p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.253C23, C4⋊C4.75D4, C8⋊2C8⋊20C2, C8⋊4Q8⋊27C2, C4⋊2Q16⋊7C2, (C2×C8).191D4, (C2×Q8).63D4, C4⋊C8.36C22, C4.D8.6C2, C4⋊Q8.74C22, C4.108(C4○D8), C4.10D8⋊30C2, C2.11(C8⋊2D4), C4.74(C8⋊C22), (C4×C8).286C22, C4⋊SD16.11C2, C4.4D8.13C2, (C4×Q8).51C22, C4⋊1D4.39C22, C4.96(C8.C22), C2.21(D4.3D4), C2.14(Q8.D4), C22.214(C4⋊D4), (C2×C4).38(C4○D4), (C2×C4).1288(C2×D4), SmallGroup(128,434)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.253C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=a-1b2, e2=b2, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a-1c, ece-1=bc, ede-1=a2d >
Subgroups: 208 in 81 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C4⋊1D4, C4⋊Q8, C2×SD16, C2×Q16, C4.D8, C4.10D8, C8⋊2C8, C8⋊4Q8, C4⋊SD16, C4⋊2Q16, C4.4D8, C42.253C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C4○D8, C8⋊C22, C8.C22, Q8.D4, C8⋊2D4, D4.3D4, C42.253C23
Character table of C42.253C23
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 16 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -√2 | √-2 | √2 | -√-2 | 0 | 0 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | √2 | √-2 | -√2 | -√-2 | 0 | 0 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | √2 | -√-2 | -√2 | √-2 | 0 | 0 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | -√2 | -√-2 | √2 | √-2 | 0 | 0 | complex lifted from C4○D8 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | 0 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.3D4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 0 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.3D4 |
►On 64 points
Generators in S64
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 62 37 58)(34 63 38 59)(35 64 39 60)(36 57 40 61)
(1 61 18 38)(2 62 19 39)(3 63 20 40)(4 64 21 33)(5 57 22 34)(6 58 23 35)(7 59 24 36)(8 60 17 37)(9 49 47 31)(10 50 48 32)(11 51 41 25)(12 52 42 26)(13 53 43 27)(14 54 44 28)(15 55 45 29)(16 56 46 30)
(2 17)(3 7)(4 23)(6 21)(8 19)(9 27)(10 52)(11 25)(12 50)(13 31)(14 56)(15 29)(16 54)(20 24)(26 48)(28 46)(30 44)(32 42)(33 35)(34 57)(36 63)(37 39)(38 61)(40 59)(41 51)(43 49)(45 55)(47 53)(58 64)(60 62)
(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 29 18 55)(2 26 19 52)(3 31 20 49)(4 28 21 54)(5 25 22 51)(6 30 23 56)(7 27 24 53)(8 32 17 50)(9 63 47 40)(10 60 48 37)(11 57 41 34)(12 62 42 39)(13 59 43 36)(14 64 44 33)(15 61 45 38)(16 58 46 35)
G:=sub<Sym(64)| (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,62,37,58)(34,63,38,59)(35,64,39,60)(36,57,40,61), (1,61,18,38)(2,62,19,39)(3,63,20,40)(4,64,21,33)(5,57,22,34)(6,58,23,35)(7,59,24,36)(8,60,17,37)(9,49,47,31)(10,50,48,32)(11,51,41,25)(12,52,42,26)(13,53,43,27)(14,54,44,28)(15,55,45,29)(16,56,46,30), (2,17)(3,7)(4,23)(6,21)(8,19)(9,27)(10,52)(11,25)(12,50)(13,31)(14,56)(15,29)(16,54)(20,24)(26,48)(28,46)(30,44)(32,42)(33,35)(34,57)(36,63)(37,39)(38,61)(40,59)(41,51)(43,49)(45,55)(47,53)(58,64)(60,62), (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), (1,29,18,55)(2,26,19,52)(3,31,20,49)(4,28,21,54)(5,25,22,51)(6,30,23,56)(7,27,24,53)(8,32,17,50)(9,63,47,40)(10,60,48,37)(11,57,41,34)(12,62,42,39)(13,59,43,36)(14,64,44,33)(15,61,45,38)(16,58,46,35)>;
