p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.407C23, C4.1132+ 1+4, C8⋊3D4⋊14C2, C8⋊5D4⋊24C2, C4⋊C4.130D4, D4⋊D4⋊29C2, Q8⋊D4⋊12C2, C4⋊C8.65C22, C22⋊C4.22D4, D4.D4⋊11C2, D4.2D4⋊26C2, C4⋊C4.160C23, (C2×C8).329C23, (C4×C8).266C22, (C2×C4).419C24, (C2×D8).73C22, C23.291(C2×D4), C4⋊Q8.120C22, C8⋊C4.22C22, C2.45(D4○SD16), (C4×D4).109C22, (C2×D4).168C23, C4⋊1D4.67C22, C4⋊D4.44C22, C22⋊C8.54C22, (C2×Q8).156C23, (C22×C4).307C23, (C2×SD16).37C22, C4.4D4.39C22, C22.679(C22×D4), C42.C2.24C22, D4⋊C4.109C22, C42.7C22⋊11C2, C42.30C22⋊3C2, C22.34C24⋊6C2, Q8⋊C4.102C22, (C22×Q8).325C22, C23.38C23⋊16C2, C42.78C22⋊14C2, C42⋊C2.158C22, C2.90(C22.29C24), (C2×C4).548(C2×D4), (C2×C4○D4).178C22, SmallGroup(128,1953)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.407C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, cac=dad-1=a-1, eae=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >
Subgroups: 428 in 194 conjugacy classes, 84 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C22×Q8, C2×C4○D4, C42.7C22, Q8⋊D4, D4⋊D4, D4.D4, D4.2D4, C42.78C22, C42.30C22, C8⋊5D4, C8⋊3D4, C23.38C23, C22.34C24, C42.407C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, D4○SD16, C42.407C23
Character table of C42.407C23
| 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 | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 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 | 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 | -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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -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 | 0 | 0 | -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 | 0 | 0 | -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 | 0 | 0 | -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 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 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 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -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 27 53 9)(2 28 54 10)(3 25 55 11)(4 26 56 12)(5 39 52 24)(6 40 49 21)(7 37 50 22)(8 38 51 23)(13 41 31 57)(14 42 32 58)(15 43 29 59)(16 44 30 60)(17 61 36 45)(18 62 33 46)(19 63 34 47)(20 64 35 48)
(1 4)(2 3)(5 21)(6 24)(7 23)(8 22)(9 26)(10 25)(11 28)(12 27)(13 14)(15 16)(17 46)(18 45)(19 48)(20 47)(29 30)(31 32)(33 61)(34 64)(35 63)(36 62)(37 51)(38 50)(39 49)(40 52)(41 58)(42 57)(43 60)(44 59)(53 56)(54 55)
(1 61 53 45)(2 64 54 48)(3 63 55 47)(4 62 56 46)(5 58 52 42)(6 57 49 41)(7 60 50 44)(8 59 51 43)(9 36 27 17)(10 35 28 20)(11 34 25 19)(12 33 26 18)(13 40 31 21)(14 39 32 24)(15 38 29 23)(16 37 30 22)
(1 41)(2 58)(3 43)(4 60)(5 48)(6 61)(7 46)(8 63)(9 13)(10 32)(11 15)(12 30)(14 28)(16 26)(17 21)(18 37)(19 23)(20 39)(22 33)(24 35)(25 29)(27 31)(34 38)(36 40)(42 54)(44 56)(45 49)(47 51)(50 62)(52 64)(53 57)(55 59)
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), (1,27,53,9)(2,28,54,10)(3,25,55,11)(4,26,56,12)(5,39,52,24)(6,40,49,21)(7,37,50,22)(8,38,51,23)(13,41,31,57)(14,42,32,58)(15,43,29,59)(16,44,30,60)(17,61,36,45)(18,62,33,46)(19,63,34,47)(20,64,35,48), (1,4)(2,3)(5,21)(6,24)(7,23)(8,22)(9,26)(10,25)(11,28)(12,27)(13,14)(15,16)(17,46)(18,45)(19,48)(20,47)(29,30)(31,32)(33,61)(34,64)(35,63)(36,62)(37,51)(38,50)(39,49)(40,52)(41,58)(42,57)(43,60)(44,59)(53,56)(54,55), (1,61,53,45)(2,64,54,48)(3,63,55,47)(4,62,56,46)(5,58,52,42)(6,57,49,41)(7,60,50,44)(8,59,51,43)(9,36,27,17)(10,35,28,20)(11,34,25,19)(12,33,26,18)(13,40,31,21)(14,39,32,24)(15,38,29,23)(16,37,30,22), (1,41)(2,58)(3,43)(4,60)(5,48)(6,61)(7,46)(8,63)(9,13)(10,32)(11,15)(12,30)(14,28)(16,26)(17,21)(18,37)(19,23)(20,39)(22,33)(24,35)(25,29)(27,31)(34,38)(36,40)(42,54)(44,56)(45,49)(47,51)(50,62)(52,64)(53,57)(55,59)>;
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), (1,27,53,9)(2,28,54,10)(3,25,55,11)(4,26,56,12)(5,39,52,24)(6,40,49,21)(7,37,50,22)(8,38,51,23)(13,41,31,57)(14,42,32,58)(15,43,29,59)(16,44,30,60)(17,61,36,45)(18,62,33,46)(19,63,34,47)(20,64,35,48), (1,4)(2,3)(5,21)(6,24)(7,23)(8,22)(9,26)(10,25)(11,28)(12,27)(13,14)(15,16)(17,46)(18,45)(19,48)(20,47)(29,30)(31,32)(33,61)(34,64)(35,63)(36,62)(37,51)(38,50)(39,49)(40,52)(41,58)(42,57)(43,60)(44,59)(53,56)(54,55), (1,61,53,45)(2,64,54,48)(3,63,55,47)(4,62,56,46)(5,58,52,42)(6,57,49,41)(7,60,50,44)(8,59,51,43)(9,36,27,17)(10,35,28,20)(11,34,25,19)(12,33,26,18)(13,40,31,21)(14,39,32,24)(15,38,29,23)(16,37,30,22), (1,41)(2,58)(3,43)(4,60)(5,48)(6,61)(7,46)(8,63)(9,13)(10,32)(11,15)(12,30)(14,28)(16,26)(17,21)(18,37)(19,23)(20,39)(22,33)(24,35)(25,29)(27,31)(34,38)(36,40)(42,54)(44,56)(45,49)(47,51)(50,62)(52,64)(53,57)(55,59) );
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)], [(1,27,53,9),(2,28,54,10),(3,25,55,11),(4,26,56,12),(5,39,52,24),(6,40,49,21),(7,37,50,22),(8,38,51,23),(13,41,31,57),(14,42,32,58),(15,43,29,59),(16,44,30,60),(17,61,36,45),(18,62,33,46),(19,63,34,47),(20,64,35,48)], [(1,4),(2,3),(5,21),(6,24),(7,23),(8,22),(9,26),(10,25),(11,28),(12,27),(13,14),(15,16),(17,46),(18,45),(19,48),(20,47),(29,30),(31,32),(33,61),(34,64),(35,63),(36,62),(37,51),(38,50),(39,49),(40,52),(41,58),(42,57),(43,60),(44,59),(53,56),(54,55)], [(1,61,53,45),(2,64,54,48),(3,63,55,47),(4,62,56,46),(5,58,52,42),(6,57,49,41),(7,60,50,44),(8,59,51,43),(9,36,27,17),(10,35,28,20),(11,34,25,19),(12,33,26,18),(13,40,31,21),(14,39,32,24),(15,38,29,23),(16,37,30,22)], [(1,41),(2,58),(3,43),(4,60),(5,48),(6,61),(7,46),(8,63),(9,13),(10,32),(11,15),(12,30),(14,28),(16,26),(17,21),(18,37),(19,23),(20,39),(22,33),(24,35),(25,29),(27,31),(34,38),(36,40),(42,54),(44,56),(45,49),(47,51),(50,62),(52,64),(53,57),(55,59)]])
Matrix representation of C42.407C23 ►in GL8(𝔽17)
| 6 | 4 | 0 | 8 | 0 | 0 | 0 | 0 |
| 13 | 6 | 9 | 0 | 0 | 0 | 0 | 0 |
| 11 | 13 | 11 | 13 | 0 | 0 | 0 | 0 |
| 4 | 11 | 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 0 | 5 |
| 0 | 0 | 0 | 0 | 3 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 3 |
| 0 | 0 | 0 | 0 | 5 | 0 | 14 | 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 6 | 4 | 0 | 8 | 0 | 0 | 0 | 0 |
| 4 | 11 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 0 | 5 |
| 0 | 0 | 0 | 0 | 14 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 3 |
| 0 | 0 | 0 | 0 | 12 | 0 | 3 | 0 |
| 10 | 16 | 11 | 6 | 0 | 0 | 0 | 0 |
| 16 | 7 | 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 16 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 15 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(17))| [6,13,11,4,0,0,0,0,4,6,13,11,0,0,0,0,0,9,11,4,0,0,0,0,8,0,13,11,0,0,0,0,0,0,0,0,0,3,0,5,0,0,0,0,14,0,12,0,0,0,0,0,0,12,0,14,0,0,0,0,5,0,3,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[6,4,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,8,6,13,0,0,0,0,8,0,13,11,0,0,0,0,0,0,0,0,0,14,0,12,0,0,0,0,14,0,12,0,0,0,0,0,0,5,0,3,0,0,0,0,5,0,3,0],[10,16,0,0,0,0,0,0,16,7,0,0,0,0,0,0,11,6,16,10,0,0,0,0,6,6,10,1,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.407C23 in GAP, Magma, Sage, TeX
C_4^2._{407}C_2^3 % in TeX
G:=Group("C4^2.407C2^3"); // GroupNames label
G:=SmallGroup(128,1953);
// by ID
G=gap.SmallGroup(128,1953);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,352,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,e*a*e=a*b^2,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*c,d*e=e*d>;
// generators/relations
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