p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.365D4, C42.721C23, C4○2(C8⋊8D4), C8⋊12(C4○D4), C4○2(C8⋊5D4), C8⋊5D4⋊31C2, C8⋊8D4⋊56C2, C4○2(C8⋊3Q8), C8⋊3Q8⋊30C2, (C2×C4)⋊11SD16, (C4×SD16)⋊33C2, C4○2(C4.4D8), C4.25(C4○D8), C4.4D8⋊46C2, C4⋊C4.106C23, (C2×C8).598C23, (C2×C4).365C24, (C4×C8).431C22, C4○2(C4.SD16), C4.SD16⋊47C2, C4.104(C2×SD16), (C4×D4).87C22, (C22×C4).618D4, C23.393(C2×D4), C4⋊Q8.289C22, (C4×Q8).84C22, C22.3(C2×SD16), (C2×D4).121C23, (C2×Q8).109C23, C2.20(C22×SD16), C4.Q8.161C22, C4⋊1D4.154C22, C4⋊D4.170C22, (C22×C8).569C22, C22.625(C22×D4), C22⋊Q8.175C22, D4⋊C4.145C22, (C22×C4).1570C23, (C2×C42).1134C22, Q8⋊C4.137C22, (C2×SD16).149C22, C23.37C23⋊12C2, C22.26C24.38C2, C2.62(C22.26C24), (C2×C4×C8)⋊36C2, C2.34(C2×C4○D8), C4.50(C2×C4○D4), (C2×C4).698(C2×D4), SmallGroup(128,1899)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.365D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 388 in 206 conjugacy classes, 100 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×SD16, C2×C4○D4, C2×C4×C8, C4×SD16, C8⋊8D4, C4.4D8, C4.SD16, C8⋊5D4, C8⋊3Q8, C22.26C24, C23.37C23, C42.365D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C4○D8, C22×D4, C2×C4○D4, C22.26C24, C22×SD16, C2×C4○D8, C42.365D4
►On 64 points
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)
(1 33 27 12)(2 34 28 13)(3 35 29 14)(4 36 30 15)(5 37 31 16)(6 38 32 9)(7 39 25 10)(8 40 26 11)(17 63 48 49)(18 64 41 50)(19 57 42 51)(20 58 43 52)(21 59 44 53)(22 60 45 54)(23 61 46 55)(24 62 47 56)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 43 21 47)(18 46 22 42)(19 41 23 45)(20 44 24 48)(25 32 29 28)(26 27 30 31)(33 40 37 36)(34 35 38 39)(49 62 53 58)(50 57 54 61)(51 60 55 64)(52 63 56 59)
G:=sub<Sym(64)| (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (1,33,27,12)(2,34,28,13)(3,35,29,14)(4,36,30,15)(5,37,31,16)(6,38,32,9)(7,39,25,10)(8,40,26,11)(17,63,48,49)(18,64,41,50)(19,57,42,51)(20,58,43,52)(21,59,44,53)(22,60,45,54)(23,61,46,55)(24,62,47,56), (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), (1,4,5,8)(2,7,6,3)(9,10,13,14)(11,16,15,12)(17,43,21,47)(18,46,22,42)(19,41,23,45)(20,44,24,48)(25,32,29,28)(26,27,30,31)(33,40,37,36)(34,35,38,39)(49,62,53,58)(50,57,54,61)(51,60,55,64)(52,63,56,59)>;
G:=Group( (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (1,33,27,12)(2,34,28,13)(3,35,29,14)(4,36,30,15)(5,37,31,16)(6,38,32,9)(7,39,25,10)(8,40,26,11)(17,63,48,49)(18,64,41,50)(19,57,42,51)(20,58,43,52)(21,59,44,53)(22,60,45,54)(23,61,46,55)(24,62,47,56), (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), (1,4,5,8)(2,7,6,3)(9,10,13,14)(11,16,15,12)(17,43,21,47)(18,46,22,42)(19,41,23,45)(20,44,24,48)(25,32,29,28)(26,27,30,31)(33,40,37,36)(34,35,38,39)(49,62,53,58)(50,57,54,61)(51,60,55,64)(52,63,56,59) );
G=PermutationGroup([[(1,55,5,51),(2,56,6,52),(3,49,7,53),(4,50,8,54),(9,43,13,47),(10,44,14,48),(11,45,15,41),(12,46,16,42),(17,39,21,35),(18,40,22,36),(19,33,23,37),(20,34,24,38),(25,59,29,63),(26,60,30,64),(27,61,31,57),(28,62,32,58)], [(1,33,27,12),(2,34,28,13),(3,35,29,14),(4,36,30,15),(5,37,31,16),(6,38,32,9),(7,39,25,10),(8,40,26,11),(17,63,48,49),(18,64,41,50),(19,57,42,51),(20,58,43,52),(21,59,44,53),(22,60,45,54),(23,61,46,55),(24,62,47,56)], [(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)], [(1,4,5,8),(2,7,6,3),(9,10,13,14),(11,16,15,12),(17,43,21,47),(18,46,22,42),(19,41,23,45),(20,44,24,48),(25,32,29,28),(26,27,30,31),(33,40,37,36),(34,35,38,39),(49,62,53,58),(50,57,54,61),(51,60,55,64),(52,63,56,59)]])
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 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | SD16 | C4○D8 |
| kernel | C42.365D4 | C2×C4×C8 | C4×SD16 | C8⋊8D4 | C4.4D8 | C4.SD16 | C8⋊5D4 | C8⋊3Q8 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C42.365D4 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[0,16,0,0,1,0,0,0,0,0,5,5,0,0,12,5],[0,16,0,0,16,0,0,0,0,0,5,12,0,0,12,12] >;
C42.365D4 in GAP, Magma, Sage, TeX
C_4^2._{365}D_4 % in TeX
G:=Group("C4^2.365D4"); // GroupNames label
G:=SmallGroup(128,1899);
// by ID
G=gap.SmallGroup(128,1899);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,184,80,4037,124]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations