p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.264D4, C42.396C23, C4.982+ (1+4), (C2×C4)⋊6SD16, C8⋊5D4⋊20C2, Q8⋊D4⋊30C2, C4.71(C2×SD16), D4.D4⋊37C2, C4⋊C4.145C23, C4⋊C8.336C22, (C2×C4).404C24, (C4×C8).264C22, (C2×C8).324C23, C4.SD16⋊25C2, C23.689(C2×D4), (C22×C4).494D4, C4⋊Q8.299C22, (C4×D4).103C22, (C2×D4).154C23, C4.25(C8.C22), (C2×Q8).141C23, C22.38(C2×SD16), C2.22(C22×SD16), C42.12C4⋊42C2, C4⋊1D4.160C22, C4⋊D4.187C22, C22⋊C8.218C22, (C2×C42).871C22, Q8⋊C4.99C22, (C2×SD16).83C22, C22.664(C22×D4), (C22×C4).1075C23, (C22×Q8).319C22, C2.75(C22.29C24), C22.26C24.40C2, (C2×C4⋊Q8)⋊40C2, (C2×C4).865(C2×D4), C2.53(C2×C8.C22), SmallGroup(128,1938)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 444 in 214 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×14], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×7], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×SD16 [×8], C22×Q8 [×2], C2×C4○D4, C42.12C4, Q8⋊D4 [×4], D4.D4 [×4], C4.SD16 [×2], C8⋊5D4 [×2], C2×C4⋊Q8, C22.26C24, C42.264D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8.C22 [×2], C22×D4, 2+ (1+4) [×2], C22.29C24, C22×SD16, C2×C8.C22, C42.264D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=c3 >
►On 64 points
(1 63 25 55)(2 64 26 56)(3 57 27 49)(4 58 28 50)(5 59 29 51)(6 60 30 52)(7 61 31 53)(8 62 32 54)(9 18 40 47)(10 19 33 48)(11 20 34 41)(12 21 35 42)(13 22 36 43)(14 23 37 44)(15 24 38 45)(16 17 39 46)
(1 37 5 33)(2 11 6 15)(3 39 7 35)(4 13 8 9)(10 25 14 29)(12 27 16 31)(17 53 21 49)(18 58 22 62)(19 55 23 51)(20 60 24 64)(26 34 30 38)(28 36 32 40)(41 52 45 56)(42 57 46 61)(43 54 47 50)(44 59 48 63)
(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 4)(3 7)(6 8)(9 38)(10 33)(11 36)(12 39)(13 34)(14 37)(15 40)(16 35)(17 21)(18 24)(20 22)(26 28)(27 31)(30 32)(41 43)(42 46)(45 47)(49 61)(50 64)(51 59)(52 62)(53 57)(54 60)(55 63)(56 58)
G:=sub<Sym(64)| (1,63,25,55)(2,64,26,56)(3,57,27,49)(4,58,28,50)(5,59,29,51)(6,60,30,52)(7,61,31,53)(8,62,32,54)(9,18,40,47)(10,19,33,48)(11,20,34,41)(12,21,35,42)(13,22,36,43)(14,23,37,44)(15,24,38,45)(16,17,39,46), (1,37,5,33)(2,11,6,15)(3,39,7,35)(4,13,8,9)(10,25,14,29)(12,27,16,31)(17,53,21,49)(18,58,22,62)(19,55,23,51)(20,60,24,64)(26,34,30,38)(28,36,32,40)(41,52,45,56)(42,57,46,61)(43,54,47,50)(44,59,48,63), (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), (2,4)(3,7)(6,8)(9,38)(10,33)(11,36)(12,39)(13,34)(14,37)(15,40)(16,35)(17,21)(18,24)(20,22)(26,28)(27,31)(30,32)(41,43)(42,46)(45,47)(49,61)(50,64)(51,59)(52,62)(53,57)(54,60)(55,63)(56,58)>;
G:=Group( (1,63,25,55)(2,64,26,56)(3,57,27,49)(4,58,28,50)(5,59,29,51)(6,60,30,52)(7,61,31,53)(8,62,32,54)(9,18,40,47)(10,19,33,48)(11,20,34,41)(12,21,35,42)(13,22,36,43)(14,23,37,44)(15,24,38,45)(16,17,39,46), (1,37,5,33)(2,11,6,15)(3,39,7,35)(4,13,8,9)(10,25,14,29)(12,27,16,31)(17,53,21,49)(18,58,22,62)(19,55,23,51)(20,60,24,64)(26,34,30,38)(28,36,32,40)(41,52,45,56)(42,57,46,61)(43,54,47,50)(44,59,48,63), (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), (2,4)(3,7)(6,8)(9,38)(10,33)(11,36)(12,39)(13,34)(14,37)(15,40)(16,35)(17,21)(18,24)(20,22)(26,28)(27,31)(30,32)(41,43)(42,46)(45,47)(49,61)(50,64)(51,59)(52,62)(53,57)(54,60)(55,63)(56,58) );
G=PermutationGroup([(1,63,25,55),(2,64,26,56),(3,57,27,49),(4,58,28,50),(5,59,29,51),(6,60,30,52),(7,61,31,53),(8,62,32,54),(9,18,40,47),(10,19,33,48),(11,20,34,41),(12,21,35,42),(13,22,36,43),(14,23,37,44),(15,24,38,45),(16,17,39,46)], [(1,37,5,33),(2,11,6,15),(3,39,7,35),(4,13,8,9),(10,25,14,29),(12,27,16,31),(17,53,21,49),(18,58,22,62),(19,55,23,51),(20,60,24,64),(26,34,30,38),(28,36,32,40),(41,52,45,56),(42,57,46,61),(43,54,47,50),(44,59,48,63)], [(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)], [(2,4),(3,7),(6,8),(9,38),(10,33),(11,36),(12,39),(13,34),(14,37),(15,40),(16,35),(17,21),(18,24),(20,22),(26,28),(27,31),(30,32),(41,43),(42,46),(45,47),(49,61),(50,64),(51,59),(52,62),(53,57),(54,60),(55,63),(56,58)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 6 |
| 0 | 0 | 1 | 0 | 6 | 0 |
| 0 | 0 | 0 | 11 | 0 | 16 |
| 0 | 0 | 11 | 0 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,11,0,0,1,0,11,0,0,0,0,6,0,16,0,0,6,0,16,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C8.C22 | 2+ (1+4) |
| kernel | C42.264D4 | C42.12C4 | Q8⋊D4 | D4.D4 | C4.SD16 | C8⋊5D4 | C2×C4⋊Q8 | C22.26C24 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{264}D_4 % in TeX
G:=Group("C4^2.264D4"); // GroupNames label
G:=SmallGroup(128,1938);
// by ID
G=gap.SmallGroup(128,1938);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,352,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations