p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.44D4, (C2×C42).18C4, (C2×C4).4M4(2), (C22×Q8).17C4, C4.15(C4.10D4), C22⋊C8.123C22, C42.6C4.13C2, C23.168(C22×C4), (C22×C4).431C23, (C2×C42).150C22, C22.20(C2×M4(2)), C2.11(C24.4C4), C2.8(C23.C23), C22.M4(2).10C2, (C2×C4×Q8).2C2, (C2×C4⋊C4).35C4, (C2×C4).1128(C2×D4), (C22×C4).70(C2×C4), C2.6(C2×C4.10D4), (C2×C4⋊C4).738C22, (C2×C4).314(C22⋊C4), C22.149(C2×C22⋊C4), SmallGroup(128,199)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.44D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 204 in 118 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×Q8, C22.M4(2), C42.6C4, C2×C4×Q8, C42.44D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C2×M4(2), C24.4C4, C23.C23, C2×C4.10D4, C42.44D4
►On 64 points
(1 59 21 55)(2 64 22 52)(3 61 23 49)(4 58 24 54)(5 63 17 51)(6 60 18 56)(7 57 19 53)(8 62 20 50)(9 41 26 36)(10 46 27 33)(11 43 28 38)(12 48 29 35)(13 45 30 40)(14 42 31 37)(15 47 32 34)(16 44 25 39)
(1 19 5 23)(2 4 6 8)(3 21 7 17)(9 32 13 28)(10 12 14 16)(11 26 15 30)(18 20 22 24)(25 27 29 31)(33 35 37 39)(34 45 38 41)(36 47 40 43)(42 44 46 48)(49 59 53 63)(50 52 54 56)(51 61 55 57)(58 60 62 64)
(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 39 19 33 5 35 23 37)(2 40 4 43 6 36 8 47)(3 42 21 44 7 46 17 48)(9 58 32 60 13 62 28 64)(10 59 12 53 14 63 16 49)(11 52 26 54 15 56 30 50)(18 41 20 34 22 45 24 38)(25 61 27 55 29 57 31 51)
G:=sub<Sym(64)| (1,59,21,55)(2,64,22,52)(3,61,23,49)(4,58,24,54)(5,63,17,51)(6,60,18,56)(7,57,19,53)(8,62,20,50)(9,41,26,36)(10,46,27,33)(11,43,28,38)(12,48,29,35)(13,45,30,40)(14,42,31,37)(15,47,32,34)(16,44,25,39), (1,19,5,23)(2,4,6,8)(3,21,7,17)(9,32,13,28)(10,12,14,16)(11,26,15,30)(18,20,22,24)(25,27,29,31)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(49,59,53,63)(50,52,54,56)(51,61,55,57)(58,60,62,64), (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), (1,39,19,33,5,35,23,37)(2,40,4,43,6,36,8,47)(3,42,21,44,7,46,17,48)(9,58,32,60,13,62,28,64)(10,59,12,53,14,63,16,49)(11,52,26,54,15,56,30,50)(18,41,20,34,22,45,24,38)(25,61,27,55,29,57,31,51)>;
G:=Group( (1,59,21,55)(2,64,22,52)(3,61,23,49)(4,58,24,54)(5,63,17,51)(6,60,18,56)(7,57,19,53)(8,62,20,50)(9,41,26,36)(10,46,27,33)(11,43,28,38)(12,48,29,35)(13,45,30,40)(14,42,31,37)(15,47,32,34)(16,44,25,39), (1,19,5,23)(2,4,6,8)(3,21,7,17)(9,32,13,28)(10,12,14,16)(11,26,15,30)(18,20,22,24)(25,27,29,31)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(49,59,53,63)(50,52,54,56)(51,61,55,57)(58,60,62,64), (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), (1,39,19,33,5,35,23,37)(2,40,4,43,6,36,8,47)(3,42,21,44,7,46,17,48)(9,58,32,60,13,62,28,64)(10,59,12,53,14,63,16,49)(11,52,26,54,15,56,30,50)(18,41,20,34,22,45,24,38)(25,61,27,55,29,57,31,51) );
G=PermutationGroup([[(1,59,21,55),(2,64,22,52),(3,61,23,49),(4,58,24,54),(5,63,17,51),(6,60,18,56),(7,57,19,53),(8,62,20,50),(9,41,26,36),(10,46,27,33),(11,43,28,38),(12,48,29,35),(13,45,30,40),(14,42,31,37),(15,47,32,34),(16,44,25,39)], [(1,19,5,23),(2,4,6,8),(3,21,7,17),(9,32,13,28),(10,12,14,16),(11,26,15,30),(18,20,22,24),(25,27,29,31),(33,35,37,39),(34,45,38,41),(36,47,40,43),(42,44,46,48),(49,59,53,63),(50,52,54,56),(51,61,55,57),(58,60,62,64)], [(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)], [(1,39,19,33,5,35,23,37),(2,40,4,43,6,36,8,47),(3,42,21,44,7,46,17,48),(9,58,32,60,13,62,28,64),(10,59,12,53,14,63,16,49),(11,52,26,54,15,56,30,50),(18,41,20,34,22,45,24,38),(25,61,27,55,29,57,31,51)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | C4.10D4 | C23.C23 |
| kernel | C42.44D4 | C22.M4(2) | C42.6C4 | C2×C4×Q8 | C2×C42 | C2×C4⋊C4 | C22×Q8 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of C42.44D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 8 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 11 | 9 | 1 | 0 |
| 0 | 0 | 8 | 11 | 0 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 2 | 8 | 0 |
| 0 | 0 | 2 | 7 | 0 | 9 |
| 0 | 0 | 4 | 2 | 7 | 2 |
| 0 | 0 | 2 | 13 | 2 | 10 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 16 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 8 | 15 | 0 |
| 0 | 0 | 9 | 6 | 0 | 15 |
| 0 | 0 | 3 | 5 | 11 | 9 |
| 0 | 0 | 12 | 3 | 8 | 11 |
G:=sub<GL(6,GF(17))| [4,8,0,0,0,0,0,13,0,0,0,0,0,0,16,0,11,8,0,0,0,16,9,11,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[11,6,0,0,0,0,6,6,0,0,0,0,0,0,10,2,4,2,0,0,2,7,2,13,0,0,8,0,7,2,0,0,0,9,2,10],[11,16,0,0,0,0,6,6,0,0,0,0,0,0,6,9,3,12,0,0,8,6,5,3,0,0,15,0,11,8,0,0,0,15,9,11] >;
C42.44D4 in GAP, Magma, Sage, TeX
C_4^2._{44}D_4 % in TeX
G:=Group("C4^2.44D4"); // GroupNames label
G:=SmallGroup(128,199);
// by ID
G=gap.SmallGroup(128,199);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,520,1123,851,172]);
// 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*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations