p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.229D4, C42.345C23, D8⋊C4⋊6C2, Q8.Q8⋊17C2, D4.Q8⋊17C2, Q16⋊C4⋊5C2, D4.6(C4○D4), D4⋊D4.1C2, C4⋊C4.64C23, C4⋊C8.48C22, (C2×C8).38C23, Q8.5(C4○D4), SD16⋊C4⋊6C2, D4.2D4⋊18C2, D4.7D4⋊19C2, C8⋊C4.5C22, (C2×C4).309C24, C42.6C4⋊3C2, Q8.D4⋊18C2, (C4×D4).76C22, (C2×D8).58C22, C23.252(C2×D4), (C22×C4).449D4, (C4×Q8).73C22, C2.D8.86C22, C4.Q8.15C22, (C2×D4).404C23, C22⋊C8.22C22, (C2×Q16).56C22, (C2×Q8).376C23, D4⋊C4.30C22, C4⋊D4.165C22, C23.19D4⋊17C2, C23.20D4⋊17C2, (C2×C42).836C22, Q8⋊C4.30C22, (C2×SD16).11C22, C22.569(C22×D4), C22⋊Q8.170C22, C2.30(D8⋊C22), C23.36C23⋊3C2, (C22×C4).1025C23, C4.4D4.130C22, C42.C2.107C22, C42⋊C2.322C22, C2.110(C22.19C24), (C4×C4○D4)⋊11C2, C4.194(C2×C4○D4), (C2×C4).1219(C2×D4), (C2×C4○D4).313C22, SmallGroup(128,1843)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.229D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=ab2, cbc-1=a2b, bd=db, dcd=a2c3 >
Subgroups: 356 in 196 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C42.6C4, SD16⋊C4, Q16⋊C4, D8⋊C4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C4×C4○D4, C23.36C23, C42.229D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.19C24, D8⋊C22, C42.229D4
►On 64 points
(1 57 5 61)(2 55 6 51)(3 59 7 63)(4 49 8 53)(9 47 13 43)(10 18 14 22)(11 41 15 45)(12 20 16 24)(17 34 21 38)(19 36 23 40)(25 50 29 54)(26 62 30 58)(27 52 31 56)(28 64 32 60)(33 42 37 46)(35 44 39 48)
(1 39 29 10)(2 36 30 15)(3 33 31 12)(4 38 32 9)(5 35 25 14)(6 40 26 11)(7 37 27 16)(8 34 28 13)(17 60 47 49)(18 57 48 54)(19 62 41 51)(20 59 42 56)(21 64 43 53)(22 61 44 50)(23 58 45 55)(24 63 46 52)
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 41)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)
G:=sub<Sym(64)| (1,57,5,61)(2,55,6,51)(3,59,7,63)(4,49,8,53)(9,47,13,43)(10,18,14,22)(11,41,15,45)(12,20,16,24)(17,34,21,38)(19,36,23,40)(25,50,29,54)(26,62,30,58)(27,52,31,56)(28,64,32,60)(33,42,37,46)(35,44,39,48), (1,39,29,10)(2,36,30,15)(3,33,31,12)(4,38,32,9)(5,35,25,14)(6,40,26,11)(7,37,27,16)(8,34,28,13)(17,60,47,49)(18,57,48,54)(19,62,41,51)(20,59,42,56)(21,64,43,53)(22,61,44,50)(23,58,45,55)(24,63,46,52), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,41)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63)>;
G:=Group( (1,57,5,61)(2,55,6,51)(3,59,7,63)(4,49,8,53)(9,47,13,43)(10,18,14,22)(11,41,15,45)(12,20,16,24)(17,34,21,38)(19,36,23,40)(25,50,29,54)(26,62,30,58)(27,52,31,56)(28,64,32,60)(33,42,37,46)(35,44,39,48), (1,39,29,10)(2,36,30,15)(3,33,31,12)(4,38,32,9)(5,35,25,14)(6,40,26,11)(7,37,27,16)(8,34,28,13)(17,60,47,49)(18,57,48,54)(19,62,41,51)(20,59,42,56)(21,64,43,53)(22,61,44,50)(23,58,45,55)(24,63,46,52), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,41)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63) );
G=PermutationGroup([[(1,57,5,61),(2,55,6,51),(3,59,7,63),(4,49,8,53),(9,47,13,43),(10,18,14,22),(11,41,15,45),(12,20,16,24),(17,34,21,38),(19,36,23,40),(25,50,29,54),(26,62,30,58),(27,52,31,56),(28,64,32,60),(33,42,37,46),(35,44,39,48)], [(1,39,29,10),(2,36,30,15),(3,33,31,12),(4,38,32,9),(5,35,25,14),(6,40,26,11),(7,37,27,16),(8,34,28,13),(17,60,47,49),(18,57,48,54),(19,62,41,51),(20,59,42,56),(21,64,43,53),(22,61,44,50),(23,58,45,55),(24,63,46,52)], [(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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,41),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(26,32),(27,31),(28,30),(33,37),(34,36),(38,40),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,64),(56,63)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4J | 4K | ··· | 4Q | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | D8⋊C22 |
| kernel | C42.229D4 | C42.6C4 | SD16⋊C4 | Q16⋊C4 | D8⋊C4 | D4⋊D4 | D4.7D4 | D4.2D4 | Q8.D4 | D4.Q8 | Q8.Q8 | C23.19D4 | C23.20D4 | C4×C4○D4 | C23.36C23 | C42 | C22×C4 | D4 | Q8 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.229D4 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 6 |
| 0 | 0 | 16 | 2 | 14 | 6 |
| 0 | 0 | 0 | 11 | 0 | 15 |
| 0 | 0 | 3 | 11 | 1 | 15 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,3,0,0,2,2,11,11,0,0,0,14,0,1,0,0,6,6,15,15],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
C42.229D4 in GAP, Magma, Sage, TeX
C_4^2._{229}D_4 % in TeX
G:=Group("C4^2.229D4"); // GroupNames label
G:=SmallGroup(128,1843);
// by ID
G=gap.SmallGroup(128,1843);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=a^2*c^3>;
// generators/relations