p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.308D4, C42.722C23, (C4×D8)⋊7C2, (C4×Q16)⋊7C2, C4○2(C8⋊7D4), C4○3(C8⋊8D4), C8⋊8D4⋊57C2, (C4×SD16)⋊34C2, C8.56(C4○D4), C8⋊7D4.12C2, C4○2(C8.5Q8), C8.5Q8⋊27C2, C4○2(C8.12D4), C4.136(C4○D8), C4○2(C8.18D4), C8.12D4⋊28C2, C8.18D4⋊42C2, C4⋊C4.107C23, (C2×C8).599C23, (C2×C4).366C24, (C4×C8).432C22, (C4×D4).88C22, C22.3(C4○D8), C23.394(C2×D4), (C22×C4).568D4, (C4×Q8).85C22, (C2×D8).132C22, (C2×D4).122C23, (C2×Q8).110C23, C2.D8.181C22, C4.Q8.162C22, C4⋊D4.171C22, (C22×C8).570C22, (C2×Q16).128C22, C22.626(C22×D4), C22⋊Q8.176C22, D4⋊C4.146C22, (C22×C4).1571C23, C23.36C23⋊5C2, (C2×C42).1135C22, Q8⋊C4.138C22, (C2×SD16).150C22, C4.4D4.143C22, C42.C2.120C22, C4○2(C42.78C22), C42.78C22⋊33C2, C2.63(C22.26C24), (C2×C4×C8)⋊31C2, C2.35(C2×C4○D8), C4.51(C2×C4○D4), (C2×C4).699(C2×D4), SmallGroup(128,1900)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 340 in 190 conjugacy classes, 92 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4×C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C42⋊2C2 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C2×C4×C8, C4×D8, C4×SD16 [×2], C4×Q16, C8⋊8D4 [×2], C8⋊7D4, C8.18D4, C42.78C22 [×2], C8.12D4, C8.5Q8, C23.36C23 [×2], C42.308D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×4], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C4○D8 [×2], C42.308D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >
►On 64 points
(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)
(1 39 31 16)(2 40 32 9)(3 33 25 10)(4 34 26 11)(5 35 27 12)(6 36 28 13)(7 37 29 14)(8 38 30 15)(17 61 46 51)(18 62 47 52)(19 63 48 53)(20 64 41 54)(21 57 42 55)(22 58 43 56)(23 59 44 49)(24 60 45 50)
(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 42 5 46)(2 45 6 41)(3 48 7 44)(4 43 8 47)(9 60 13 64)(10 63 14 59)(11 58 15 62)(12 61 16 57)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)(33 53 37 49)(34 56 38 52)(35 51 39 55)(36 54 40 50)
G:=sub<Sym(64)| (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (1,39,31,16)(2,40,32,9)(3,33,25,10)(4,34,26,11)(5,35,27,12)(6,36,28,13)(7,37,29,14)(8,38,30,15)(17,61,46,51)(18,62,47,52)(19,63,48,53)(20,64,41,54)(21,57,42,55)(22,58,43,56)(23,59,44,49)(24,60,45,50), (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), (1,42,5,46)(2,45,6,41)(3,48,7,44)(4,43,8,47)(9,60,13,64)(10,63,14,59)(11,58,15,62)(12,61,16,57)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)(33,53,37,49)(34,56,38,52)(35,51,39,55)(36,54,40,50)>;
G:=Group( (1,49,5,53)(2,50,6,54)(3,51,7,55)(4,52,8,56)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (1,39,31,16)(2,40,32,9)(3,33,25,10)(4,34,26,11)(5,35,27,12)(6,36,28,13)(7,37,29,14)(8,38,30,15)(17,61,46,51)(18,62,47,52)(19,63,48,53)(20,64,41,54)(21,57,42,55)(22,58,43,56)(23,59,44,49)(24,60,45,50), (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), (1,42,5,46)(2,45,6,41)(3,48,7,44)(4,43,8,47)(9,60,13,64)(10,63,14,59)(11,58,15,62)(12,61,16,57)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)(33,53,37,49)(34,56,38,52)(35,51,39,55)(36,54,40,50) );
G=PermutationGroup([(1,49,5,53),(2,50,6,54),(3,51,7,55),(4,52,8,56),(9,45,13,41),(10,46,14,42),(11,47,15,43),(12,48,16,44),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,61,29,57),(26,62,30,58),(27,63,31,59),(28,64,32,60)], [(1,39,31,16),(2,40,32,9),(3,33,25,10),(4,34,26,11),(5,35,27,12),(6,36,28,13),(7,37,29,14),(8,38,30,15),(17,61,46,51),(18,62,47,52),(19,63,48,53),(20,64,41,54),(21,57,42,55),(22,58,43,56),(23,59,44,49),(24,60,45,50)], [(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)], [(1,42,5,46),(2,45,6,41),(3,48,7,44),(4,43,8,47),(9,60,13,64),(10,63,14,59),(11,58,15,62),(12,61,16,57),(17,31,21,27),(18,26,22,30),(19,29,23,25),(20,32,24,28),(33,53,37,49),(34,56,38,52),(35,51,39,55),(36,54,40,50)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 4 | 9 | 0 | 0 |
| 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 16 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [4,4,0,0,9,13,0,0,0,0,0,16,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[1,1,0,0,15,16,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,2,1,0,0,0,0,4,0,0,0,0,13] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | C4○D8 |
| kernel | C42.308D4 | C2×C4×C8 | C4×D8 | C4×SD16 | C4×Q16 | C8⋊8D4 | C8⋊7D4 | C8.18D4 | C42.78C22 | C8.12D4 | C8.5Q8 | C23.36C23 | C42 | C22×C4 | C8 | C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{308}D_4 % in TeX
G:=Group("C4^2.308D4"); // GroupNames label
G:=SmallGroup(128,1900);
// by ID
G=gap.SmallGroup(128,1900);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,184,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations