p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.2C42, C22.7D8, C23.52D4, C22.4Q16, C22.9SD16, C4⋊C4⋊4C4, (C2×C8)⋊4C4, C4.3(C4⋊C4), (C2×C4).12Q8, (C2×C4).111D4, (C22×C8).2C2, C2.2(C4.Q8), C2.2(C2.D8), C2.2(D4⋊C4), C22.15(C4⋊C4), C4.19(C22⋊C4), C2.2(Q8⋊C4), C22.25(C22⋊C4), C2.5(C2.C42), (C22×C4).102C22, (C2×C4⋊C4).2C2, (C2×C4).39(C2×C4), SmallGroup(64,21)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.4Q16
G = < a,b,c,d | a2=b2=c8=1, d2=bc4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Subgroups: 93 in 57 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×C4⋊C4, C22×C8, C22.4Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16
Character table of C22.4Q16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | i | i | -i | -i | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | -i | -i | -i | i | i | i | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -1 | -i | -1 | 1 | i | i | -i | i | -i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | 1 | 1 | -1 | i | -1 | -i | i | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | i | -i | 1 | i | 1 | -1 | i | i | -i | i | -i | -i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | 1 | 1 | -1 | -i | -1 | i | -i | i | -i | -i | -i | i | i | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | -1 | -1 | 1 | i | 1 | -i | i | i | -i | -i | -i | i | i | i | -i | linear of order 4 |
| ρ12 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | -i | i | 1 | -i | 1 | -1 | -i | -i | i | -i | i | i | -i | i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -1 | -1 | 1 | -i | 1 | i | -i | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | -1 | i | -1 | 1 | -i | -i | i | -i | i | i | -i | i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | i | i | i | -i | -i | -i | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | -i | -i | i | i | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ21 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ22 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ26 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ27 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ28 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
►Regular action on 64 points
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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 23 31 50)(2 64 32 35)(3 21 25 56)(4 62 26 33)(5 19 27 54)(6 60 28 39)(7 17 29 52)(8 58 30 37)(9 63 46 34)(10 20 47 55)(11 61 48 40)(12 18 41 53)(13 59 42 38)(14 24 43 51)(15 57 44 36)(16 22 45 49)
G:=sub<Sym(64)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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), (1,23,31,50)(2,64,32,35)(3,21,25,56)(4,62,26,33)(5,19,27,54)(6,60,28,39)(7,17,29,52)(8,58,30,37)(9,63,46,34)(10,20,47,55)(11,61,48,40)(12,18,41,53)(13,59,42,38)(14,24,43,51)(15,57,44,36)(16,22,45,49)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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), (1,23,31,50)(2,64,32,35)(3,21,25,56)(4,62,26,33)(5,19,27,54)(6,60,28,39)(7,17,29,52)(8,58,30,37)(9,63,46,34)(10,20,47,55)(11,61,48,40)(12,18,41,53)(13,59,42,38)(14,24,43,51)(15,57,44,36)(16,22,45,49) );
G=PermutationGroup([[(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45),(33,55),(34,56),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,56),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(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)], [(1,23,31,50),(2,64,32,35),(3,21,25,56),(4,62,26,33),(5,19,27,54),(6,60,28,39),(7,17,29,52),(8,58,30,37),(9,63,46,34),(10,20,47,55),(11,61,48,40),(12,18,41,53),(13,59,42,38),(14,24,43,51),(15,57,44,36),(16,22,45,49)]])
C22.4Q16 is a maximal subgroup of
C24.132D4 C24.152D4 C4×D4⋊C4 C4×Q8⋊C4 D4⋊C42 Q8⋊C42 C4×C4.Q8 C4×C2.D8 C8⋊C42 C42.98D4 C42.99D4 C42.100D4 C42.101D4 C24.133D4 C24.67D4 C24.157D4 C24.69D4 C42.55Q8 C42.56Q8 C42.24Q8 C24.159D4 C24.71D4 C2.(C4×D8) Q8⋊(C4⋊C4) D4⋊(C4⋊C4) Q8⋊C4⋊C4 C24.160D4 C24.73D4 C24.74D4 (C2×SD16)⋊14C4 (C2×C4)⋊9Q16 (C2×C4)⋊9D8 (C2×SD16)⋊15C4 C24.135D4 C24.75D4 C24.76D4 C2.D8⋊4C4 C4.Q8⋊9C4 C4.Q8⋊10C4 C2.D8⋊5C4 D4⋊C4⋊C4 C4.67(C4×D4) C4.68(C4×D4) C2.(C4×Q16) C2.(C8⋊8D4) C2.(C8⋊7D4) C2.(C8⋊D4) C2.(C8⋊2D4) C42.29Q8 C42.30Q8 C42.31Q8 C42.121D4 C42.122D4 C42.123D4 C42.436D4 C42.125D4 C23⋊2D8 C23⋊3SD16 C23⋊2Q16 (C2×D4)⋊Q8 (C2×Q8)⋊Q8 C4⋊C4.84D4 C4⋊C4.85D4 C24.84D4 C24.85D4 C24.86D4 (C2×C4)⋊3D8 (C2×C4)⋊5SD16 (C2×C4)⋊3Q16 C4⋊C4⋊Q8 (C2×C8)⋊Q8 C4⋊C4.106D4 (C2×Q8).8Q8 (C2×C4).23D8 (C2×C8).52D4 (C2×C4).24D8 (C2×C4).19Q16 C42⋊8C4⋊C2 (C2×Q8).109D4 C24.88D4 C24.89D4 (C2×C4).26D8 (C2×C4).21Q16 C4.(C4⋊Q8) (C2×C4).28D8 (C2×C4).23Q16 C4⋊C4.Q8 D10.18D8 D10.10D8
C23.D4p: C22.SD32 C23.32D8 C23.22D8 C23.36D8 C23.37D8 C23.38D8 C23.23D8 C24.83D4 ...
(C2×C4p).Q8: C8⋊7(C4⋊C4) C8⋊5(C4⋊C4) C4.(C4×Q8) C8⋊(C4⋊C4) C42.437D4 C42.124D4 C2.(C8⋊Q8) (C2×C8).1Q8 ...
C22.4Q16 is a maximal quotient of
C42.385D4 C42.46Q8 C42.5Q8 C23.8D8 C42.27D4 C23.30D8 C42.8Q8 C42.389D4 C42.10Q8
C4p.C42: C8.7C42 C8.8C42 C8.9C42 C8.11C42 C23.9D8 C8.13C42 C8.C42 C8.2C42 ...
Matrix representation of C22.4Q16 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 |
| 13 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,12,12,0,0,5,12],[13,0,0,0,0,1,0,0,0,0,12,5,0,0,5,5] >;
C22.4Q16 in GAP, Magma, Sage, TeX
C_2^2._4Q_{16} % in TeX
G:=Group("C2^2.4Q16"); // GroupNames label
G:=SmallGroup(64,21);
// by ID
G=gap.SmallGroup(64,21);
# by ID
G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,158,1444,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^2=b*c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
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