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(C8.18D4), C4.136(C4○D8), C4○2(C8.12D4), 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, C4.Q8.162C22, C2.D8.181C22, C4⋊D4.171C22, (C22×C8).570C22, (C2×Q16).128C22, C22.626(C22×D4), C22⋊Q8.176C22, D4⋊C4.146C22, (C2×C42).1135C22, (C22×C4).1571C23, C23.36C23⋊5C2, 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, C4.51(C2×C4○D4), C2.35(C2×C4○D8), (C2×C4).699(C2×D4), SmallGroup(128,1900)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.308D4
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 >
Subgroups: 340 in 190 conjugacy classes, 92 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×D8, C2×SD16, C2×Q16, C2×C4×C8, C4×D8, C4×SD16, C4×Q16, C8⋊8D4, C8⋊7D4, C8.18D4, C42.78C22, C8.12D4, C8.5Q8, C23.36C23, C42.308D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, C22.26C24, C2×C4○D8, C42.308D4
►On 64 points
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(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 10)(2 40 32 11)(3 33 25 12)(4 34 26 13)(5 35 27 14)(6 36 28 15)(7 37 29 16)(8 38 30 9)(17 61 48 55)(18 62 41 56)(19 63 42 49)(20 64 43 50)(21 57 44 51)(22 58 45 52)(23 59 46 53)(24 60 47 54)
(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 44 5 48)(2 47 6 43)(3 42 7 46)(4 45 8 41)(9 62 13 58)(10 57 14 61)(11 60 15 64)(12 63 16 59)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)(33 49 37 53)(34 52 38 56)(35 55 39 51)(36 50 40 54)
G:=sub<Sym(64)| (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(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), (1,39,31,10)(2,40,32,11)(3,33,25,12)(4,34,26,13)(5,35,27,14)(6,36,28,15)(7,37,29,16)(8,38,30,9)(17,61,48,55)(18,62,41,56)(19,63,42,49)(20,64,43,50)(21,57,44,51)(22,58,45,52)(23,59,46,53)(24,60,47,54), (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), (1,44,5,48)(2,47,6,43)(3,42,7,46)(4,45,8,41)(9,62,13,58)(10,57,14,61)(11,60,15,64)(12,63,16,59)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)(33,49,37,53)(34,52,38,56)(35,55,39,51)(36,50,40,54)>;
G:=Group( (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(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), (1,39,31,10)(2,40,32,11)(3,33,25,12)(4,34,26,13)(5,35,27,14)(6,36,28,15)(7,37,29,16)(8,38,30,9)(17,61,48,55)(18,62,41,56)(19,63,42,49)(20,64,43,50)(21,57,44,51)(22,58,45,52)(23,59,46,53)(24,60,47,54), (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), (1,44,5,48)(2,47,6,43)(3,42,7,46)(4,45,8,41)(9,62,13,58)(10,57,14,61)(11,60,15,64)(12,63,16,59)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)(33,49,37,53)(34,52,38,56)(35,55,39,51)(36,50,40,54) );
G=PermutationGroup([[(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(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)], [(1,39,31,10),(2,40,32,11),(3,33,25,12),(4,34,26,13),(5,35,27,14),(6,36,28,15),(7,37,29,16),(8,38,30,9),(17,61,48,55),(18,62,41,56),(19,63,42,49),(20,64,43,50),(21,57,44,51),(22,58,45,52),(23,59,46,53),(24,60,47,54)], [(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)], [(1,44,5,48),(2,47,6,43),(3,42,7,46),(4,45,8,41),(9,62,13,58),(10,57,14,61),(11,60,15,64),(12,63,16,59),(17,31,21,27),(18,26,22,30),(19,29,23,25),(20,32,24,28),(33,49,37,53),(34,52,38,56),(35,55,39,51),(36,50,40,54)]])
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 |
Matrix representation of C42.308D4 ►in 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] >;
C42.308D4 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