p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.261D4, C42.726C23, C8⋊3(C4○D4), C8⋊2D4⋊8C2, C8⋊3Q8⋊7C2, C8⋊4D4⋊20C2, D8⋊C4⋊18C2, C4.4D8⋊44C2, C4⋊C4.123C23, C4.24(C8⋊C22), (C4×M4(2))⋊11C2, (C2×C4).382C24, (C2×C8).463C23, (C4×C8).187C22, (C2×D8).67C22, C23.269(C2×D4), (C22×C4).480D4, C4⋊Q8.297C22, C4.Q8.33C22, (C4×D4).102C22, (C2×D4).136C23, C8⋊C4.139C22, C4⋊1D4.159C22, C4⋊D4.179C22, (C2×C42).868C22, C22.642(C22×D4), D4⋊C4.140C22, (C22×C4).1060C23, C22.26C24⋊15C2, (C2×M4(2)).290C22, C2.79(C22.26C24), C4.67(C2×C4○D4), C2.48(C2×C8⋊C22), (C2×C4).1225(C2×D4), SmallGroup(128,1916)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 484 in 223 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×6], C4 [×7], C22, C22 [×15], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×24], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], C4.Q8 [×4], C2×C42, C4×D4 [×4], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×D8 [×4], C2×C4○D4 [×2], C4×M4(2), D8⋊C4 [×4], C8⋊2D4 [×4], C4.4D8 [×2], C8⋊4D4, C8⋊3Q8, C22.26C24 [×2], C42.261D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8⋊C22 [×2], C42.261D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=a2b, bd=db, dcd=a2c3 >
►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 35 21 39)(18 40 22 36)(19 37 23 33)(20 34 24 38)(25 59 29 63)(26 64 30 60)(27 61 31 57)(28 58 32 62)
(1 40 31 12)(2 37 32 9)(3 34 25 14)(4 39 26 11)(5 36 27 16)(6 33 28 13)(7 38 29 10)(8 35 30 15)(17 64 49 47)(18 61 50 44)(19 58 51 41)(20 63 52 46)(21 60 53 43)(22 57 54 48)(23 62 55 45)(24 59 56 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 53)(2 52)(3 51)(4 50)(5 49)(6 56)(7 55)(8 54)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 64)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 42)(34 41)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)
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)(17,35,21,39)(18,40,22,36)(19,37,23,33)(20,34,24,38)(25,59,29,63)(26,64,30,60)(27,61,31,57)(28,58,32,62), (1,40,31,12)(2,37,32,9)(3,34,25,14)(4,39,26,11)(5,36,27,16)(6,33,28,13)(7,38,29,10)(8,35,30,15)(17,64,49,47)(18,61,50,44)(19,58,51,41)(20,63,52,46)(21,60,53,43)(22,57,54,48)(23,62,55,45)(24,59,56,42), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,56)(7,55)(8,54)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,64)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)(33,42)(34,41)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)>;
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)(17,35,21,39)(18,40,22,36)(19,37,23,33)(20,34,24,38)(25,59,29,63)(26,64,30,60)(27,61,31,57)(28,58,32,62), (1,40,31,12)(2,37,32,9)(3,34,25,14)(4,39,26,11)(5,36,27,16)(6,33,28,13)(7,38,29,10)(8,35,30,15)(17,64,49,47)(18,61,50,44)(19,58,51,41)(20,63,52,46)(21,60,53,43)(22,57,54,48)(23,62,55,45)(24,59,56,42), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,56)(7,55)(8,54)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,64)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)(33,42)(34,41)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43) );
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),(17,35,21,39),(18,40,22,36),(19,37,23,33),(20,34,24,38),(25,59,29,63),(26,64,30,60),(27,61,31,57),(28,58,32,62)], [(1,40,31,12),(2,37,32,9),(3,34,25,14),(4,39,26,11),(5,36,27,16),(6,33,28,13),(7,38,29,10),(8,35,30,15),(17,64,49,47),(18,61,50,44),(19,58,51,41),(20,63,52,46),(21,60,53,43),(22,57,54,48),(23,62,55,45),(24,59,56,42)], [(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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,56),(7,55),(8,54),(9,63),(10,62),(11,61),(12,60),(13,59),(14,58),(15,57),(16,64),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28),(33,42),(34,41),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 13 | 4 | 16 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 1 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 4 | 13 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 6 |
| 0 | 0 | 12 | 5 | 6 | 11 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 11 | 5 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 11 | 6 |
| 0 | 0 | 12 | 5 | 0 | 11 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 3 | 11 | 5 | 7 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,1,0,0,0,0,4,0,0,0,0,16,16,0,0,0,0,0,2,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,16,0,0,13,0,16,0,0,0,1,16,0,4,0,0,15,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,5,12,3,3,0,0,12,5,14,11,0,0,0,6,0,5,0,0,6,11,0,7],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,5,12,14,3,0,0,12,5,14,11,0,0,11,0,0,5,0,0,6,11,0,7] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | 4L | 4M | 4N | 4O | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 |
| kernel | C42.261D4 | C4×M4(2) | D8⋊C4 | C8⋊2D4 | C4.4D8 | C8⋊4D4 | C8⋊3Q8 | C22.26C24 | C42 | C22×C4 | C8 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{261}D_4 % in TeX
G:=Group("C4^2.261D4"); // GroupNames label
G:=SmallGroup(128,1916);
// by ID
G=gap.SmallGroup(128,1916);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,520,521,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=a^2*c^3>;
// generators/relations