p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.425D4, C23.31M4(2), (C22×C4)⋊8C8, C4⋊2(C22⋊C8), C22⋊1(C4⋊C8), (C23×C4).36C4, (C2×C42).36C4, C23.32(C2×C8), (C22×C4).77Q8, C23.66(C4⋊C4), C24.115(C2×C4), (C22×C4).543D4, (C2×C4).60M4(2), (C22×C8).8C22, C4.179(C4⋊D4), C4.108(C22⋊Q8), (C22×C42).16C2, C22.34(C22×C8), C2.3(C4⋊M4(2)), C2.4(C42.6C4), C2.4(C24.4C4), (C2×C42).995C22, (C23×C4).633C22, C22.7C42⋊2C2, C23.263(C22×C4), C22.45(C2×M4(2)), C2.4(C23.7Q8), C2.5(C42.12C4), (C22×C4).1617C23, C22.54(C42⋊C2), (C2×C4⋊C8)⋊9C2, C2.8(C2×C4⋊C8), (C2×C4).84(C2×C8), (C2×C4).82(C4⋊C4), C2.17(C2×C22⋊C8), C22.63(C2×C4⋊C4), (C2×C4).335(C2×Q8), (C2×C4).1513(C2×D4), (C2×C22⋊C8).13C2, (C2×C4).925(C4○D4), (C22×C4).479(C2×C4), (C2×C4).354(C22⋊C4), C22.124(C2×C22⋊C4), SmallGroup(128,529)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.425D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 316 in 200 conjugacy classes, 92 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C2×C8, C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C22×C8, C23×C4, C22.7C42, C2×C22⋊C8, C2×C4⋊C8, C22×C42, C42.425D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C4⋊C8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C22×C8, C2×M4(2), C23.7Q8, C2×C22⋊C8, C24.4C4, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.6C4, C42.425D4
►On 64 points
(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 37 21 31)(10 32 22 38)(11 39 23 25)(12 26 24 40)(13 33 17 27)(14 28 18 34)(15 35 19 29)(16 30 20 36)
(1 37 5 33)(2 38 6 34)(3 39 7 35)(4 40 8 36)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 63 21 59)(18 64 22 60)(19 57 23 61)(20 58 24 62)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)
(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 30 37 50 5 26 33 54)(2 49 38 25 6 53 34 29)(3 32 39 52 7 28 35 56)(4 51 40 27 8 55 36 31)(9 58 43 24 13 62 47 20)(10 23 44 61 14 19 48 57)(11 60 45 18 15 64 41 22)(12 17 46 63 16 21 42 59)
G:=sub<Sym(64)| (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,37,21,31)(10,32,22,38)(11,39,23,25)(12,26,24,40)(13,33,17,27)(14,28,18,34)(15,35,19,29)(16,30,20,36), (1,37,5,33)(2,38,6,34)(3,39,7,35)(4,40,8,36)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,63,21,59)(18,64,22,60)(19,57,23,61)(20,58,24,62)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52), (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), (1,30,37,50,5,26,33,54)(2,49,38,25,6,53,34,29)(3,32,39,52,7,28,35,56)(4,51,40,27,8,55,36,31)(9,58,43,24,13,62,47,20)(10,23,44,61,14,19,48,57)(11,60,45,18,15,64,41,22)(12,17,46,63,16,21,42,59)>;
G:=Group( (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,37,21,31)(10,32,22,38)(11,39,23,25)(12,26,24,40)(13,33,17,27)(14,28,18,34)(15,35,19,29)(16,30,20,36), (1,37,5,33)(2,38,6,34)(3,39,7,35)(4,40,8,36)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,63,21,59)(18,64,22,60)(19,57,23,61)(20,58,24,62)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52), (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), (1,30,37,50,5,26,33,54)(2,49,38,25,6,53,34,29)(3,32,39,52,7,28,35,56)(4,51,40,27,8,55,36,31)(9,58,43,24,13,62,47,20)(10,23,44,61,14,19,48,57)(11,60,45,18,15,64,41,22)(12,17,46,63,16,21,42,59) );
G=PermutationGroup([[(1,63,55,47),(2,48,56,64),(3,57,49,41),(4,42,50,58),(5,59,51,43),(6,44,52,60),(7,61,53,45),(8,46,54,62),(9,37,21,31),(10,32,22,38),(11,39,23,25),(12,26,24,40),(13,33,17,27),(14,28,18,34),(15,35,19,29),(16,30,20,36)], [(1,37,5,33),(2,38,6,34),(3,39,7,35),(4,40,8,36),(9,43,13,47),(10,44,14,48),(11,45,15,41),(12,46,16,42),(17,63,21,59),(18,64,22,60),(19,57,23,61),(20,58,24,62),(25,53,29,49),(26,54,30,50),(27,55,31,51),(28,56,32,52)], [(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)], [(1,30,37,50,5,26,33,54),(2,49,38,25,6,53,34,29),(3,32,39,52,7,28,35,56),(4,51,40,27,8,55,36,31),(9,58,43,24,13,62,47,20),(10,23,44,61,14,19,48,57),(11,60,45,18,15,64,41,22),(12,17,46,63,16,21,42,59)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | Q8 | M4(2) | C4○D4 | M4(2) |
| kernel | C42.425D4 | C22.7C42 | C2×C22⋊C8 | C2×C4⋊C8 | C22×C42 | C2×C42 | C23×C4 | C22×C4 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C23 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 16 | 4 | 2 | 2 | 8 | 4 | 4 |
Matrix representation of C42.425D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,4],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[9,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,16,0],[8,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0] >;
C42.425D4 in GAP, Magma, Sage, TeX
C_4^2._{425}D_4 % in TeX
G:=Group("C4^2.425D4"); // GroupNames label
G:=SmallGroup(128,529);
// by ID
G=gap.SmallGroup(128,529);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations