p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.72D4, C42.153C23, (C4×D4).5C4, (C4×Q8).5C4, C4⋊D4.12C4, C42.94(C2×C4), C22⋊Q8.12C4, (C22×C4).128D4, C8⋊C4.88C22, C42.6C4⋊38C2, C23.59(C22⋊C4), (C2×C42).197C22, C42.C2.99C22, C42.2C22⋊11C2, C42.C22⋊12C2, C4.4D4.118C22, C2.35(C42⋊C22), C23.36C23.12C2, C2.13(M4(2).8C22), C4⋊C4.30(C2×C4), (C2×D4).25(C2×C4), (C2×Q8).25(C2×C4), (C2×C4).1181(C2×D4), (C2×C4).98(C22⋊C4), (C2×C4).147(C22×C4), (C22×C4).219(C2×C4), C22.211(C2×C22⋊C4), SmallGroup(128,267)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.72D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=a2bc3 >
Subgroups: 204 in 100 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C42.C22, C42.2C22, C42.6C4, C23.36C23, C42.72D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, M4(2).8C22, C42⋊C22, C42.72D4
Character table of C42.72D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 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 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 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 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -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 | 1 | -1 | 1 | -1 | 1 | -1 | -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 | 1 | -1 | 1 | 1 | -1 | 1 | 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 | 1 | 1 | -1 | 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 | -1 | 1 | 1 | 1 | -1 | -1 | -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 | -1 | 1 | -1 | 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 | 1 | -1 | 1 | 1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | i | i | i | -i | i | -i | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2).8C22 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2).8C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
►On 64 points
Generators in S64
(1 35 5 39)(2 52 6 56)(3 37 7 33)(4 54 8 50)(9 23 13 19)(10 59 14 63)(11 17 15 21)(12 61 16 57)(18 42 22 46)(20 44 24 48)(25 49 29 53)(26 34 30 38)(27 51 31 55)(28 36 32 40)(41 64 45 60)(43 58 47 62)
(1 22 27 57)(2 19 28 62)(3 24 29 59)(4 21 30 64)(5 18 31 61)(6 23 32 58)(7 20 25 63)(8 17 26 60)(9 36 43 52)(10 33 44 49)(11 38 45 54)(12 35 46 51)(13 40 47 56)(14 37 48 53)(15 34 41 50)(16 39 42 55)
(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 17 22 26 27 60 57 8)(2 29 19 59 28 3 62 24)(4 5 21 18 30 31 64 61)(6 25 23 63 32 7 58 20)(9 10 36 33 43 44 52 49)(11 46 38 51 45 12 54 35)(13 14 40 37 47 48 56 53)(15 42 34 55 41 16 50 39)
G:=sub<Sym(64)| (1,35,5,39)(2,52,6,56)(3,37,7,33)(4,54,8,50)(9,23,13,19)(10,59,14,63)(11,17,15,21)(12,61,16,57)(18,42,22,46)(20,44,24,48)(25,49,29,53)(26,34,30,38)(27,51,31,55)(28,36,32,40)(41,64,45,60)(43,58,47,62), (1,22,27,57)(2,19,28,62)(3,24,29,59)(4,21,30,64)(5,18,31,61)(6,23,32,58)(7,20,25,63)(8,17,26,60)(9,36,43,52)(10,33,44,49)(11,38,45,54)(12,35,46,51)(13,40,47,56)(14,37,48,53)(15,34,41,50)(16,39,42,55), (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), (1,17,22,26,27,60,57,8)(2,29,19,59,28,3,62,24)(4,5,21,18,30,31,64,61)(6,25,23,63,32,7,58,20)(9,10,36,33,43,44,52,49)(11,46,38,51,45,12,54,35)(13,14,40,37,47,48,56,53)(15,42,34,55,41,16,50,39)>;
G:=Group( (1,35,5,39)(2,52,6,56)(3,37,7,33)(4,54,8,50)(9,23,13,19)(10,59,14,63)(11,17,15,21)(12,61,16,57)(18,42,22,46)(20,44,24,48)(25,49,29,53)(26,34,30,38)(27,51,31,55)(28,36,32,40)(41,64,45,60)(43,58,47,62), (1,22,27,57)(2,19,28,62)(3,24,29,59)(4,21,30,64)(5,18,31,61)(6,23,32,58)(7,20,25,63)(8,17,26,60)(9,36,43,52)(10,33,44,49)(11,38,45,54)(12,35,46,51)(13,40,47,56)(14,37,48,53)(15,34,41,50)(16,39,42,55), (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), (1,17,22,26,27,60,57,8)(2,29,19,59,28,3,62,24)(4,5,21,18,30,31,64,61)(6,25,23,63,32,7,58,20)(9,10,36,33,43,44,52,49)(11,46,38,51,45,12,54,35)(13,14,40,37,47,48,56,53)(15,42,34,55,41,16,50,39) );
G=PermutationGroup([[(1,35,5,39),(2,52,6,56),(3,37,7,33),(4,54,8,50),(9,23,13,19),(10,59,14,63),(11,17,15,21),(12,61,16,57),(18,42,22,46),(20,44,24,48),(25,49,29,53),(26,34,30,38),(27,51,31,55),(28,36,32,40),(41,64,45,60),(43,58,47,62)], [(1,22,27,57),(2,19,28,62),(3,24,29,59),(4,21,30,64),(5,18,31,61),(6,23,32,58),(7,20,25,63),(8,17,26,60),(9,36,43,52),(10,33,44,49),(11,38,45,54),(12,35,46,51),(13,40,47,56),(14,37,48,53),(15,34,41,50),(16,39,42,55)], [(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)], [(1,17,22,26,27,60,57,8),(2,29,19,59,28,3,62,24),(4,5,21,18,30,31,64,61),(6,25,23,63,32,7,58,20),(9,10,36,33,43,44,52,49),(11,46,38,51,45,12,54,35),(13,14,40,37,47,48,56,53),(15,42,34,55,41,16,50,39)]])
Matrix representation of C42.72D4 ►in GL8(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 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 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.72D4 in GAP, Magma, Sage, TeX
C_4^2._{72}D_4 % in TeX
G:=Group("C4^2.72D4"); // GroupNames label
G:=SmallGroup(128,267);
// by ID
G=gap.SmallGroup(128,267);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations
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