD16⋊C4), Q8.4(C22×C4), C4⋊C4.360C23, (C2×C8).411C23, (C2×C4).200C24, C23.381(C2×D4), (C22×C4).168D4, SD16⋊C4⋊53C2, (C2×D8).156C22, (C4×D4).291C22, (C2×D4).369C23, (C4×Q8).274C22, (C2×Q8).342C23, C4.Q8.125C22, C2.D8.211C22, C8⋊C4.111C22, C23.24D4⋊37C2, C23.25D4⋊23C2, C2.3(D8⋊C22), (C22×C8).439C22, (C2×C42).765C22, (C2×Q16).151C22, C22.144(C22×D4), D4⋊C4.195C22, (C22×C4).1516C23, Q8⋊C4.195C22, (C2×SD16).106C22, C42⋊C2.296C22, C2.60(C2×C4×D4), (C2×C8)⋊13(C2×C4), C4○D4⋊8(C2×C4), (C4×C4○D4)⋊4C2, (C2×C8⋊C4)⋊6C2, C4.8(C2×C4○D4), (C2×C4○D8).14C2, (C2×C4)○(Q16⋊C4), (C2×C4).1212(C2×D4), (C2×C4).692(C4○D4), (C2×C4).471(C22×C4), (C2×C4○D4).290C22, SmallGroup(128,1675)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.383D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b2c3 >
Subgroups: 396 in 246 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C8⋊C4, C23.24D4, C23.25D4, SD16⋊C4, Q16⋊C4, D8⋊C4, C4×C4○D4, C2×C4○D8, C42.383D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D8⋊C22, C42.383D4
►On 64 points
(1 10 60 34)(2 15 61 39)(3 12 62 36)(4 9 63 33)(5 14 64 38)(6 11 57 35)(7 16 58 40)(8 13 59 37)(17 53 41 30)(18 50 42 27)(19 55 43 32)(20 52 44 29)(21 49 45 26)(22 54 46 31)(23 51 47 28)(24 56 48 25)
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 48 13 44)(10 41 14 45)(11 42 15 46)(12 43 16 47)(17 38 21 34)(18 39 22 35)(19 40 23 36)(20 33 24 37)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 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 21 60 45)(2 44 61 20)(3 19 62 43)(4 42 63 18)(5 17 64 41)(6 48 57 24)(7 23 58 47)(8 46 59 22)(9 31 33 54)(10 53 34 30)(11 29 35 52)(12 51 36 28)(13 27 37 50)(14 49 38 26)(15 25 39 56)(16 55 40 32)
G:=sub<Sym(64)| (1,10,60,34)(2,15,61,39)(3,12,62,36)(4,9,63,33)(5,14,64,38)(6,11,57,35)(7,16,58,40)(8,13,59,37)(17,53,41,30)(18,50,42,27)(19,55,43,32)(20,52,44,29)(21,49,45,26)(22,54,46,31)(23,51,47,28)(24,56,48,25), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,48,13,44)(10,41,14,45)(11,42,15,46)(12,43,16,47)(17,38,21,34)(18,39,22,35)(19,40,23,36)(20,33,24,37)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (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), (1,21,60,45)(2,44,61,20)(3,19,62,43)(4,42,63,18)(5,17,64,41)(6,48,57,24)(7,23,58,47)(8,46,59,22)(9,31,33,54)(10,53,34,30)(11,29,35,52)(12,51,36,28)(13,27,37,50)(14,49,38,26)(15,25,39,56)(16,55,40,32)>;
G:=Group( (1,10,60,34)(2,15,61,39)(3,12,62,36)(4,9,63,33)(5,14,64,38)(6,11,57,35)(7,16,58,40)(8,13,59,37)(17,53,41,30)(18,50,42,27)(19,55,43,32)(20,52,44,29)(21,49,45,26)(22,54,46,31)(23,51,47,28)(24,56,48,25), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,48,13,44)(10,41,14,45)(11,42,15,46)(12,43,16,47)(17,38,21,34)(18,39,22,35)(19,40,23,36)(20,33,24,37)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (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), (1,21,60,45)(2,44,61,20)(3,19,62,43)(4,42,63,18)(5,17,64,41)(6,48,57,24)(7,23,58,47)(8,46,59,22)(9,31,33,54)(10,53,34,30)(11,29,35,52)(12,51,36,28)(13,27,37,50)(14,49,38,26)(15,25,39,56)(16,55,40,32) );
G=PermutationGroup([[(1,10,60,34),(2,15,61,39),(3,12,62,36),(4,9,63,33),(5,14,64,38),(6,11,57,35),(7,16,58,40),(8,13,59,37),(17,53,41,30),(18,50,42,27),(19,55,43,32),(20,52,44,29),(21,49,45,26),(22,54,46,31),(23,51,47,28),(24,56,48,25)], [(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(9,48,13,44),(10,41,14,45),(11,42,15,46),(12,43,16,47),(17,38,21,34),(18,39,22,35),(19,40,23,36),(20,33,24,37),(25,59,29,63),(26,60,30,64),(27,61,31,57),(28,62,32,58)], [(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)], [(1,21,60,45),(2,44,61,20),(3,19,62,43),(4,42,63,18),(5,17,64,41),(6,48,57,24),(7,23,58,47),(8,46,59,22),(9,31,33,54),(10,53,34,30),(11,29,35,52),(12,51,36,28),(13,27,37,50),(14,49,38,26),(15,25,39,56),(16,55,40,32)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4Z | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D8⋊C22 |
| kernel | C42.383D4 | C2×C8⋊C4 | C23.24D4 | C23.25D4 | SD16⋊C4 | Q16⋊C4 | D8⋊C4 | C4×C4○D4 | C2×C4○D8 | C4○D8 | C42 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C42.383D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 6 | 11 |
| 0 | 0 | 4 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 |
| 0 | 0 | 0 | 0 | 9 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 4 | 1 |
| 0 | 0 | 6 | 3 | 6 | 6 |
| 0 | 0 | 2 | 15 | 6 | 3 |
| 0 | 0 | 4 | 0 | 1 | 14 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 11 | 6 |
| 0 | 0 | 16 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,6,6,13,9,0,0,11,0,4,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,11,6,2,4,0,0,9,3,15,0,0,0,4,6,6,1,0,0,1,6,3,14],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,11,11,1,0,0,0,6,0,16,16] >;
C42.383D4 in GAP, Magma, Sage, TeX
C_4^2._{383}D_4 % in TeX
G:=Group("C4^2.383D4"); // GroupNames label
G:=SmallGroup(128,1675);
// by ID
G=gap.SmallGroup(128,1675);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,248,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations