p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.664C23, C8⋊C8⋊11C2, (C2×C8).33D4, C8⋊5D4⋊19C2, C4.7(C4○D8), C8⋊4D4.7C2, C4.4D8⋊38C2, C2.8(C8⋊3D4), C4.3(C8⋊C22), C4⋊Q8.88C22, (C4×C8).257C22, C4.SD16⋊10C2, C4⋊1D4.48C22, C2.12(C8.12D4), C22.65(C4⋊1D4), (C2×C4).721(C2×D4), SmallGroup(128,449)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.664C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=ab2, e2=b, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a-1c, ece-1=b-1c, ede-1=a2d >
Subgroups: 304 in 104 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, D8, SD16, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C8⋊C8, C4.4D8, C4.SD16, C8⋊5D4, C8⋊4D4, C42.664C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4⋊1D4, C4○D8, C8⋊C22, C8.12D4, C8⋊3D4, C42.664C23
Character table of C42.664C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -√2 | -√2 | √-2 | -√-2 | 0 | √2 | -√-2 | -2i | √2 | √-2 | 2i | 0 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | √2 | √2 | -√-2 | √-2 | 0 | -√2 | √-2 | -2i | -√2 | -√-2 | 2i | 0 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | √2 | √2 | √-2 | -√-2 | 0 | -√2 | -√-2 | 2i | -√2 | √-2 | -2i | 0 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | √2 | -√2 | √-2 | -√-2 | -2i | √2 | √-2 | 0 | -√2 | -√-2 | 0 | 2i | complex lifted from C4○D8 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -√2 | √2 | √-2 | -√-2 | 2i | -√2 | √-2 | 0 | √2 | -√-2 | 0 | -2i | complex lifted from C4○D8 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -√2 | √2 | -√-2 | √-2 | -2i | -√2 | -√-2 | 0 | √2 | √-2 | 0 | 2i | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -√2 | -√2 | -√-2 | √-2 | 0 | √2 | √-2 | 2i | √2 | -√-2 | -2i | 0 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | √2 | -√2 | -√-2 | √-2 | 2i | √2 | -√-2 | 0 | -√2 | √-2 | 0 | -2i | complex lifted from C4○D8 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
►On 64 points
Generators in S64
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)
(1 61 31 40)(2 62 32 33)(3 63 25 34)(4 64 26 35)(5 57 27 36)(6 58 28 37)(7 59 29 38)(8 60 30 39)(9 21 45 56)(10 22 46 49)(11 23 47 50)(12 24 48 51)(13 17 41 52)(14 18 42 53)(15 19 43 54)(16 20 44 55)
(2 26)(3 7)(4 32)(6 30)(8 28)(9 17)(10 55)(11 23)(12 53)(13 21)(14 51)(15 19)(16 49)(18 48)(20 46)(22 44)(24 42)(25 29)(33 35)(34 59)(36 57)(37 39)(38 63)(40 61)(41 56)(43 54)(45 52)(47 50)(58 60)(62 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)
(1 15 61 19 31 43 40 54)(2 12 62 24 32 48 33 51)(3 9 63 21 25 45 34 56)(4 14 64 18 26 42 35 53)(5 11 57 23 27 47 36 50)(6 16 58 20 28 44 37 55)(7 13 59 17 29 41 38 52)(8 10 60 22 30 46 39 49)
G:=sub<Sym(64)| (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,47,13,43)(10,48,14,44)(11,41,15,45)(12,42,16,46)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(33,64,37,60)(34,57,38,61)(35,58,39,62)(36,59,40,63), (1,61,31,40)(2,62,32,33)(3,63,25,34)(4,64,26,35)(5,57,27,36)(6,58,28,37)(7,59,29,38)(8,60,30,39)(9,21,45,56)(10,22,46,49)(11,23,47,50)(12,24,48,51)(13,17,41,52)(14,18,42,53)(15,19,43,54)(16,20,44,55), (2,26)(3,7)(4,32)(6,30)(8,28)(9,17)(10,55)(11,23)(12,53)(13,21)(14,51)(15,19)(16,49)(18,48)(20,46)(22,44)(24,42)(25,29)(33,35)(34,59)(36,57)(37,39)(38,63)(40,61)(41,56)(43,54)(45,52)(47,50)(58,60)(62,64), (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), (1,15,61,19,31,43,40,54)(2,12,62,24,32,48,33,51)(3,9,63,21,25,45,34,56)(4,14,64,18,26,42,35,53)(5,11,57,23,27,47,36,50)(6,16,58,20,28,44,37,55)(7,13,59,17,29,41,38,52)(8,10,60,22,30,46,39,49)>;
G:=Group( (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,47,13,43)(10,48,14,44)(11,41,15,45)(12,42,16,46)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(33,64,37,60)(34,57,38,61)(35,58,39,62)(36,59,40,63), (1,61,31,40)(2,62,32,33)(3,63,25,34)(4,64,26,35)(5,57,27,36)(6,58,28,37)(7,59,29,38)(8,60,30,39)(9,21,45,56)(10,22,46,49)(11,23,47,50)(12,24,48,51)(13,17,41,52)(14,18,42,53)(15,19,43,54)(16,20,44,55), (2,26)(3,7)(4,32)(6,30)(8,28)(9,17)(10,55)(11,23)(12,53)(13,21)(14,51)(15,19)(16,49)(18,48)(20,46)(22,44)(24,42)(25,29)(33,35)(34,59)(36,57)(37,39)(38,63)(40,61)(41,56)(43,54)(45,52)(47,50)(58,60)(62,64), (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), (1,15,61,19,31,43,40,54)(2,12,62,24,32,48,33,51)(3,9,63,21,25,45,34,56)(4,14,64,18,26,42,35,53)(5,11,57,23,27,47,36,50)(6,16,58,20,28,44,37,55)(7,13,59,17,29,41,38,52)(8,10,60,22,30,46,39,49) );
G=PermutationGroup([[(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,47,13,43),(10,48,14,44),(11,41,15,45),(12,42,16,46),(17,54,21,50),(18,55,22,51),(19,56,23,52),(20,49,24,53),(33,64,37,60),(34,57,38,61),(35,58,39,62),(36,59,40,63)], [(1,61,31,40),(2,62,32,33),(3,63,25,34),(4,64,26,35),(5,57,27,36),(6,58,28,37),(7,59,29,38),(8,60,30,39),(9,21,45,56),(10,22,46,49),(11,23,47,50),(12,24,48,51),(13,17,41,52),(14,18,42,53),(15,19,43,54),(16,20,44,55)], [(2,26),(3,7),(4,32),(6,30),(8,28),(9,17),(10,55),(11,23),(12,53),(13,21),(14,51),(15,19),(16,49),(18,48),(20,46),(22,44),(24,42),(25,29),(33,35),(34,59),(36,57),(37,39),(38,63),(40,61),(41,56),(43,54),(45,52),(47,50),(58,60),(62,64)], [(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)], [(1,15,61,19,31,43,40,54),(2,12,62,24,32,48,33,51),(3,9,63,21,25,45,34,56),(4,14,64,18,26,42,35,53),(5,11,57,23,27,47,36,50),(6,16,58,20,28,44,37,55),(7,13,59,17,29,41,38,52),(8,10,60,22,30,46,39,49)]])
Matrix representation of C42.664C23 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 6 |
| 0 | 0 | 12 | 5 | 11 | 0 |
| 0 | 0 | 6 | 0 | 12 | 12 |
| 0 | 0 | 0 | 6 | 5 | 12 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,5,12,6,0,0,0,5,5,0,6,0,0,0,11,12,5,0,0,6,0,12,12],[10,12,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.664C23 in GAP, Magma, Sage, TeX
C_4^2._{664}C_2^3 % in TeX
G:=Group("C4^2.664C2^3"); // GroupNames label
G:=SmallGroup(128,449);
// by ID
G=gap.SmallGroup(128,449);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,387,100,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a*b^2,e^2=b,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1*c,e*c*e^-1=b^-1*c,e*d*e^-1=a^2*d>;
// generators/relations
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