p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.256D4, C42.384C23, C4○3(C8⋊Q8), C8⋊Q8⋊38C2, C4○3(C8⋊D4), (C4×Q16)⋊29C2, C8.7(C4○D4), C8⋊D4.4C2, (C4×SD16)⋊16C2, C4○2(C8.D4), C4○2(C8.2D4), C8.D4⋊36C2, C8.2D4⋊25C2, (C4×M4(2))⋊8C2, C4⋊C4.111C23, (C2×C4).370C24, (C2×C8).277C23, (C4×C8).182C22, (C4×D4).91C22, (C22×C4).476D4, C23.266(C2×D4), C4⋊Q8.293C22, (C4×Q8).88C22, (C2×D4).125C23, (C2×Q8).113C23, C8⋊C4.127C22, C4.Q8.135C22, C2.D8.219C22, C4⋊D4.174C22, C4.144(C8.C22), (C2×C42).863C22, (C2×Q16).158C22, C22.630(C22×D4), C22⋊Q8.179C22, D4⋊C4.203C22, C2.43(D8⋊C22), (C22×C4).1050C23, Q8⋊C4.205C22, (C2×SD16).117C22, C4.4D4.145C22, C42.C2.122C22, C4○3(C42.28C22), C4○2(C42.30C22), C42.30C22⋊24C2, C42.28C22⋊39C2, C23.37C23⋊13C2, (C2×M4(2)).280C22, C23.36C23.22C2, C2.67(C22.26C24), C4.55(C2×C4○D4), (C2×C4).523(C2×D4), C2.45(C2×C8.C22), SmallGroup(128,1904)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 316 in 182 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], 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], M4(2) [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C4×C8 [×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], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C4×M4(2), C4×SD16 [×2], C4×Q16 [×2], C8⋊D4 [×2], C8.D4 [×2], C42.28C22, C42.30C22, C8.2D4, C8⋊Q8, C23.36C23, C23.37C23, C42.256D4
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, C42.256D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c3 >
►On 64 points
(1 48 5 44)(2 45 6 41)(3 42 7 46)(4 47 8 43)(9 55 13 51)(10 52 14 56)(11 49 15 53)(12 54 16 50)(17 34 21 38)(18 39 22 35)(19 36 23 40)(20 33 24 37)(25 63 29 59)(26 60 30 64)(27 57 31 61)(28 62 32 58)
(1 40 27 14)(2 33 28 15)(3 34 29 16)(4 35 30 9)(5 36 31 10)(6 37 32 11)(7 38 25 12)(8 39 26 13)(17 63 54 46)(18 64 55 47)(19 57 56 48)(20 58 49 41)(21 59 50 42)(22 60 51 43)(23 61 52 44)(24 62 53 45)
(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 13)(10 16)(12 14)(17 52)(18 55)(19 50)(20 53)(21 56)(22 51)(23 54)(24 49)(25 31)(27 29)(28 32)(34 36)(35 39)(38 40)(41 58)(42 61)(43 64)(44 59)(45 62)(46 57)(47 60)(48 63)
G:=sub<Sym(64)| (1,48,5,44)(2,45,6,41)(3,42,7,46)(4,47,8,43)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,63,29,59)(26,60,30,64)(27,57,31,61)(28,62,32,58), (1,40,27,14)(2,33,28,15)(3,34,29,16)(4,35,30,9)(5,36,31,10)(6,37,32,11)(7,38,25,12)(8,39,26,13)(17,63,54,46)(18,64,55,47)(19,57,56,48)(20,58,49,41)(21,59,50,42)(22,60,51,43)(23,61,52,44)(24,62,53,45), (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)(9,13)(10,16)(12,14)(17,52)(18,55)(19,50)(20,53)(21,56)(22,51)(23,54)(24,49)(25,31)(27,29)(28,32)(34,36)(35,39)(38,40)(41,58)(42,61)(43,64)(44,59)(45,62)(46,57)(47,60)(48,63)>;
G:=Group( (1,48,5,44)(2,45,6,41)(3,42,7,46)(4,47,8,43)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,63,29,59)(26,60,30,64)(27,57,31,61)(28,62,32,58), (1,40,27,14)(2,33,28,15)(3,34,29,16)(4,35,30,9)(5,36,31,10)(6,37,32,11)(7,38,25,12)(8,39,26,13)(17,63,54,46)(18,64,55,47)(19,57,56,48)(20,58,49,41)(21,59,50,42)(22,60,51,43)(23,61,52,44)(24,62,53,45), (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)(9,13)(10,16)(12,14)(17,52)(18,55)(19,50)(20,53)(21,56)(22,51)(23,54)(24,49)(25,31)(27,29)(28,32)(34,36)(35,39)(38,40)(41,58)(42,61)(43,64)(44,59)(45,62)(46,57)(47,60)(48,63) );
G=PermutationGroup([(1,48,5,44),(2,45,6,41),(3,42,7,46),(4,47,8,43),(9,55,13,51),(10,52,14,56),(11,49,15,53),(12,54,16,50),(17,34,21,38),(18,39,22,35),(19,36,23,40),(20,33,24,37),(25,63,29,59),(26,60,30,64),(27,57,31,61),(28,62,32,58)], [(1,40,27,14),(2,33,28,15),(3,34,29,16),(4,35,30,9),(5,36,31,10),(6,37,32,11),(7,38,25,12),(8,39,26,13),(17,63,54,46),(18,64,55,47),(19,57,56,48),(20,58,49,41),(21,59,50,42),(22,60,51,43),(23,61,52,44),(24,62,53,45)], [(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),(9,13),(10,16),(12,14),(17,52),(18,55),(19,50),(20,53),(21,56),(22,51),(23,54),(24,49),(25,31),(27,29),(28,32),(34,36),(35,39),(38,40),(41,58),(42,61),(43,64),(44,59),(45,62),(46,57),(47,60),(48,63)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 6 | 4 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 3 | 15 | 15 |
| 0 | 0 | 14 | 1 | 0 | 2 |
| 0 | 0 | 5 | 5 | 4 | 8 |
| 0 | 0 | 0 | 12 | 14 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 13 | 9 |
| 0 | 0 | 4 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 14 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [6,4,0,0,0,0,4,11,0,0,0,0,0,0,16,14,5,0,0,0,3,1,5,12,0,0,15,0,4,14,0,0,15,2,8,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,13,4,0,0,4,0,13,0,0,0,0,0,13,4,0,0,0,0,9,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,1,16,0,0,0,0,0,15,0,1],[1,14,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 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 | C42.256D4 | C4×M4(2) | C4×SD16 | C4×Q16 | C8⋊D4 | C8.D4 | C42.28C22 | C42.30C22 | C8.2D4 | C8⋊Q8 | C23.36C23 | C23.37C23 | C42 | C22×C4 | C8 | C4 | 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._{256}D_4 % in TeX
G:=Group("C4^2.256D4"); // GroupNames label
G:=SmallGroup(128,1904);
// by ID
G=gap.SmallGroup(128,1904);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,184,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations