p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.415D4, C42.166C23, C4⋊Q8.23C4, (C2×C4).12Q16, C4.36(C2×Q16), (C2×C4).29SD16, C4.54(C2×SD16), C4⋊C8.204C22, C42.107(C2×C4), C4.6Q16⋊20C2, C4.7(C4.D4), (C22×C4).238D4, C4⋊Q8.239C22, (C22×Q8).10C4, C4.17(Q8⋊C4), C4.105(C8.C22), C4⋊M4(2).15C2, (C2×C42).210C22, C23.183(C22⋊C4), C42.12C4.25C2, C22.22(Q8⋊C4), C2.12(C23.38D4), (C2×C4⋊Q8).4C2, (C2×Q8).30(C2×C4), (C2×C4).1237(C2×D4), C2.14(C2×Q8⋊C4), C2.19(C2×C4.D4), (C2×C4).160(C22×C4), (C22×C4).232(C2×C4), (C2×C4).247(C22⋊C4), C22.224(C2×C22⋊C4), SmallGroup(128,280)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.415D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 236 in 120 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C4.6Q16, C4⋊M4(2), C42.12C4, C2×C4⋊Q8, C42.415D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4.D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C8.C22, C2×C4.D4, C2×Q8⋊C4, C23.38D4, C42.415D4
►On 64 points
(1 33 62 50)(2 51 63 34)(3 35 64 52)(4 53 57 36)(5 37 58 54)(6 55 59 38)(7 39 60 56)(8 49 61 40)(9 42 32 18)(10 19 25 43)(11 44 26 20)(12 21 27 45)(13 46 28 22)(14 23 29 47)(15 48 30 24)(16 17 31 41)
(1 52 58 39)(2 40 59 53)(3 54 60 33)(4 34 61 55)(5 56 62 35)(6 36 63 49)(7 50 64 37)(8 38 57 51)(9 24 28 44)(10 45 29 17)(11 18 30 46)(12 47 31 19)(13 20 32 48)(14 41 25 21)(15 22 26 42)(16 43 27 23)
(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 48 52 13 58 20 39 32)(2 16 40 43 59 27 53 23)(3 46 54 11 60 18 33 30)(4 14 34 41 61 25 55 21)(5 44 56 9 62 24 35 28)(6 12 36 47 63 31 49 19)(7 42 50 15 64 22 37 26)(8 10 38 45 57 29 51 17)
G:=sub<Sym(64)| (1,33,62,50)(2,51,63,34)(3,35,64,52)(4,53,57,36)(5,37,58,54)(6,55,59,38)(7,39,60,56)(8,49,61,40)(9,42,32,18)(10,19,25,43)(11,44,26,20)(12,21,27,45)(13,46,28,22)(14,23,29,47)(15,48,30,24)(16,17,31,41), (1,52,58,39)(2,40,59,53)(3,54,60,33)(4,34,61,55)(5,56,62,35)(6,36,63,49)(7,50,64,37)(8,38,57,51)(9,24,28,44)(10,45,29,17)(11,18,30,46)(12,47,31,19)(13,20,32,48)(14,41,25,21)(15,22,26,42)(16,43,27,23), (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), (1,48,52,13,58,20,39,32)(2,16,40,43,59,27,53,23)(3,46,54,11,60,18,33,30)(4,14,34,41,61,25,55,21)(5,44,56,9,62,24,35,28)(6,12,36,47,63,31,49,19)(7,42,50,15,64,22,37,26)(8,10,38,45,57,29,51,17)>;
G:=Group( (1,33,62,50)(2,51,63,34)(3,35,64,52)(4,53,57,36)(5,37,58,54)(6,55,59,38)(7,39,60,56)(8,49,61,40)(9,42,32,18)(10,19,25,43)(11,44,26,20)(12,21,27,45)(13,46,28,22)(14,23,29,47)(15,48,30,24)(16,17,31,41), (1,52,58,39)(2,40,59,53)(3,54,60,33)(4,34,61,55)(5,56,62,35)(6,36,63,49)(7,50,64,37)(8,38,57,51)(9,24,28,44)(10,45,29,17)(11,18,30,46)(12,47,31,19)(13,20,32,48)(14,41,25,21)(15,22,26,42)(16,43,27,23), (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), (1,48,52,13,58,20,39,32)(2,16,40,43,59,27,53,23)(3,46,54,11,60,18,33,30)(4,14,34,41,61,25,55,21)(5,44,56,9,62,24,35,28)(6,12,36,47,63,31,49,19)(7,42,50,15,64,22,37,26)(8,10,38,45,57,29,51,17) );
G=PermutationGroup([[(1,33,62,50),(2,51,63,34),(3,35,64,52),(4,53,57,36),(5,37,58,54),(6,55,59,38),(7,39,60,56),(8,49,61,40),(9,42,32,18),(10,19,25,43),(11,44,26,20),(12,21,27,45),(13,46,28,22),(14,23,29,47),(15,48,30,24),(16,17,31,41)], [(1,52,58,39),(2,40,59,53),(3,54,60,33),(4,34,61,55),(5,56,62,35),(6,36,63,49),(7,50,64,37),(8,38,57,51),(9,24,28,44),(10,45,29,17),(11,18,30,46),(12,47,31,19),(13,20,32,48),(14,41,25,21),(15,22,26,42),(16,43,27,23)], [(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)], [(1,48,52,13,58,20,39,32),(2,16,40,43,59,27,53,23),(3,46,54,11,60,18,33,30),(4,14,34,41,61,25,55,21),(5,44,56,9,62,24,35,28),(6,12,36,47,63,31,49,19),(7,42,50,15,64,22,37,26),(8,10,38,45,57,29,51,17)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4.D4 | C8.C22 |
| kernel | C42.415D4 | C4.6Q16 | C4⋊M4(2) | C42.12C4 | C2×C4⋊Q8 | C4⋊Q8 | C22×Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.415D4 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 | 1 | 15 |
| 0 | 0 | 7 | 14 | 1 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 3 |
| 0 | 0 | 0 | 4 | 10 | 14 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 11 | 16 | 2 | 13 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 14 | 2 | 15 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 4 | 14 | 3 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,1,0,7,0,0,15,1,14,14,0,0,0,0,1,1,0,0,0,0,15,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,0,0,0,11,0,0,13,4,5,16,0,0,0,10,0,2,0,0,3,14,0,13],[11,14,0,0,0,0,6,0,0,0,0,0,0,0,0,0,8,8,0,0,0,14,0,4,0,0,15,2,0,14,0,0,0,15,0,3] >;
C42.415D4 in GAP, Magma, Sage, TeX
C_4^2._{415}D_4 % in TeX
G:=Group("C4^2.415D4"); // GroupNames label
G:=SmallGroup(128,280);
// by ID
G=gap.SmallGroup(128,280);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,758,1123,1018,248,1971,242]);
// 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,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations