p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.279D4, C42.413C23, C4.512- (1+4), C8⋊3Q8⋊20C2, D4⋊2Q8⋊35C2, C4.4D8⋊28C2, (C2×C4).57SD16, C4.72(C2×SD16), C4⋊C8.337C22, C4⋊C4.166C23, C4.28(C8⋊C22), (C4×C8).267C22, (C2×C4).425C24, (C2×C8).331C23, C23.698(C2×D4), (C22×C4).508D4, C4⋊Q8.309C22, C4.Q8.83C22, (C4×D4).112C22, (C2×D4).174C23, C22.39(C2×SD16), C2.24(C22×SD16), C42.12C4⋊44C2, C4⋊1D4.170C22, C23.46D4⋊30C2, C4⋊D4.197C22, C22⋊C8.220C22, (C2×C42).886C22, C22.685(C22×D4), D4⋊C4.110C22, (C22×C4).1090C23, C22.26C24.44C2, C2.73(C23.38C23), (C2×C4⋊Q8)⋊43C2, (C2×C4).869(C2×D4), C2.59(C2×C8⋊C22), (C2×C4⋊C4).646C22, SmallGroup(128,1959)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 412 in 202 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 [×10], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×9], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C42.12C4, D4⋊2Q8 [×4], C23.46D4 [×4], C4.4D8 [×2], C8⋊3Q8 [×2], C2×C4⋊Q8, C22.26C24, C42.279D4
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], C23.38C23, C22×SD16, C2×C8⋊C22, C42.279D4
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=a2b2c3 >
►On 64 points
(1 41 25 59)(2 42 26 60)(3 43 27 61)(4 44 28 62)(5 45 29 63)(6 46 30 64)(7 47 31 57)(8 48 32 58)(9 52 35 21)(10 53 36 22)(11 54 37 23)(12 55 38 24)(13 56 39 17)(14 49 40 18)(15 50 33 19)(16 51 34 20)
(1 15 5 11)(2 38 6 34)(3 9 7 13)(4 40 8 36)(10 28 14 32)(12 30 16 26)(17 61 21 57)(18 48 22 44)(19 63 23 59)(20 42 24 46)(25 33 29 37)(27 35 31 39)(41 50 45 54)(43 52 47 56)(49 58 53 62)(51 60 55 64)
(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 32)(3 7)(4 30)(6 28)(8 26)(9 39)(10 12)(11 37)(13 35)(14 16)(15 33)(17 21)(18 51)(20 49)(22 55)(24 53)(27 31)(34 40)(36 38)(41 59)(42 48)(43 57)(44 46)(45 63)(47 61)(52 56)(58 60)(62 64)
G:=sub<Sym(64)| (1,41,25,59)(2,42,26,60)(3,43,27,61)(4,44,28,62)(5,45,29,63)(6,46,30,64)(7,47,31,57)(8,48,32,58)(9,52,35,21)(10,53,36,22)(11,54,37,23)(12,55,38,24)(13,56,39,17)(14,49,40,18)(15,50,33,19)(16,51,34,20), (1,15,5,11)(2,38,6,34)(3,9,7,13)(4,40,8,36)(10,28,14,32)(12,30,16,26)(17,61,21,57)(18,48,22,44)(19,63,23,59)(20,42,24,46)(25,33,29,37)(27,35,31,39)(41,50,45,54)(43,52,47,56)(49,58,53,62)(51,60,55,64), (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), (2,32)(3,7)(4,30)(6,28)(8,26)(9,39)(10,12)(11,37)(13,35)(14,16)(15,33)(17,21)(18,51)(20,49)(22,55)(24,53)(27,31)(34,40)(36,38)(41,59)(42,48)(43,57)(44,46)(45,63)(47,61)(52,56)(58,60)(62,64)>;
G:=Group( (1,41,25,59)(2,42,26,60)(3,43,27,61)(4,44,28,62)(5,45,29,63)(6,46,30,64)(7,47,31,57)(8,48,32,58)(9,52,35,21)(10,53,36,22)(11,54,37,23)(12,55,38,24)(13,56,39,17)(14,49,40,18)(15,50,33,19)(16,51,34,20), (1,15,5,11)(2,38,6,34)(3,9,7,13)(4,40,8,36)(10,28,14,32)(12,30,16,26)(17,61,21,57)(18,48,22,44)(19,63,23,59)(20,42,24,46)(25,33,29,37)(27,35,31,39)(41,50,45,54)(43,52,47,56)(49,58,53,62)(51,60,55,64), (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), (2,32)(3,7)(4,30)(6,28)(8,26)(9,39)(10,12)(11,37)(13,35)(14,16)(15,33)(17,21)(18,51)(20,49)(22,55)(24,53)(27,31)(34,40)(36,38)(41,59)(42,48)(43,57)(44,46)(45,63)(47,61)(52,56)(58,60)(62,64) );
G=PermutationGroup([(1,41,25,59),(2,42,26,60),(3,43,27,61),(4,44,28,62),(5,45,29,63),(6,46,30,64),(7,47,31,57),(8,48,32,58),(9,52,35,21),(10,53,36,22),(11,54,37,23),(12,55,38,24),(13,56,39,17),(14,49,40,18),(15,50,33,19),(16,51,34,20)], [(1,15,5,11),(2,38,6,34),(3,9,7,13),(4,40,8,36),(10,28,14,32),(12,30,16,26),(17,61,21,57),(18,48,22,44),(19,63,23,59),(20,42,24,46),(25,33,29,37),(27,35,31,39),(41,50,45,54),(43,52,47,56),(49,58,53,62),(51,60,55,64)], [(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)], [(2,32),(3,7),(4,30),(6,28),(8,26),(9,39),(10,12),(11,37),(13,35),(14,16),(15,33),(17,21),(18,51),(20,49),(22,55),(24,53),(27,31),(34,40),(36,38),(41,59),(42,48),(43,57),(44,46),(45,63),(47,61),(52,56),(58,60),(62,64)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 15 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 | 2 | 2 |
| 0 | 0 | 6 | 4 | 15 | 2 |
| 0 | 0 | 4 | 11 | 13 | 6 |
| 0 | 0 | 6 | 4 | 11 | 13 |
| 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 | 1 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,2,0,1,0,0,15,0,16,0],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,4,6,4,6,0,0,11,4,11,4,0,0,2,15,13,11,0,0,2,2,6,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,16,0,0,0,0,16,0,0,0,0,0,0,1] >;
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 | C42.279D4 | C42.12C4 | D4⋊2Q8 | C23.46D4 | C4.4D8 | C8⋊3Q8 | 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._{279}D_4 % in TeX
G:=Group("C4^2.279D4"); // GroupNames label
G:=SmallGroup(128,1959);
// by ID
G=gap.SmallGroup(128,1959);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,436,675,4037,1027,124]);
// 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=a^2*b^2*c^3>;
// generators/relations