p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.98D4, D4⋊1(C4⋊C4), (C4×D4)⋊13C4, (C2×C4).133D8, (C2×D4).22Q8, C4⋊3(D4⋊C4), C42⋊9C4⋊1C2, (C2×D4).198D4, C2.1(C4⋊D8), C22.31(C2×D8), C42.136(C2×C4), C22.4Q16⋊1C2, C2.1(D4⋊2Q8), C2.1(D4⋊Q8), (C2×C4).103SD16, C23.740(C2×D4), (C22×C4).266D4, C4.93(C22⋊Q8), C4.118(C4⋊D4), C2.1(D4.D4), (C22×C8).10C22, C22.46(C2×SD16), C4.31(C42⋊C2), C22.59(C8⋊C22), (C2×C42).247C22, C22.67(C22⋊Q8), C22.105(C4⋊D4), (C22×C4).1324C23, (C22×D4).456C22, C22.48(C8.C22), C2.11(C23.7Q8), C2.21(C23.36D4), C4.1(C2×C4⋊C4), (C2×C4⋊C8)⋊11C2, (C2×C4×D4).15C2, C4⋊C4.189(C2×C4), (C2×C4).259(C2×Q8), (C2×D4⋊C4).1C2, (C2×D4).205(C2×C4), (C2×C4).1314(C2×D4), C2.18(C2×D4⋊C4), (C2×C4⋊C4).32C22, (C2×C4).860(C4○D4), (C2×C4).362(C22×C4), (C2×C4).358(C22⋊C4), C22.247(C2×C22⋊C4), SmallGroup(128,534)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.98D4
G = < a,b,c,d | a4=b4=c4=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=a-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 404 in 186 conjugacy classes, 76 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C23×C4, C22×D4, C22.4Q16, C42⋊9C4, C2×D4⋊C4, C2×C4⋊C8, C2×C4×D4, C42.98D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.7Q8, C2×D4⋊C4, C23.36D4, C4⋊D8, D4.D4, D4⋊Q8, D4⋊2Q8, C42.98D4
►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 32 35 20)(10 21 36 25)(11 26 37 22)(12 23 38 27)(13 28 39 24)(14 17 40 29)(15 30 33 18)(16 19 34 31)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 12 39 54)(2 53 40 11)(3 10 33 52)(4 51 34 9)(5 16 35 50)(6 49 36 15)(7 14 37 56)(8 55 38 13)(17 26 64 45)(18 44 57 25)(19 32 58 43)(20 42 59 31)(21 30 60 41)(22 48 61 29)(23 28 62 47)(24 46 63 27)
(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)
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)(9,32,35,20)(10,21,36,25)(11,26,37,22)(12,23,38,27)(13,28,39,24)(14,17,40,29)(15,30,33,18)(16,19,34,31), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,12,39,54)(2,53,40,11)(3,10,33,52)(4,51,34,9)(5,16,35,50)(6,49,36,15)(7,14,37,56)(8,55,38,13)(17,26,64,45)(18,44,57,25)(19,32,58,43)(20,42,59,31)(21,30,60,41)(22,48,61,29)(23,28,62,47)(24,46,63,27), (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)>;
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)(9,32,35,20)(10,21,36,25)(11,26,37,22)(12,23,38,27)(13,28,39,24)(14,17,40,29)(15,30,33,18)(16,19,34,31), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,12,39,54)(2,53,40,11)(3,10,33,52)(4,51,34,9)(5,16,35,50)(6,49,36,15)(7,14,37,56)(8,55,38,13)(17,26,64,45)(18,44,57,25)(19,32,58,43)(20,42,59,31)(21,30,60,41)(22,48,61,29)(23,28,62,47)(24,46,63,27), (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) );
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),(9,32,35,20),(10,21,36,25),(11,26,37,22),(12,23,38,27),(13,28,39,24),(14,17,40,29),(15,30,33,18),(16,19,34,31)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,12,39,54),(2,53,40,11),(3,10,33,52),(4,51,34,9),(5,16,35,50),(6,49,36,15),(7,14,37,56),(8,55,38,13),(17,26,64,45),(18,44,57,25),(19,32,58,43),(20,42,59,31),(21,30,60,41),(22,48,61,29),(23,28,62,47),(24,46,63,27)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | Q8 | D8 | SD16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.98D4 | C22.4Q16 | C42⋊9C4 | C2×D4⋊C4 | C2×C4⋊C8 | C2×C4×D4 | C4×D4 | C42 | C22×C4 | C2×D4 | C2×D4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C42.98D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 7 | 4 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 15 |
| 0 | 0 | 0 | 0 | 9 | 6 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,7,0,0,0,0,0,4],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,6,7,0,0,0,0,2,11],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,11,9,0,0,0,0,15,6] >;
C42.98D4 in GAP, Magma, Sage, TeX
C_4^2._{98}D_4 % in TeX
G:=Group("C4^2.98D4"); // GroupNames label
G:=SmallGroup(128,534);
// by ID
G=gap.SmallGroup(128,534);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations