p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊33D4, C23.718C24, C24.101C23, C22.4912+ 1+4, C42⋊5C4⋊37C2, C23⋊2D4⋊48C2, (C2×C42).730C22, (C22×C4).229C23, C22.450(C22×D4), C23.10D4⋊109C2, (C22×D4).295C22, C2.70(C22.29C24), C2.45(C22.54C24), C2.C42.421C22, (C2×C4⋊1D4)⋊11C2, (C2×C4).435(C2×D4), (C2×C42⋊2C2)⋊27C2, (C2×C4⋊C4).527C22, (C2×C22⋊C4).337C22, SmallGroup(128,1550)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊33D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 804 in 315 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C4⋊1D4, C22×D4, C42⋊5C4, C23⋊2D4, C23.10D4, C2×C42⋊2C2, C2×C4⋊1D4, C42⋊33D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, C22.54C24, C42⋊33D4
Character table of C42⋊33D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | -4 | 4 | 4 | -4 | 0 | 0 | 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 | 4 | -4 | -4 | -4 | 0 | 0 | 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 | -4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ24 | 4 | 4 | 4 | -4 | -4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ25 | 4 | -4 | -4 | -4 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | 4 | -4 | -4 | -4 | -4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
►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 43 22 10)(2 44 23 11)(3 41 24 12)(4 42 21 9)(5 25 38 54)(6 26 39 55)(7 27 40 56)(8 28 37 53)(13 64 46 36)(14 61 47 33)(15 62 48 34)(16 63 45 35)(17 31 50 60)(18 32 51 57)(19 29 52 58)(20 30 49 59)
(1 18 6 63)(2 50 7 34)(3 20 8 61)(4 52 5 36)(9 31 54 48)(10 59 55 14)(11 29 56 46)(12 57 53 16)(13 44 58 27)(15 42 60 25)(17 40 62 23)(19 38 64 21)(22 51 39 35)(24 49 37 33)(26 47 43 30)(28 45 41 32)
(1 59)(2 31)(3 57)(4 29)(5 46)(6 14)(7 48)(8 16)(9 50)(10 18)(11 52)(12 20)(13 38)(15 40)(17 42)(19 44)(21 58)(22 30)(23 60)(24 32)(25 62)(26 35)(27 64)(28 33)(34 54)(36 56)(37 45)(39 47)(41 49)(43 51)(53 61)(55 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), (1,43,22,10)(2,44,23,11)(3,41,24,12)(4,42,21,9)(5,25,38,54)(6,26,39,55)(7,27,40,56)(8,28,37,53)(13,64,46,36)(14,61,47,33)(15,62,48,34)(16,63,45,35)(17,31,50,60)(18,32,51,57)(19,29,52,58)(20,30,49,59), (1,18,6,63)(2,50,7,34)(3,20,8,61)(4,52,5,36)(9,31,54,48)(10,59,55,14)(11,29,56,46)(12,57,53,16)(13,44,58,27)(15,42,60,25)(17,40,62,23)(19,38,64,21)(22,51,39,35)(24,49,37,33)(26,47,43,30)(28,45,41,32), (1,59)(2,31)(3,57)(4,29)(5,46)(6,14)(7,48)(8,16)(9,50)(10,18)(11,52)(12,20)(13,38)(15,40)(17,42)(19,44)(21,58)(22,30)(23,60)(24,32)(25,62)(26,35)(27,64)(28,33)(34,54)(36,56)(37,45)(39,47)(41,49)(43,51)(53,61)(55,63)>;
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), (1,43,22,10)(2,44,23,11)(3,41,24,12)(4,42,21,9)(5,25,38,54)(6,26,39,55)(7,27,40,56)(8,28,37,53)(13,64,46,36)(14,61,47,33)(15,62,48,34)(16,63,45,35)(17,31,50,60)(18,32,51,57)(19,29,52,58)(20,30,49,59), (1,18,6,63)(2,50,7,34)(3,20,8,61)(4,52,5,36)(9,31,54,48)(10,59,55,14)(11,29,56,46)(12,57,53,16)(13,44,58,27)(15,42,60,25)(17,40,62,23)(19,38,64,21)(22,51,39,35)(24,49,37,33)(26,47,43,30)(28,45,41,32), (1,59)(2,31)(3,57)(4,29)(5,46)(6,14)(7,48)(8,16)(9,50)(10,18)(11,52)(12,20)(13,38)(15,40)(17,42)(19,44)(21,58)(22,30)(23,60)(24,32)(25,62)(26,35)(27,64)(28,33)(34,54)(36,56)(37,45)(39,47)(41,49)(43,51)(53,61)(55,63) );
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)], [(1,43,22,10),(2,44,23,11),(3,41,24,12),(4,42,21,9),(5,25,38,54),(6,26,39,55),(7,27,40,56),(8,28,37,53),(13,64,46,36),(14,61,47,33),(15,62,48,34),(16,63,45,35),(17,31,50,60),(18,32,51,57),(19,29,52,58),(20,30,49,59)], [(1,18,6,63),(2,50,7,34),(3,20,8,61),(4,52,5,36),(9,31,54,48),(10,59,55,14),(11,29,56,46),(12,57,53,16),(13,44,58,27),(15,42,60,25),(17,40,62,23),(19,38,64,21),(22,51,39,35),(24,49,37,33),(26,47,43,30),(28,45,41,32)], [(1,59),(2,31),(3,57),(4,29),(5,46),(6,14),(7,48),(8,16),(9,50),(10,18),(11,52),(12,20),(13,38),(15,40),(17,42),(19,44),(21,58),(22,30),(23,60),(24,32),(25,62),(26,35),(27,64),(28,33),(34,54),(36,56),(37,45),(39,47),(41,49),(43,51),(53,61),(55,63)]])
Matrix representation of C42⋊33D4 ►in GL10(ℤ)
| 1 | 0 | 0 | 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 | 0 | 0 | 1 | 0 | 0 | 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 | 0 | 0 | -1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 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 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | -2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -2 | 0 | 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 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 1 | 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 | 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 |
| 0 | 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 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(10,Integers())| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,2,0,1],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;
C42⋊33D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{33}D_4 % in TeX
G:=Group("C4^2:33D4"); // GroupNames label
G:=SmallGroup(128,1550);
// by ID
G=gap.SmallGroup(128,1550);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,794,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations
Export