p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.30C23, C4.192- 1+4, C4.382+ 1+4, C8⋊8D4⋊32C2, C4⋊C4.145D4, C8⋊2D4.7C2, Q8⋊Q8⋊16C2, D4⋊2Q8⋊16C2, C8.D4⋊18C2, C4⋊C8.91C22, C22⋊C4.37D4, C23.98(C2×D4), D4.2D4⋊33C2, C4⋊C4.202C23, (C2×C4).461C24, (C2×C8).342C23, Q8.D4⋊33C2, (C2×D8).78C22, C4⋊Q8.132C22, C4.Q8.98C22, C2.55(D4○SD16), (C4×D4).139C22, (C2×D4).201C23, C4⋊D4.55C22, (C2×Q8).189C23, (C4×Q8).136C22, (C2×Q16).79C22, C22⋊Q8.55C22, (C22×C8).353C22, Q8⋊C4.64C22, (C2×SD16).92C22, C4.4D4.48C22, C22.721(C22×D4), D4⋊C4.116C22, C42.6C22⋊17C2, (C22×C4).1116C23, (C2×M4(2)).99C22, C42⋊C2.179C22, C22.36C24⋊11C2, C2.80(C22.31C24), (C2×C4).585(C2×D4), SmallGroup(128,1995)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.30C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, ede=a2b2d >
Subgroups: 348 in 173 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C42.6C22, D4.2D4, Q8.D4, C8⋊8D4, C8⋊2D4, C8.D4, Q8⋊Q8, D4⋊2Q8, C22.36C24, C42.30C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○SD16, C42.30C23
Character table of C42.30C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 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 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| 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 28 17 22)(2 25 18 23)(3 26 19 24)(4 27 20 21)(5 12 15 62)(6 9 16 63)(7 10 13 64)(8 11 14 61)(29 35 37 44)(30 36 38 41)(31 33 39 42)(32 34 40 43)(45 60 53 50)(46 57 54 51)(47 58 55 52)(48 59 56 49)
(1 55 17 47)(2 46 18 54)(3 53 19 45)(4 48 20 56)(5 35 15 44)(6 43 16 34)(7 33 13 42)(8 41 14 36)(9 40 63 32)(10 31 64 39)(11 38 61 30)(12 29 62 37)(21 59 27 49)(22 52 28 58)(23 57 25 51)(24 50 26 60)
(1 37 17 29)(2 30 18 38)(3 39 19 31)(4 32 20 40)(5 60 15 50)(6 51 16 57)(7 58 13 52)(8 49 14 59)(9 46 63 54)(10 55 64 47)(11 48 61 56)(12 53 62 45)(21 43 27 34)(22 35 28 44)(23 41 25 36)(24 33 26 42)
(5 64)(6 61)(7 62)(8 63)(9 14)(10 15)(11 16)(12 13)(21 27)(22 28)(23 25)(24 26)(29 39)(30 40)(31 37)(32 38)(33 35)(34 36)(41 43)(42 44)(45 50)(46 51)(47 52)(48 49)(53 60)(54 57)(55 58)(56 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), (1,28,17,22)(2,25,18,23)(3,26,19,24)(4,27,20,21)(5,12,15,62)(6,9,16,63)(7,10,13,64)(8,11,14,61)(29,35,37,44)(30,36,38,41)(31,33,39,42)(32,34,40,43)(45,60,53,50)(46,57,54,51)(47,58,55,52)(48,59,56,49), (1,55,17,47)(2,46,18,54)(3,53,19,45)(4,48,20,56)(5,35,15,44)(6,43,16,34)(7,33,13,42)(8,41,14,36)(9,40,63,32)(10,31,64,39)(11,38,61,30)(12,29,62,37)(21,59,27,49)(22,52,28,58)(23,57,25,51)(24,50,26,60), (1,37,17,29)(2,30,18,38)(3,39,19,31)(4,32,20,40)(5,60,15,50)(6,51,16,57)(7,58,13,52)(8,49,14,59)(9,46,63,54)(10,55,64,47)(11,48,61,56)(12,53,62,45)(21,43,27,34)(22,35,28,44)(23,41,25,36)(24,33,26,42), (5,64)(6,61)(7,62)(8,63)(9,14)(10,15)(11,16)(12,13)(21,27)(22,28)(23,25)(24,26)(29,39)(30,40)(31,37)(32,38)(33,35)(34,36)(41,43)(42,44)(45,50)(46,51)(47,52)(48,49)(53,60)(54,57)(55,58)(56,59)>;
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), (1,28,17,22)(2,25,18,23)(3,26,19,24)(4,27,20,21)(5,12,15,62)(6,9,16,63)(7,10,13,64)(8,11,14,61)(29,35,37,44)(30,36,38,41)(31,33,39,42)(32,34,40,43)(45,60,53,50)(46,57,54,51)(47,58,55,52)(48,59,56,49), (1,55,17,47)(2,46,18,54)(3,53,19,45)(4,48,20,56)(5,35,15,44)(6,43,16,34)(7,33,13,42)(8,41,14,36)(9,40,63,32)(10,31,64,39)(11,38,61,30)(12,29,62,37)(21,59,27,49)(22,52,28,58)(23,57,25,51)(24,50,26,60), (1,37,17,29)(2,30,18,38)(3,39,19,31)(4,32,20,40)(5,60,15,50)(6,51,16,57)(7,58,13,52)(8,49,14,59)(9,46,63,54)(10,55,64,47)(11,48,61,56)(12,53,62,45)(21,43,27,34)(22,35,28,44)(23,41,25,36)(24,33,26,42), (5,64)(6,61)(7,62)(8,63)(9,14)(10,15)(11,16)(12,13)(21,27)(22,28)(23,25)(24,26)(29,39)(30,40)(31,37)(32,38)(33,35)(34,36)(41,43)(42,44)(45,50)(46,51)(47,52)(48,49)(53,60)(54,57)(55,58)(56,59) );
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)], [(1,28,17,22),(2,25,18,23),(3,26,19,24),(4,27,20,21),(5,12,15,62),(6,9,16,63),(7,10,13,64),(8,11,14,61),(29,35,37,44),(30,36,38,41),(31,33,39,42),(32,34,40,43),(45,60,53,50),(46,57,54,51),(47,58,55,52),(48,59,56,49)], [(1,55,17,47),(2,46,18,54),(3,53,19,45),(4,48,20,56),(5,35,15,44),(6,43,16,34),(7,33,13,42),(8,41,14,36),(9,40,63,32),(10,31,64,39),(11,38,61,30),(12,29,62,37),(21,59,27,49),(22,52,28,58),(23,57,25,51),(24,50,26,60)], [(1,37,17,29),(2,30,18,38),(3,39,19,31),(4,32,20,40),(5,60,15,50),(6,51,16,57),(7,58,13,52),(8,49,14,59),(9,46,63,54),(10,55,64,47),(11,48,61,56),(12,53,62,45),(21,43,27,34),(22,35,28,44),(23,41,25,36),(24,33,26,42)], [(5,64),(6,61),(7,62),(8,63),(9,14),(10,15),(11,16),(12,13),(21,27),(22,28),(23,25),(24,26),(29,39),(30,40),(31,37),(32,38),(33,35),(34,36),(41,43),(42,44),(45,50),(46,51),(47,52),(48,49),(53,60),(54,57),(55,58),(56,59)]])
Matrix representation of C42.30C23 ►in GL8(𝔽17)
| 16 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 11 | 16 | 15 |
| 0 | 0 | 0 | 0 | 14 | 10 | 1 | 1 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 13 | 11 | 2 | 15 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 2 | 0 | 0 | 0 | 0 |
| 4 | 2 | 2 | 15 | 0 | 0 | 0 | 0 |
| 15 | 13 | 15 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 9 | 3 |
| 0 | 0 | 0 | 0 | 0 | 10 | 2 | 16 |
| 0 | 0 | 0 | 0 | 8 | 9 | 14 | 10 |
| 0 | 0 | 0 | 0 | 16 | 15 | 14 | 10 |
| 14 | 8 | 0 | 14 | 0 | 0 | 0 | 0 |
| 3 | 13 | 7 | 10 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 4 | 13 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 14 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 16 | 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 | 14 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 0 | 16 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,1,1,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,2,14,14,0,0,0,0,16,1,11,10,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[16,1,0,16,0,0,0,0,15,1,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[13,2,4,15,0,0,0,0,11,6,2,13,0,0,0,0,2,0,2,15,0,0,0,0,15,2,15,13,0,0,0,0,0,0,0,0,0,0,8,16,0,0,0,0,1,10,9,15,0,0,0,0,9,2,14,14,0,0,0,0,3,16,10,10],[14,3,3,0,0,0,0,0,8,13,3,4,0,0,0,0,0,7,0,13,0,0,0,0,14,10,14,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,3,4,13,0,0,0,0,1,0,0,0,0,0,0,0,0,2,14,14],[1,16,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,14,14,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
C42.30C23 in GAP, Magma, Sage, TeX
C_4^2._{30}C_2^3 % in TeX
G:=Group("C4^2.30C2^3"); // GroupNames label
G:=SmallGroup(128,1995);
// by ID
G=gap.SmallGroup(128,1995);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,e*d*e=a^2*b^2*d>;
// generators/relations
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