p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.449D4, C42.336C23, C4○D4.9Q8, Q8.4(C2×Q8), D4.4(C2×Q8), Q8.Q8⋊13C2, D4.Q8⋊13C2, C4⋊C4.43C23, C4⋊C8.44C22, (C2×C8).27C23, C4.31(C22×Q8), C4⋊M4(2)⋊9C2, (C2×C4).278C24, (C22×C4).433D4, C23.660(C2×D4), C4.90(C22⋊Q8), C4.Q8.10C22, C2.D8.81C22, (C4×D4).318C22, (C2×D4).396C23, (C4×Q8).299C22, (C2×Q8).367C23, M4(2)⋊C4⋊18C2, D4⋊C4.25C22, (C2×C42).824C22, (C22×C4).997C23, Q8⋊C4.26C22, C23.36D4.2C2, C22.538(C22×D4), C22.55(C22⋊Q8), C2.20(D8⋊C22), (C2×M4(2)).67C22, C42.C2.104C22, C42⋊C2.315C22, C4.88(C2×C4○D4), (C4×C4○D4).25C2, (C2×C4).102(C2×Q8), C2.59(C2×C22⋊Q8), (C2×C4).1215(C2×D4), (C2×C42.C2)⋊32C2, (C2×C4).295(C4○D4), (C2×C4⋊C4).604C22, (C2×C4○D4).309C22, SmallGroup(128,1812)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 192 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×21], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×2], C2×M4(2) [×2], C2×C4○D4, C23.36D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], D4.Q8 [×4], Q8.Q8 [×4], C4×C4○D4, C2×C42.C2, C42.449D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D8⋊C22 [×2], C42.449D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2c3 >
►On 64 points
(1 51 45 63)(2 60 46 56)(3 53 47 57)(4 62 48 50)(5 55 41 59)(6 64 42 52)(7 49 43 61)(8 58 44 54)(9 27 40 17)(10 22 33 32)(11 29 34 19)(12 24 35 26)(13 31 36 21)(14 18 37 28)(15 25 38 23)(16 20 39 30)
(1 26 5 30)(2 31 6 27)(3 28 7 32)(4 25 8 29)(9 56 13 52)(10 53 14 49)(11 50 15 54)(12 55 16 51)(17 46 21 42)(18 43 22 47)(19 48 23 44)(20 45 24 41)(33 57 37 61)(34 62 38 58)(35 59 39 63)(36 64 40 60)
(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 48 41 8)(2 7 42 47)(3 46 43 6)(4 5 44 45)(9 14 36 33)(10 40 37 13)(11 12 38 39)(15 16 34 35)(17 22 31 28)(18 27 32 21)(19 20 25 26)(23 24 29 30)(49 60 57 52)(50 51 58 59)(53 64 61 56)(54 55 62 63)
G:=sub<Sym(64)| (1,51,45,63)(2,60,46,56)(3,53,47,57)(4,62,48,50)(5,55,41,59)(6,64,42,52)(7,49,43,61)(8,58,44,54)(9,27,40,17)(10,22,33,32)(11,29,34,19)(12,24,35,26)(13,31,36,21)(14,18,37,28)(15,25,38,23)(16,20,39,30), (1,26,5,30)(2,31,6,27)(3,28,7,32)(4,25,8,29)(9,56,13,52)(10,53,14,49)(11,50,15,54)(12,55,16,51)(17,46,21,42)(18,43,22,47)(19,48,23,44)(20,45,24,41)(33,57,37,61)(34,62,38,58)(35,59,39,63)(36,64,40,60), (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), (1,48,41,8)(2,7,42,47)(3,46,43,6)(4,5,44,45)(9,14,36,33)(10,40,37,13)(11,12,38,39)(15,16,34,35)(17,22,31,28)(18,27,32,21)(19,20,25,26)(23,24,29,30)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63)>;
G:=Group( (1,51,45,63)(2,60,46,56)(3,53,47,57)(4,62,48,50)(5,55,41,59)(6,64,42,52)(7,49,43,61)(8,58,44,54)(9,27,40,17)(10,22,33,32)(11,29,34,19)(12,24,35,26)(13,31,36,21)(14,18,37,28)(15,25,38,23)(16,20,39,30), (1,26,5,30)(2,31,6,27)(3,28,7,32)(4,25,8,29)(9,56,13,52)(10,53,14,49)(11,50,15,54)(12,55,16,51)(17,46,21,42)(18,43,22,47)(19,48,23,44)(20,45,24,41)(33,57,37,61)(34,62,38,58)(35,59,39,63)(36,64,40,60), (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), (1,48,41,8)(2,7,42,47)(3,46,43,6)(4,5,44,45)(9,14,36,33)(10,40,37,13)(11,12,38,39)(15,16,34,35)(17,22,31,28)(18,27,32,21)(19,20,25,26)(23,24,29,30)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63) );
G=PermutationGroup([(1,51,45,63),(2,60,46,56),(3,53,47,57),(4,62,48,50),(5,55,41,59),(6,64,42,52),(7,49,43,61),(8,58,44,54),(9,27,40,17),(10,22,33,32),(11,29,34,19),(12,24,35,26),(13,31,36,21),(14,18,37,28),(15,25,38,23),(16,20,39,30)], [(1,26,5,30),(2,31,6,27),(3,28,7,32),(4,25,8,29),(9,56,13,52),(10,53,14,49),(11,50,15,54),(12,55,16,51),(17,46,21,42),(18,43,22,47),(19,48,23,44),(20,45,24,41),(33,57,37,61),(34,62,38,58),(35,59,39,63),(36,64,40,60)], [(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)], [(1,48,41,8),(2,7,42,47),(3,46,43,6),(4,5,44,45),(9,14,36,33),(10,40,37,13),(11,12,38,39),(15,16,34,35),(17,22,31,28),(18,27,32,21),(19,20,25,26),(23,24,29,30),(49,60,57,52),(50,51,58,59),(53,64,61,56),(54,55,62,63)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 | 7 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,13,0,0,0,0,0,0,13,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,7,5,0,0,0,0,7,0,0,0,0,0,0,0,10,12,0,0,0,0,10,0],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,7,5,0,0,0,0,7,10,0,0,0,0,0,0,10,12,0,0,0,0,10,7] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | D8⋊C22 |
| kernel | C42.449D4 | C23.36D4 | C4⋊M4(2) | M4(2)⋊C4 | D4.Q8 | Q8.Q8 | C4×C4○D4 | C2×C42.C2 | C42 | C22×C4 | C4○D4 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{449}D_4 % in TeX
G:=Group("C4^2.449D4"); // GroupNames label
G:=SmallGroup(128,1812);
// by ID
G=gap.SmallGroup(128,1812);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,2019,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;
// generators/relations