p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.54D4, C42.618C23, Q8⋊C8⋊37C2, C4⋊Q8.13C4, C4.6(C8○D4), C22⋊Q8.8C4, C42.68(C2×C4), (C4×C8).10C22, (C4×Q8).8C22, C4⋊C8.202C22, (C22×C4).208D4, C4.130(C8.C22), C23.50(C22⋊C4), C42.6C4.16C2, (C2×C42).171C22, C42.12C4.19C2, C2.6(C23.38D4), C2.14(C42⋊C22), C23.37C23.4C2, C4⋊C4.58(C2×C4), (C2×Q8).50(C2×C4), (C2×C4).1454(C2×D4), (C2×C4).84(C22⋊C4), (C22×C4).193(C2×C4), (C2×C4).323(C22×C4), C22.173(C2×C22⋊C4), C2.23((C22×C8)⋊C2), SmallGroup(128,229)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.54D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 180 in 101 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, Q8⋊C8, C42.12C4, C42.6C4, C23.37C23, C42.54D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8○D4, C8.C22, (C22×C8)⋊C2, C23.38D4, C42⋊C22, C42.54D4
►On 64 points
(1 34 59 56)(2 49 60 35)(3 36 61 50)(4 51 62 37)(5 38 63 52)(6 53 64 39)(7 40 57 54)(8 55 58 33)(9 21 30 45)(10 46 31 22)(11 23 32 47)(12 48 25 24)(13 17 26 41)(14 42 27 18)(15 19 28 43)(16 44 29 20)
(1 36 63 54)(2 37 64 55)(3 38 57 56)(4 39 58 49)(5 40 59 50)(6 33 60 51)(7 34 61 52)(8 35 62 53)(9 43 26 23)(10 44 27 24)(11 45 28 17)(12 46 29 18)(13 47 30 19)(14 48 31 20)(15 41 32 21)(16 42 25 22)
(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 24 36 10 63 44 54 27)(2 30 37 19 64 13 55 47)(3 42 38 25 57 22 56 16)(4 11 39 45 58 28 49 17)(5 20 40 14 59 48 50 31)(6 26 33 23 60 9 51 43)(7 46 34 29 61 18 52 12)(8 15 35 41 62 32 53 21)
G:=sub<Sym(64)| (1,34,59,56)(2,49,60,35)(3,36,61,50)(4,51,62,37)(5,38,63,52)(6,53,64,39)(7,40,57,54)(8,55,58,33)(9,21,30,45)(10,46,31,22)(11,23,32,47)(12,48,25,24)(13,17,26,41)(14,42,27,18)(15,19,28,43)(16,44,29,20), (1,36,63,54)(2,37,64,55)(3,38,57,56)(4,39,58,49)(5,40,59,50)(6,33,60,51)(7,34,61,52)(8,35,62,53)(9,43,26,23)(10,44,27,24)(11,45,28,17)(12,46,29,18)(13,47,30,19)(14,48,31,20)(15,41,32,21)(16,42,25,22), (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), (1,24,36,10,63,44,54,27)(2,30,37,19,64,13,55,47)(3,42,38,25,57,22,56,16)(4,11,39,45,58,28,49,17)(5,20,40,14,59,48,50,31)(6,26,33,23,60,9,51,43)(7,46,34,29,61,18,52,12)(8,15,35,41,62,32,53,21)>;
G:=Group( (1,34,59,56)(2,49,60,35)(3,36,61,50)(4,51,62,37)(5,38,63,52)(6,53,64,39)(7,40,57,54)(8,55,58,33)(9,21,30,45)(10,46,31,22)(11,23,32,47)(12,48,25,24)(13,17,26,41)(14,42,27,18)(15,19,28,43)(16,44,29,20), (1,36,63,54)(2,37,64,55)(3,38,57,56)(4,39,58,49)(5,40,59,50)(6,33,60,51)(7,34,61,52)(8,35,62,53)(9,43,26,23)(10,44,27,24)(11,45,28,17)(12,46,29,18)(13,47,30,19)(14,48,31,20)(15,41,32,21)(16,42,25,22), (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), (1,24,36,10,63,44,54,27)(2,30,37,19,64,13,55,47)(3,42,38,25,57,22,56,16)(4,11,39,45,58,28,49,17)(5,20,40,14,59,48,50,31)(6,26,33,23,60,9,51,43)(7,46,34,29,61,18,52,12)(8,15,35,41,62,32,53,21) );
G=PermutationGroup([[(1,34,59,56),(2,49,60,35),(3,36,61,50),(4,51,62,37),(5,38,63,52),(6,53,64,39),(7,40,57,54),(8,55,58,33),(9,21,30,45),(10,46,31,22),(11,23,32,47),(12,48,25,24),(13,17,26,41),(14,42,27,18),(15,19,28,43),(16,44,29,20)], [(1,36,63,54),(2,37,64,55),(3,38,57,56),(4,39,58,49),(5,40,59,50),(6,33,60,51),(7,34,61,52),(8,35,62,53),(9,43,26,23),(10,44,27,24),(11,45,28,17),(12,46,29,18),(13,47,30,19),(14,48,31,20),(15,41,32,21),(16,42,25,22)], [(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)], [(1,24,36,10,63,44,54,27),(2,30,37,19,64,13,55,47),(3,42,38,25,57,22,56,16),(4,11,39,45,58,28,49,17),(5,20,40,14,59,48,50,31),(6,26,33,23,60,9,51,43),(7,46,34,29,61,18,52,12),(8,15,35,41,62,32,53,21)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C8○D4 | C8.C22 | C42⋊C22 |
| kernel | C42.54D4 | Q8⋊C8 | C42.12C4 | C42.6C4 | C23.37C23 | C22⋊Q8 | C4⋊Q8 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.54D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 16 |
| 0 | 0 | 5 | 12 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 13 |
| 0 | 0 | 7 | 10 | 13 | 0 |
| 0 | 0 | 2 | 0 | 7 | 10 |
| 0 | 0 | 0 | 15 | 7 | 7 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 2 |
| 0 | 0 | 10 | 7 | 2 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,5,5,0,0,16,0,5,12,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,7,2,0,0,0,10,10,0,15,0,0,0,13,7,7,0,0,13,0,10,7],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,2,0,7,10,0,0,0,15,7,7,0,0,0,0,0,2,0,0,0,0,2,0] >;
C42.54D4 in GAP, Magma, Sage, TeX
C_4^2._{54}D_4 % in TeX
G:=Group("C4^2.54D4"); // GroupNames label
G:=SmallGroup(128,229);
// by ID
G=gap.SmallGroup(128,229);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1059,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations