p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.455D4, C42.597C23, C4○D4⋊1C8, D4⋊4(C2×C8), Q8⋊4(C2×C8), C4.44C4≀C2, C4○2(D4⋊C8), D4⋊C8⋊37C2, C4○2(Q8⋊C8), Q8⋊C8⋊43C2, C42○(D4⋊C8), (C4×D4).12C4, C42○(Q8⋊C8), C4.3(C22×C8), (C4×Q8).12C4, C4.3(C2×M4(2)), C4.118(C4○D8), C4⋊C8.244C22, C4.36(C22⋊C8), (C4×C8).362C22, C42.256(C2×C4), (C22×C4).539D4, (C2×C4).44M4(2), C42.12C4⋊5C2, C42⋊C2.15C4, (C4×D4).261C22, C22.2(C22⋊C8), (C4×Q8).248C22, C23.92(C22⋊C4), (C2×C42).1034C22, C2.1(C23.24D4), (C2×C4×C8)⋊2C2, C2.3(C2×C4≀C2), (C2×C4).52(C2×C8), (C4×C4○D4).1C2, C4⋊C4.175(C2×C4), (C2×C4○D4).13C4, C2.12(C2×C22⋊C8), (C2×D4).189(C2×C4), (C2×C4).1440(C2×D4), (C2×Q8).172(C2×C4), (C2×C4).302(C22×C4), (C22×C4).395(C2×C4), C22.96(C2×C22⋊C4), (C2×C4).389(C22⋊C4), SmallGroup(128,208)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.455D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 236 in 138 conjugacy classes, 64 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×C4○D4, D4⋊C8, Q8⋊C8, C2×C4×C8, C42.12C4, C4×C4○D4, C42.455D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C4≀C2, C2×C22⋊C4, C22×C8, C2×M4(2), C4○D8, C2×C22⋊C8, C23.24D4, C2×C4≀C2, C42.455D4
►On 64 points
(1 39 31 47)(2 40 32 48)(3 33 25 41)(4 34 26 42)(5 35 27 43)(6 36 28 44)(7 37 29 45)(8 38 30 46)(9 59 49 24)(10 60 50 17)(11 61 51 18)(12 62 52 19)(13 63 53 20)(14 64 54 21)(15 57 55 22)(16 58 56 23)
(1 22 27 61)(2 23 28 62)(3 24 29 63)(4 17 30 64)(5 18 31 57)(6 19 32 58)(7 20 25 59)(8 21 26 60)(9 45 53 33)(10 46 54 34)(11 47 55 35)(12 48 56 36)(13 41 49 37)(14 42 50 38)(15 43 51 39)(16 44 52 40)
(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 22 26 27 60 61 8)(2 25 23 59 28 7 62 20)(3 58 24 6 29 19 63 32)(4 5 17 18 30 31 64 57)(9 36 45 12 53 48 33 56)(10 11 46 47 54 55 34 35)(13 40 41 16 49 44 37 52)(14 15 42 43 50 51 38 39)
G:=sub<Sym(64)| (1,39,31,47)(2,40,32,48)(3,33,25,41)(4,34,26,42)(5,35,27,43)(6,36,28,44)(7,37,29,45)(8,38,30,46)(9,59,49,24)(10,60,50,17)(11,61,51,18)(12,62,52,19)(13,63,53,20)(14,64,54,21)(15,57,55,22)(16,58,56,23), (1,22,27,61)(2,23,28,62)(3,24,29,63)(4,17,30,64)(5,18,31,57)(6,19,32,58)(7,20,25,59)(8,21,26,60)(9,45,53,33)(10,46,54,34)(11,47,55,35)(12,48,56,36)(13,41,49,37)(14,42,50,38)(15,43,51,39)(16,44,52,40), (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), (1,21,22,26,27,60,61,8)(2,25,23,59,28,7,62,20)(3,58,24,6,29,19,63,32)(4,5,17,18,30,31,64,57)(9,36,45,12,53,48,33,56)(10,11,46,47,54,55,34,35)(13,40,41,16,49,44,37,52)(14,15,42,43,50,51,38,39)>;
G:=Group( (1,39,31,47)(2,40,32,48)(3,33,25,41)(4,34,26,42)(5,35,27,43)(6,36,28,44)(7,37,29,45)(8,38,30,46)(9,59,49,24)(10,60,50,17)(11,61,51,18)(12,62,52,19)(13,63,53,20)(14,64,54,21)(15,57,55,22)(16,58,56,23), (1,22,27,61)(2,23,28,62)(3,24,29,63)(4,17,30,64)(5,18,31,57)(6,19,32,58)(7,20,25,59)(8,21,26,60)(9,45,53,33)(10,46,54,34)(11,47,55,35)(12,48,56,36)(13,41,49,37)(14,42,50,38)(15,43,51,39)(16,44,52,40), (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), (1,21,22,26,27,60,61,8)(2,25,23,59,28,7,62,20)(3,58,24,6,29,19,63,32)(4,5,17,18,30,31,64,57)(9,36,45,12,53,48,33,56)(10,11,46,47,54,55,34,35)(13,40,41,16,49,44,37,52)(14,15,42,43,50,51,38,39) );
G=PermutationGroup([[(1,39,31,47),(2,40,32,48),(3,33,25,41),(4,34,26,42),(5,35,27,43),(6,36,28,44),(7,37,29,45),(8,38,30,46),(9,59,49,24),(10,60,50,17),(11,61,51,18),(12,62,52,19),(13,63,53,20),(14,64,54,21),(15,57,55,22),(16,58,56,23)], [(1,22,27,61),(2,23,28,62),(3,24,29,63),(4,17,30,64),(5,18,31,57),(6,19,32,58),(7,20,25,59),(8,21,26,60),(9,45,53,33),(10,46,54,34),(11,47,55,35),(12,48,56,36),(13,41,49,37),(14,42,50,38),(15,43,51,39),(16,44,52,40)], [(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)], [(1,21,22,26,27,60,61,8),(2,25,23,59,28,7,62,20),(3,58,24,6,29,19,63,32),(4,5,17,18,30,31,64,57),(9,36,45,12,53,48,33,56),(10,11,46,47,54,55,34,35),(13,40,41,16,49,44,37,52),(14,15,42,43,50,51,38,39)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | D4 | M4(2) | C4≀C2 | C4○D8 |
| kernel | C42.455D4 | D4⋊C8 | Q8⋊C8 | C2×C4×C8 | C42.12C4 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C42.455D4 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 13 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 9 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 14 | 14 |
| 8 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(17))| [1,0,0,0,13,0,0,0,13],[13,0,0,0,13,0,0,0,13],[9,0,0,0,0,14,0,10,14],[8,0,0,0,0,3,0,10,0] >;
C42.455D4 in GAP, Magma, Sage, TeX
C_4^2._{455}D_4 % in TeX
G:=Group("C4^2.455D4"); // GroupNames label
G:=SmallGroup(128,208);
// by ID
G=gap.SmallGroup(128,208);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,570,136,172]);
// 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,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations