p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.75C23, C4.1512+ 1+4, C4⋊C4.187D4, (C4×Q16)⋊35C2, C8⋊4Q8⋊22C2, C8⋊5D4.5C2, Q8⋊3Q8⋊14C2, C4⋊2Q16⋊45C2, C8.21(C4○D4), C8.2D4⋊21C2, (C2×Q8).255D4, C4⋊SD16.3C2, D4.D4⋊28C2, C4⋊C4.451C23, C4⋊C8.153C22, (C4×C8).210C22, (C2×C4).592C24, (C2×C8).381C23, Q8.D4⋊48C2, C4⋊Q8.218C22, SD16⋊C4⋊50C2, C8⋊C4.79C22, C2.46(Q8⋊6D4), (C4×D4).225C22, (C2×D4).286C23, C4.83(C8.C22), (C2×Q8).271C23, (C2×Q16).95C22, (C4×Q8).215C22, C2.D8.229C22, C2.114(D4○SD16), C4⋊1D4.110C22, Q8⋊C4.96C22, (C2×SD16).77C22, C4.4D4.92C22, C22.852(C22×D4), D4⋊C4.100C22, C22.53C24.6C2, C4.170(C2×C4○D4), (C2×C4).656(C2×D4), C2.93(C2×C8.C22), SmallGroup(128,2132)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.75C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b2, e2=a2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >
Subgroups: 344 in 181 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, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C2×Q16, C4×Q16, SD16⋊C4, C8⋊4Q8, C4⋊SD16, D4.D4, C4⋊2Q16, Q8.D4, C8⋊5D4, C8.2D4, Q8⋊3Q8, C22.53C24, C42.75C23
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, C42.75C23
Character table of C42.75C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -4 | 0 | 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 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 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 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 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 |
►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 25 31 36)(2 26 32 33)(3 27 29 34)(4 28 30 35)(5 42 56 19)(6 43 53 20)(7 44 54 17)(8 41 55 18)(9 15 51 40)(10 16 52 37)(11 13 49 38)(12 14 50 39)(21 62 46 60)(22 63 47 57)(23 64 48 58)(24 61 45 59)
(1 61 31 59)(2 64 32 58)(3 63 29 57)(4 62 30 60)(5 52 56 10)(6 51 53 9)(7 50 54 12)(8 49 55 11)(13 18 38 41)(14 17 39 44)(15 20 40 43)(16 19 37 42)(21 35 46 28)(22 34 47 27)(23 33 48 26)(24 36 45 25)
(1 30 29 2)(3 32 31 4)(5 18 54 43)(6 42 55 17)(7 20 56 41)(8 44 53 19)(9 52 49 12)(10 11 50 51)(13 14 40 37)(15 16 38 39)(21 63 48 59)(22 58 45 62)(23 61 46 57)(24 60 47 64)(25 28 34 33)(26 36 35 27)
(1 9 3 11)(2 12 4 10)(5 64 7 62)(6 63 8 61)(13 25 15 27)(14 28 16 26)(17 21 19 23)(18 24 20 22)(29 49 31 51)(30 52 32 50)(33 39 35 37)(34 38 36 40)(41 45 43 47)(42 48 44 46)(53 57 55 59)(54 60 56 58)
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), (1,25,31,36)(2,26,32,33)(3,27,29,34)(4,28,30,35)(5,42,56,19)(6,43,53,20)(7,44,54,17)(8,41,55,18)(9,15,51,40)(10,16,52,37)(11,13,49,38)(12,14,50,39)(21,62,46,60)(22,63,47,57)(23,64,48,58)(24,61,45,59), (1,61,31,59)(2,64,32,58)(3,63,29,57)(4,62,30,60)(5,52,56,10)(6,51,53,9)(7,50,54,12)(8,49,55,11)(13,18,38,41)(14,17,39,44)(15,20,40,43)(16,19,37,42)(21,35,46,28)(22,34,47,27)(23,33,48,26)(24,36,45,25), (1,30,29,2)(3,32,31,4)(5,18,54,43)(6,42,55,17)(7,20,56,41)(8,44,53,19)(9,52,49,12)(10,11,50,51)(13,14,40,37)(15,16,38,39)(21,63,48,59)(22,58,45,62)(23,61,46,57)(24,60,47,64)(25,28,34,33)(26,36,35,27), (1,9,3,11)(2,12,4,10)(5,64,7,62)(6,63,8,61)(13,25,15,27)(14,28,16,26)(17,21,19,23)(18,24,20,22)(29,49,31,51)(30,52,32,50)(33,39,35,37)(34,38,36,40)(41,45,43,47)(42,48,44,46)(53,57,55,59)(54,60,56,58)>;
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), (1,25,31,36)(2,26,32,33)(3,27,29,34)(4,28,30,35)(5,42,56,19)(6,43,53,20)(7,44,54,17)(8,41,55,18)(9,15,51,40)(10,16,52,37)(11,13,49,38)(12,14,50,39)(21,62,46,60)(22,63,47,57)(23,64,48,58)(24,61,45,59), (1,61,31,59)(2,64,32,58)(3,63,29,57)(4,62,30,60)(5,52,56,10)(6,51,53,9)(7,50,54,12)(8,49,55,11)(13,18,38,41)(14,17,39,44)(15,20,40,43)(16,19,37,42)(21,35,46,28)(22,34,47,27)(23,33,48,26)(24,36,45,25), (1,30,29,2)(3,32,31,4)(5,18,54,43)(6,42,55,17)(7,20,56,41)(8,44,53,19)(9,52,49,12)(10,11,50,51)(13,14,40,37)(15,16,38,39)(21,63,48,59)(22,58,45,62)(23,61,46,57)(24,60,47,64)(25,28,34,33)(26,36,35,27), (1,9,3,11)(2,12,4,10)(5,64,7,62)(6,63,8,61)(13,25,15,27)(14,28,16,26)(17,21,19,23)(18,24,20,22)(29,49,31,51)(30,52,32,50)(33,39,35,37)(34,38,36,40)(41,45,43,47)(42,48,44,46)(53,57,55,59)(54,60,56,58) );
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)], [(1,25,31,36),(2,26,32,33),(3,27,29,34),(4,28,30,35),(5,42,56,19),(6,43,53,20),(7,44,54,17),(8,41,55,18),(9,15,51,40),(10,16,52,37),(11,13,49,38),(12,14,50,39),(21,62,46,60),(22,63,47,57),(23,64,48,58),(24,61,45,59)], [(1,61,31,59),(2,64,32,58),(3,63,29,57),(4,62,30,60),(5,52,56,10),(6,51,53,9),(7,50,54,12),(8,49,55,11),(13,18,38,41),(14,17,39,44),(15,20,40,43),(16,19,37,42),(21,35,46,28),(22,34,47,27),(23,33,48,26),(24,36,45,25)], [(1,30,29,2),(3,32,31,4),(5,18,54,43),(6,42,55,17),(7,20,56,41),(8,44,53,19),(9,52,49,12),(10,11,50,51),(13,14,40,37),(15,16,38,39),(21,63,48,59),(22,58,45,62),(23,61,46,57),(24,60,47,64),(25,28,34,33),(26,36,35,27)], [(1,9,3,11),(2,12,4,10),(5,64,7,62),(6,63,8,61),(13,25,15,27),(14,28,16,26),(17,21,19,23),(18,24,20,22),(29,49,31,51),(30,52,32,50),(33,39,35,37),(34,38,36,40),(41,45,43,47),(42,48,44,46),(53,57,55,59),(54,60,56,58)]])
Matrix representation of C42.75C23 ►in GL8(𝔽17)
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 6 | 16 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 16 | 6 | 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 | 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 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 5 | 12 |
| 0 | 0 | 0 | 0 | 4 | 13 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 5 | 4 | 4 |
| 0 | 0 | 0 | 0 | 5 | 5 | 4 | 13 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 1 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 | 11 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,6,0,0,16,0,0,0,0,16,0,0,6,0,0,0,0,0,16,11,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[6,0,13,0,0,0,0,0,0,6,0,13,0,0,0,0,13,0,11,0,0,0,0,0,0,13,0,11,0,0,0,0,0,0,0,0,4,4,12,5,0,0,0,0,4,13,5,5,0,0,0,0,5,12,4,4,0,0,0,0,12,12,4,13],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,11,0,0,1,0,0,0,0,1,0,0,11,0,0,0,0,0,16,11,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.75C23 in GAP, Magma, Sage, TeX
C_4^2._{75}C_2^3 % in TeX
G:=Group("C4^2.75C2^3"); // GroupNames label
G:=SmallGroup(128,2132);
// by ID
G=gap.SmallGroup(128,2132);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,436,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b^2,e^2=a^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=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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