p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.72C23, C4.1482+ 1+4, (C4×D8)⋊35C2, C8⋊6(C4○D4), C4⋊D8⋊42C2, C8⋊3D4⋊21C2, C8⋊5D4⋊11C2, C4⋊C4.184D4, C8⋊4Q8⋊19C2, C4⋊SD16⋊26C2, Q8⋊6D4⋊13C2, (C2×Q8).252D4, D4.2D4⋊48C2, D4.D4⋊27C2, C4⋊C4.448C23, C4⋊C8.150C22, C4.53(C8⋊C22), (C2×C4).589C24, (C4×C8).207C22, (C2×C8).380C23, (C2×D8).95C22, C4⋊Q8.216C22, SD16⋊C4⋊49C2, C8⋊C4.76C22, C2.43(Q8⋊6D4), (C4×D4).222C22, (C2×D4).283C23, (C2×Q8).268C23, (C4×Q8).212C22, C2.D8.228C22, C2.113(D4○SD16), D4⋊C4.98C22, C4⋊1D4.108C22, Q8⋊C4.94C22, (C2×SD16).75C22, C4.4D4.89C22, C22.849(C22×D4), C22.50C24⋊15C2, C4.167(C2×C4○D4), (C2×C4).653(C2×D4), C2.92(C2×C8⋊C22), SmallGroup(128,2129)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.72C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=a2, ab=ba, cac=eae-1=a-1, dad-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >
Subgroups: 440 in 203 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×D8, C2×SD16, C2×C4○D4, C4×D8, SD16⋊C4, C8⋊4Q8, C4⋊D8, C4⋊SD16, D4.D4, D4.2D4, C8⋊5D4, C8⋊3D4, Q8⋊6D4, C22.50C24, C42.72C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2+ 1+4, Q8⋊6D4, C2×C8⋊C22, D4○SD16, C42.72C23
Character table of C42.72C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 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 | 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 | -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 | -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 | -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 | 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 | -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 | 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 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ10 | 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 | -1 | linear of order 2 |
| ρ11 | 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 | 1 | linear of order 2 |
| ρ12 | 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 | 1 | linear of order 2 |
| ρ13 | 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 | -1 | linear of order 2 |
| ρ14 | 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 | -1 | linear of order 2 |
| ρ15 | 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 | 1 | linear of order 2 |
| ρ16 | 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 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2i | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
►On 64 points
Generators in S64
(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 59 11 41)(2 60 12 42)(3 57 9 43)(4 58 10 44)(5 62 50 20)(6 63 51 17)(7 64 52 18)(8 61 49 19)(13 25 31 47)(14 26 32 48)(15 27 29 45)(16 28 30 46)(21 55 40 34)(22 56 37 35)(23 53 38 36)(24 54 39 33)
(1 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 48)(10 47)(11 46)(12 45)(13 58)(14 57)(15 60)(16 59)(17 22)(18 21)(19 24)(20 23)(29 42)(30 41)(31 44)(32 43)(37 63)(38 62)(39 61)(40 64)(49 54)(50 53)(51 56)(52 55)
(1 63 3 61)(2 18 4 20)(5 42 7 44)(6 57 8 59)(9 19 11 17)(10 62 12 64)(13 23 15 21)(14 39 16 37)(22 32 24 30)(25 36 27 34)(26 54 28 56)(29 40 31 38)(33 46 35 48)(41 51 43 49)(45 55 47 53)(50 60 52 58)
(1 27 3 25)(2 26 4 28)(5 37 7 39)(6 40 8 38)(9 47 11 45)(10 46 12 48)(13 41 15 43)(14 44 16 42)(17 55 19 53)(18 54 20 56)(21 49 23 51)(22 52 24 50)(29 57 31 59)(30 60 32 58)(33 62 35 64)(34 61 36 63)
G:=sub<Sym(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), (1,59,11,41)(2,60,12,42)(3,57,9,43)(4,58,10,44)(5,62,50,20)(6,63,51,17)(7,64,52,18)(8,61,49,19)(13,25,31,47)(14,26,32,48)(15,27,29,45)(16,28,30,46)(21,55,40,34)(22,56,37,35)(23,53,38,36)(24,54,39,33), (1,28)(2,27)(3,26)(4,25)(5,36)(6,35)(7,34)(8,33)(9,48)(10,47)(11,46)(12,45)(13,58)(14,57)(15,60)(16,59)(17,22)(18,21)(19,24)(20,23)(29,42)(30,41)(31,44)(32,43)(37,63)(38,62)(39,61)(40,64)(49,54)(50,53)(51,56)(52,55), (1,63,3,61)(2,18,4,20)(5,42,7,44)(6,57,8,59)(9,19,11,17)(10,62,12,64)(13,23,15,21)(14,39,16,37)(22,32,24,30)(25,36,27,34)(26,54,28,56)(29,40,31,38)(33,46,35,48)(41,51,43,49)(45,55,47,53)(50,60,52,58), (1,27,3,25)(2,26,4,28)(5,37,7,39)(6,40,8,38)(9,47,11,45)(10,46,12,48)(13,41,15,43)(14,44,16,42)(17,55,19,53)(18,54,20,56)(21,49,23,51)(22,52,24,50)(29,57,31,59)(30,60,32,58)(33,62,35,64)(34,61,36,63)>;
G:=Group( (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), (1,59,11,41)(2,60,12,42)(3,57,9,43)(4,58,10,44)(5,62,50,20)(6,63,51,17)(7,64,52,18)(8,61,49,19)(13,25,31,47)(14,26,32,48)(15,27,29,45)(16,28,30,46)(21,55,40,34)(22,56,37,35)(23,53,38,36)(24,54,39,33), (1,28)(2,27)(3,26)(4,25)(5,36)(6,35)(7,34)(8,33)(9,48)(10,47)(11,46)(12,45)(13,58)(14,57)(15,60)(16,59)(17,22)(18,21)(19,24)(20,23)(29,42)(30,41)(31,44)(32,43)(37,63)(38,62)(39,61)(40,64)(49,54)(50,53)(51,56)(52,55), (1,63,3,61)(2,18,4,20)(5,42,7,44)(6,57,8,59)(9,19,11,17)(10,62,12,64)(13,23,15,21)(14,39,16,37)(22,32,24,30)(25,36,27,34)(26,54,28,56)(29,40,31,38)(33,46,35,48)(41,51,43,49)(45,55,47,53)(50,60,52,58), (1,27,3,25)(2,26,4,28)(5,37,7,39)(6,40,8,38)(9,47,11,45)(10,46,12,48)(13,41,15,43)(14,44,16,42)(17,55,19,53)(18,54,20,56)(21,49,23,51)(22,52,24,50)(29,57,31,59)(30,60,32,58)(33,62,35,64)(34,61,36,63) );
G=PermutationGroup([[(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)], [(1,59,11,41),(2,60,12,42),(3,57,9,43),(4,58,10,44),(5,62,50,20),(6,63,51,17),(7,64,52,18),(8,61,49,19),(13,25,31,47),(14,26,32,48),(15,27,29,45),(16,28,30,46),(21,55,40,34),(22,56,37,35),(23,53,38,36),(24,54,39,33)], [(1,28),(2,27),(3,26),(4,25),(5,36),(6,35),(7,34),(8,33),(9,48),(10,47),(11,46),(12,45),(13,58),(14,57),(15,60),(16,59),(17,22),(18,21),(19,24),(20,23),(29,42),(30,41),(31,44),(32,43),(37,63),(38,62),(39,61),(40,64),(49,54),(50,53),(51,56),(52,55)], [(1,63,3,61),(2,18,4,20),(5,42,7,44),(6,57,8,59),(9,19,11,17),(10,62,12,64),(13,23,15,21),(14,39,16,37),(22,32,24,30),(25,36,27,34),(26,54,28,56),(29,40,31,38),(33,46,35,48),(41,51,43,49),(45,55,47,53),(50,60,52,58)], [(1,27,3,25),(2,26,4,28),(5,37,7,39),(6,40,8,38),(9,47,11,45),(10,46,12,48),(13,41,15,43),(14,44,16,42),(17,55,19,53),(18,54,20,56),(21,49,23,51),(22,52,24,50),(29,57,31,59),(30,60,32,58),(33,62,35,64),(34,61,36,63)]])
Matrix representation of C42.72C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 8 | 9 |
| 0 | 0 | 13 | 4 | 8 | 0 |
| 0 | 0 | 4 | 13 | 0 | 13 |
| 0 | 0 | 8 | 13 | 4 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 0 | 0 | 0 | 16 | 1 | 0 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 8 | 9 |
| 0 | 0 | 8 | 13 | 0 | 9 |
| 0 | 0 | 13 | 4 | 4 | 0 |
| 0 | 0 | 0 | 4 | 8 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,0,13,0,0,0,0,0,0,4,13,4,8,0,0,0,4,13,13,0,0,8,8,0,4,0,0,9,0,13,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,16,0],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,4,8,13,0,0,0,0,13,4,4,0,0,8,0,4,8,0,0,9,9,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42.72C23 in GAP, Magma, Sage, TeX
C_4^2._{72}C_2^3 % in TeX
G:=Group("C4^2.72C2^3"); // GroupNames label
G:=SmallGroup(128,2129);
// by ID
G=gap.SmallGroup(128,2129);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,100,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=e^2=a^2,a*b=b*a,c*a*c=e*a*e^-1=a^-1,d*a*d^-1=a*b^2,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,c*e=e*c,d*e=e*d>;
// generators/relations
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