p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.258D4, C42.393C23, C8⋊Q8⋊11C2, C8.2D4⋊8C2, C8⋊8D4.4C2, C8.31(C4○D4), Q16⋊C4⋊16C2, C8.18D4⋊28C2, C4⋊C4.120C23, (C2×C8).281C23, (C2×C4).379C24, (C4×D4).99C22, (C22×C4).478D4, C23.398(C2×D4), C4⋊Q8.295C22, SD16⋊C4⋊24C2, (C4×Q8).96C22, C2.D8.98C22, C4.Q8.31C22, (C2×D4).133C23, (C2×Q16).66C22, (C2×Q8).121C23, C8⋊C4.136C22, C4⋊D4.176C22, (C22×C8).281C22, (C2×C42).865C22, (C2×SD16).27C22, C22.639(C22×D4), C22.1(C8.C22), C22⋊Q8.181C22, D4⋊C4.137C22, C2.45(D8⋊C22), (C22×C4).1575C23, Q8⋊C4.130C22, C4.4D4.147C22, C42.C2.124C22, C42.30C22⋊22C2, C42.28C22⋊36C2, C23.37C23⋊14C2, C23.36C23.23C2, C2.76(C22.26C24), (C2×C8⋊C4)⋊12C2, C4.64(C2×C4○D4), (C2×C4).703(C2×D4), C2.46(C2×C8.C22), SmallGroup(128,1913)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 316 in 183 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C8⋊C4 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×2], C2×Q16 [×2], C2×C8⋊C4, SD16⋊C4 [×2], Q16⋊C4 [×2], C8⋊8D4 [×2], C8.18D4 [×2], C42.28C22, C42.30C22, C8.2D4, C8⋊Q8, C23.36C23, C23.37C23, C42.258D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8.C22, D8⋊C22, C42.258D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=ab2, cbc-1=a2b, bd=db, dcd=c3 >
►On 64 points
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 50 13 54)(10 51 14 55)(11 52 15 56)(12 53 16 49)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)
(1 33 27 9)(2 38 28 14)(3 35 29 11)(4 40 30 16)(5 37 31 13)(6 34 32 10)(7 39 25 15)(8 36 26 12)(17 63 52 47)(18 60 53 44)(19 57 54 41)(20 62 55 46)(21 59 56 43)(22 64 49 48)(23 61 50 45)(24 58 51 42)
(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 3)(2 6)(5 7)(9 11)(10 14)(13 15)(17 50)(18 53)(19 56)(20 51)(21 54)(22 49)(23 52)(24 55)(25 31)(27 29)(28 32)(33 35)(34 38)(37 39)(41 59)(42 62)(43 57)(44 60)(45 63)(46 58)(47 61)(48 64)
G:=sub<Sym(64)| (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,50,13,54)(10,51,14,55)(11,52,15,56)(12,53,16,49)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (1,33,27,9)(2,38,28,14)(3,35,29,11)(4,40,30,16)(5,37,31,13)(6,34,32,10)(7,39,25,15)(8,36,26,12)(17,63,52,47)(18,60,53,44)(19,57,54,41)(20,62,55,46)(21,59,56,43)(22,64,49,48)(23,61,50,45)(24,58,51,42), (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), (1,3)(2,6)(5,7)(9,11)(10,14)(13,15)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(25,31)(27,29)(28,32)(33,35)(34,38)(37,39)(41,59)(42,62)(43,57)(44,60)(45,63)(46,58)(47,61)(48,64)>;
G:=Group( (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,50,13,54)(10,51,14,55)(11,52,15,56)(12,53,16,49)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (1,33,27,9)(2,38,28,14)(3,35,29,11)(4,40,30,16)(5,37,31,13)(6,34,32,10)(7,39,25,15)(8,36,26,12)(17,63,52,47)(18,60,53,44)(19,57,54,41)(20,62,55,46)(21,59,56,43)(22,64,49,48)(23,61,50,45)(24,58,51,42), (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), (1,3)(2,6)(5,7)(9,11)(10,14)(13,15)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(25,31)(27,29)(28,32)(33,35)(34,38)(37,39)(41,59)(42,62)(43,57)(44,60)(45,63)(46,58)(47,61)(48,64) );
G=PermutationGroup([(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,50,13,54),(10,51,14,55),(11,52,15,56),(12,53,16,49),(17,39,21,35),(18,40,22,36),(19,33,23,37),(20,34,24,38),(25,59,29,63),(26,60,30,64),(27,61,31,57),(28,62,32,58)], [(1,33,27,9),(2,38,28,14),(3,35,29,11),(4,40,30,16),(5,37,31,13),(6,34,32,10),(7,39,25,15),(8,36,26,12),(17,63,52,47),(18,60,53,44),(19,57,54,41),(20,62,55,46),(21,59,56,43),(22,64,49,48),(23,61,50,45),(24,58,51,42)], [(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)], [(1,3),(2,6),(5,7),(9,11),(10,14),(13,15),(17,50),(18,53),(19,56),(20,51),(21,54),(22,49),(23,52),(24,55),(25,31),(27,29),(28,32),(33,35),(34,38),(37,39),(41,59),(42,62),(43,57),(44,60),(45,63),(46,58),(47,61),(48,64)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 8 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,8,13,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8.C22 | D8⋊C22 |
| kernel | C42.258D4 | C2×C8⋊C4 | SD16⋊C4 | Q16⋊C4 | C8⋊8D4 | C8.18D4 | C42.28C22 | C42.30C22 | C8.2D4 | C8⋊Q8 | C23.36C23 | C23.37C23 | C42 | C22×C4 | C8 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{258}D_4 % in TeX
G:=Group("C4^2.258D4"); // GroupNames label
G:=SmallGroup(128,1913);
// by ID
G=gap.SmallGroup(128,1913);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,184,521,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations