p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.241D4, C42.708C23, C4⋊C4.92C23, (C4×C8).351C22, (C2×C8).458C23, (C2×C4).337C24, C4.SD16⋊41C2, (C22×C4).461D4, C23.679(C2×D4), C4⋊Q8.275C22, (C2×Q8).92C23, C4.78(C4.4D4), C4.20(C8.C22), (C4×M4(2)).31C2, C8⋊C4.169C22, (C2×C42).848C22, C22.597(C22×D4), (C22×C4).1035C23, Q8⋊C4.125C22, C23.38D4.11C2, C22.32(C4.4D4), C42.C2.112C22, (C22×Q8).302C22, C42⋊C2.142C22, C42.30C22⋊18C2, (C2×M4(2)).374C22, C23.37C23.30C2, C4.46(C2×C4○D4), (C2×C4⋊Q8).47C2, (C2×C4).515(C2×D4), C2.48(C2×C4.4D4), C2.39(C2×C8.C22), (C2×C4).301(C4○D4), SmallGroup(128,1871)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.241D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >
Subgroups: 324 in 190 conjugacy classes, 96 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C4×M4(2), C23.38D4, C4.SD16, C42.30C22, C2×C4⋊Q8, C23.37C23, C42.241D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C8.C22, C22×D4, C2×C4○D4, C2×C4.4D4, C2×C8.C22, C42.241D4
►On 64 points
(1 34 25 13)(2 35 26 14)(3 36 27 15)(4 37 28 16)(5 38 29 9)(6 39 30 10)(7 40 31 11)(8 33 32 12)(17 62 46 51)(18 63 47 52)(19 64 48 53)(20 57 41 54)(21 58 42 55)(22 59 43 56)(23 60 44 49)(24 61 45 50)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)
(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 55 25 58)(2 61 26 50)(3 53 27 64)(4 59 28 56)(5 51 29 62)(6 57 30 54)(7 49 31 60)(8 63 32 52)(9 17 38 46)(10 41 39 20)(11 23 40 44)(12 47 33 18)(13 21 34 42)(14 45 35 24)(15 19 36 48)(16 43 37 22)
G:=sub<Sym(64)| (1,34,25,13)(2,35,26,14)(3,36,27,15)(4,37,28,16)(5,38,29,9)(6,39,30,10)(7,40,31,11)(8,33,32,12)(17,62,46,51)(18,63,47,52)(19,64,48,53)(20,57,41,54)(21,58,42,55)(22,59,43,56)(23,60,44,49)(24,61,45,50), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60), (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), (1,55,25,58)(2,61,26,50)(3,53,27,64)(4,59,28,56)(5,51,29,62)(6,57,30,54)(7,49,31,60)(8,63,32,52)(9,17,38,46)(10,41,39,20)(11,23,40,44)(12,47,33,18)(13,21,34,42)(14,45,35,24)(15,19,36,48)(16,43,37,22)>;
G:=Group( (1,34,25,13)(2,35,26,14)(3,36,27,15)(4,37,28,16)(5,38,29,9)(6,39,30,10)(7,40,31,11)(8,33,32,12)(17,62,46,51)(18,63,47,52)(19,64,48,53)(20,57,41,54)(21,58,42,55)(22,59,43,56)(23,60,44,49)(24,61,45,50), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60), (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), (1,55,25,58)(2,61,26,50)(3,53,27,64)(4,59,28,56)(5,51,29,62)(6,57,30,54)(7,49,31,60)(8,63,32,52)(9,17,38,46)(10,41,39,20)(11,23,40,44)(12,47,33,18)(13,21,34,42)(14,45,35,24)(15,19,36,48)(16,43,37,22) );
G=PermutationGroup([[(1,34,25,13),(2,35,26,14),(3,36,27,15),(4,37,28,16),(5,38,29,9),(6,39,30,10),(7,40,31,11),(8,33,32,12),(17,62,46,51),(18,63,47,52),(19,64,48,53),(20,57,41,54),(21,58,42,55),(22,59,43,56),(23,60,44,49),(24,61,45,50)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12),(17,23,21,19),(18,20,22,24),(25,27,29,31),(26,32,30,28),(33,39,37,35),(34,36,38,40),(41,43,45,47),(42,48,46,44),(49,55,53,51),(50,52,54,56),(57,59,61,63),(58,64,62,60)], [(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)], [(1,55,25,58),(2,61,26,50),(3,53,27,64),(4,59,28,56),(5,51,29,62),(6,57,30,54),(7,49,31,60),(8,63,32,52),(9,17,38,46),(10,41,39,20),(11,23,40,44),(12,47,33,18),(13,21,34,42),(14,45,35,24),(15,19,36,48),(16,43,37,22)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8.C22 |
| kernel | C42.241D4 | C4×M4(2) | C23.38D4 | C4.SD16 | C42.30C22 | C2×C4⋊Q8 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 8 | 4 |
Matrix representation of C42.241D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 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 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,16,0,0,0],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,4,0,0,0,0,0,0,13,0,0] >;
C42.241D4 in GAP, Magma, Sage, TeX
C_4^2._{241}D_4 % in TeX
G:=Group("C4^2.241D4"); // GroupNames label
G:=SmallGroup(128,1871);
// by ID
G=gap.SmallGroup(128,1871);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,1018,521,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations