p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.74C23, C4.1502+ 1+4, C8⋊7(C4○D4), C4⋊D8⋊43C2, C8⋊3D4⋊22C2, C8⋊4D4⋊23C2, C4⋊C4.186D4, C8⋊4Q8⋊21C2, D8⋊C4⋊30C2, C4⋊SD16⋊27C2, Q8⋊6D4⋊14C2, (C4×SD16)⋊25C2, C2.72(D4○D8), (C2×Q8).254D4, D4.2D4⋊49C2, C4⋊C4.450C23, C4⋊C8.152C22, C4.81(C8⋊C22), (C2×C4).591C24, (C4×C8).209C22, (C2×C8).223C23, (C2×D8).96C22, C8⋊C4.78C22, C2.45(Q8⋊6D4), (C4×D4).224C22, (C2×D4).285C23, (C4×Q8).214C22, (C2×Q8).270C23, C4.Q8.142C22, D4⋊C4.99C22, C4⋊1D4.109C22, C4.4D4.91C22, C22.851(C22×D4), C22.53C24⋊8C2, Q8⋊C4.169C22, (C2×SD16).126C22, C4.169(C2×C4○D4), (C2×C4).655(C2×D4), C2.93(C2×C8⋊C22), SmallGroup(128,2131)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.74C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=e2=a2b2, ab=ba, cac-1=eae-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >
Subgroups: 480 in 209 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, C4○D4, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22.D4, C4.4D4, C4.4D4, C4⋊1D4, C4⋊1D4, C2×D8, C2×SD16, C2×SD16, C2×C4○D4, C4×SD16, D8⋊C4, C8⋊4Q8, C4⋊D8, C4⋊D8, C4⋊SD16, D4.2D4, C8⋊4D4, C8⋊3D4, Q8⋊6D4, C22.53C24, C42.74C23
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○D8, C42.74C23
Character table of C42.74C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | -2 | -2 | -2 | -2 | -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 | 0 | -2 | -2 | 2 | 2 | 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 | 0 | -2 | -2 | -2 | -2 | 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 | 0 | -2 | -2 | 2 | 2 | -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 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 2i | 0 | 0 | -2i | -2i | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 0 | -2i | 0 | 0 | 2i | 2i | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 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 C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -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 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 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 C8⋊C22 |
| 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 | orthogonal lifted from D4○D8 |
| 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 | orthogonal lifted from D4○D8 |
►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 29 12 38)(2 30 9 39)(3 31 10 40)(4 32 11 37)(5 59 23 42)(6 60 24 43)(7 57 21 44)(8 58 22 41)(13 35 25 52)(14 36 26 49)(15 33 27 50)(16 34 28 51)(17 54 62 45)(18 55 63 46)(19 56 64 47)(20 53 61 48)
(1 46 12 55)(2 54 9 45)(3 48 10 53)(4 56 11 47)(5 35 23 52)(6 51 24 34)(7 33 21 50)(8 49 22 36)(13 42 25 59)(14 58 26 41)(15 44 27 57)(16 60 28 43)(17 39 62 30)(18 29 63 38)(19 37 64 32)(20 31 61 40)
(1 25 10 15)(2 14 11 28)(3 27 12 13)(4 16 9 26)(5 53 21 46)(6 45 22 56)(7 55 23 48)(8 47 24 54)(17 58 64 43)(18 42 61 57)(19 60 62 41)(20 44 63 59)(29 35 40 50)(30 49 37 34)(31 33 38 52)(32 51 39 36)
(1 15 10 25)(2 26 11 16)(3 13 12 27)(4 28 9 14)(5 63 21 20)(6 17 22 64)(7 61 23 18)(8 19 24 62)(29 33 40 52)(30 49 37 34)(31 35 38 50)(32 51 39 36)(41 47 60 54)(42 55 57 48)(43 45 58 56)(44 53 59 46)
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), (1,29,12,38)(2,30,9,39)(3,31,10,40)(4,32,11,37)(5,59,23,42)(6,60,24,43)(7,57,21,44)(8,58,22,41)(13,35,25,52)(14,36,26,49)(15,33,27,50)(16,34,28,51)(17,54,62,45)(18,55,63,46)(19,56,64,47)(20,53,61,48), (1,46,12,55)(2,54,9,45)(3,48,10,53)(4,56,11,47)(5,35,23,52)(6,51,24,34)(7,33,21,50)(8,49,22,36)(13,42,25,59)(14,58,26,41)(15,44,27,57)(16,60,28,43)(17,39,62,30)(18,29,63,38)(19,37,64,32)(20,31,61,40), (1,25,10,15)(2,14,11,28)(3,27,12,13)(4,16,9,26)(5,53,21,46)(6,45,22,56)(7,55,23,48)(8,47,24,54)(17,58,64,43)(18,42,61,57)(19,60,62,41)(20,44,63,59)(29,35,40,50)(30,49,37,34)(31,33,38,52)(32,51,39,36), (1,15,10,25)(2,26,11,16)(3,13,12,27)(4,28,9,14)(5,63,21,20)(6,17,22,64)(7,61,23,18)(8,19,24,62)(29,33,40,52)(30,49,37,34)(31,35,38,50)(32,51,39,36)(41,47,60,54)(42,55,57,48)(43,45,58,56)(44,53,59,46)>;
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), (1,29,12,38)(2,30,9,39)(3,31,10,40)(4,32,11,37)(5,59,23,42)(6,60,24,43)(7,57,21,44)(8,58,22,41)(13,35,25,52)(14,36,26,49)(15,33,27,50)(16,34,28,51)(17,54,62,45)(18,55,63,46)(19,56,64,47)(20,53,61,48), (1,46,12,55)(2,54,9,45)(3,48,10,53)(4,56,11,47)(5,35,23,52)(6,51,24,34)(7,33,21,50)(8,49,22,36)(13,42,25,59)(14,58,26,41)(15,44,27,57)(16,60,28,43)(17,39,62,30)(18,29,63,38)(19,37,64,32)(20,31,61,40), (1,25,10,15)(2,14,11,28)(3,27,12,13)(4,16,9,26)(5,53,21,46)(6,45,22,56)(7,55,23,48)(8,47,24,54)(17,58,64,43)(18,42,61,57)(19,60,62,41)(20,44,63,59)(29,35,40,50)(30,49,37,34)(31,33,38,52)(32,51,39,36), (1,15,10,25)(2,26,11,16)(3,13,12,27)(4,28,9,14)(5,63,21,20)(6,17,22,64)(7,61,23,18)(8,19,24,62)(29,33,40,52)(30,49,37,34)(31,35,38,50)(32,51,39,36)(41,47,60,54)(42,55,57,48)(43,45,58,56)(44,53,59,46) );
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)], [(1,29,12,38),(2,30,9,39),(3,31,10,40),(4,32,11,37),(5,59,23,42),(6,60,24,43),(7,57,21,44),(8,58,22,41),(13,35,25,52),(14,36,26,49),(15,33,27,50),(16,34,28,51),(17,54,62,45),(18,55,63,46),(19,56,64,47),(20,53,61,48)], [(1,46,12,55),(2,54,9,45),(3,48,10,53),(4,56,11,47),(5,35,23,52),(6,51,24,34),(7,33,21,50),(8,49,22,36),(13,42,25,59),(14,58,26,41),(15,44,27,57),(16,60,28,43),(17,39,62,30),(18,29,63,38),(19,37,64,32),(20,31,61,40)], [(1,25,10,15),(2,14,11,28),(3,27,12,13),(4,16,9,26),(5,53,21,46),(6,45,22,56),(7,55,23,48),(8,47,24,54),(17,58,64,43),(18,42,61,57),(19,60,62,41),(20,44,63,59),(29,35,40,50),(30,49,37,34),(31,33,38,52),(32,51,39,36)], [(1,15,10,25),(2,26,11,16),(3,13,12,27),(4,28,9,14),(5,63,21,20),(6,17,22,64),(7,61,23,18),(8,19,24,62),(29,33,40,52),(30,49,37,34),(31,35,38,50),(32,51,39,36),(41,47,60,54),(42,55,57,48),(43,45,58,56),(44,53,59,46)]])
Matrix representation of C42.74C23 ►in GL6(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 16 | 1 |
| 0 | 0 | 11 | 10 | 0 | 16 |
| 0 | 0 | 11 | 7 | 0 | 13 |
| 0 | 0 | 6 | 1 | 6 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 1 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,6,11,11,6,0,0,11,10,7,1,0,0,16,0,0,6,0,0,1,16,13,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,0,0,0,0,1,1,0,0,0,0,0,16],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C42.74C23 in GAP, Magma, Sage, TeX
C_4^2._{74}C_2^3 % in TeX
G:=Group("C4^2.74C2^3"); // GroupNames label
G:=SmallGroup(128,2131);
// by ID
G=gap.SmallGroup(128,2131);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,100,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=e^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,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
Export