G:=Group( (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,62,37,58)(34,63,38,59)(35,64,39,60)(36,57,40,61), (1,61,18,38)(2,62,19,39)(3,63,20,40)(4,64,21,33)(5,57,22,34)(6,58,23,35)(7,59,24,36)(8,60,17,37)(9,49,47,31)(10,50,48,32)(11,51,41,25)(12,52,42,26)(13,53,43,27)(14,54,44,28)(15,55,45,29)(16,56,46,30), (2,17)(3,7)(4,23)(6,21)(8,19)(9,27)(10,52)(11,25)(12,50)(13,31)(14,56)(15,29)(16,54)(20,24)(26,48)(28,46)(30,44)(32,42)(33,35)(34,57)(36,63)(37,39)(38,61)(40,59)(41,51)(43,49)(45,55)(47,53)(58,64)(60,62), (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), (1,29,18,55)(2,26,19,52)(3,31,20,49)(4,28,21,54)(5,25,22,51)(6,30,23,56)(7,27,24,53)(8,32,17,50)(9,63,47,40)(10,60,48,37)(11,57,41,34)(12,62,42,39)(13,59,43,36)(14,64,44,33)(15,61,45,38)(16,58,46,35) );
G=PermutationGroup([[(1,24,5,20),(2,17,6,21),(3,18,7,22),(4,19,8,23),(9,45,13,41),(10,46,14,42),(11,47,15,43),(12,48,16,44),(25,49,29,53),(26,50,30,54),(27,51,31,55),(28,52,32,56),(33,62,37,58),(34,63,38,59),(35,64,39,60),(36,57,40,61)], [(1,61,18,38),(2,62,19,39),(3,63,20,40),(4,64,21,33),(5,57,22,34),(6,58,23,35),(7,59,24,36),(8,60,17,37),(9,49,47,31),(10,50,48,32),(11,51,41,25),(12,52,42,26),(13,53,43,27),(14,54,44,28),(15,55,45,29),(16,56,46,30)], [(2,17),(3,7),(4,23),(6,21),(8,19),(9,27),(10,52),(11,25),(12,50),(13,31),(14,56),(15,29),(16,54),(20,24),(26,48),(28,46),(30,44),(32,42),(33,35),(34,57),(36,63),(37,39),(38,61),(40,59),(41,51),(43,49),(45,55),(47,53),(58,64),(60,62)], [(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)], [(1,29,18,55),(2,26,19,52),(3,31,20,49),(4,28,21,54),(5,25,22,51),(6,30,23,56),(7,27,24,53),(8,32,17,50),(9,63,47,40),(10,60,48,37),(11,57,41,34),(12,62,42,39),(13,59,43,36),(14,64,44,33),(15,61,45,38),(16,58,46,35)]])
Matrix representation of C42.253C23 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 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 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 5 |
| 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 5 |
| 0 | 0 | 5 | 0 | 12 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 12 | 5 |
| 0 | 0 | 4 | 13 | 5 | 5 |
| 0 | 0 | 5 | 12 | 4 | 4 |
| 0 | 0 | 12 | 12 | 4 | 13 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,12,0,5,0,0,5,0,12,0,0,0,0,12,0,12,0,0,5,0,5,0],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,4,4,5,12,0,0,4,13,12,12,0,0,12,5,4,4,0,0,5,5,4,13] >;
C42.253C23 in GAP, Magma, Sage, TeX
C_4^2._{253}C_2^3 % in TeX
G:=Group("C4^2.253C2^3"); // GroupNames label
G:=SmallGroup(128,434);
// by ID
G=gap.SmallGroup(128,434);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,387,352,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^-1*b^2,e^2=b^2,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^-1*c,e*c*e^-1=b*c,e*d*e^-1=a^2*d>;
// generators/relations
Export