Aliases: C4.A4⋊2C4, (C2×C4).15S4, C4.3(A4⋊C4), Q8⋊Dic3⋊7C2, C4○D4⋊2Dic3, (C2×Q8).18D6, C22.20(C2×S4), Q8.5(C2×Dic3), C2.2(C4.3S4), C2.2(C4.S4), SL2(𝔽3)⋊5(C2×C4), (C2×SL2(𝔽3)).18C22, C2.10(C2×A4⋊C4), (C2×C4.A4).2C2, (C2×C4○D4).5S3, SmallGroup(192,985)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — (C2×C4).S4 |
Generators and relations for (C2×C4).S4
G = < a,b,c,d,e,f | a2=b4=e3=1, c2=d2=b2, f2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >
Subgroups: 291 in 79 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C2×C6, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×Dic3, C2×C12, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C4⋊Dic3, C2×SL2(𝔽3), C4.A4, C23.36D4, Q8⋊Dic3, C2×C4.A4, (C2×C4).S4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, C4.S4, C4.3S4, C2×A4⋊C4, (C2×C4).S4
Character table of (C2×C4).S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 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 | i | -i | -i | i | -1 | 1 | -1 | i | -i | -i | i | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | i | i | -i | -i | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | i | -i | -1 | 1 | -1 | -i | i | i | -i | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | -2 | 2 | -2 | 2 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | -1 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | -3 | -3 | 3 | 1 | -1 | 0 | 3 | -3 | -1 | 1 | i | i | -i | -i | 0 | 0 | 0 | -i | i | -i | i | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ18 | 3 | -3 | -3 | 3 | -1 | 1 | 0 | -3 | 3 | -1 | 1 | -i | i | i | -i | 0 | 0 | 0 | i | -i | -i | i | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ19 | 3 | -3 | -3 | 3 | -1 | 1 | 0 | -3 | 3 | -1 | 1 | i | -i | -i | i | 0 | 0 | 0 | -i | i | i | -i | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ20 | 3 | -3 | -3 | 3 | 1 | -1 | 0 | 3 | -3 | -1 | 1 | -i | -i | i | i | 0 | 0 | 0 | i | -i | i | -i | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | orthogonal lifted from C4.3S4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | orthogonal lifted from C4.3S4 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | symplectic lifted from C4.S4, Schur index 2 |
►On 64 points
Generators in S64
(1 7)(2 8)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 36)(14 33)(15 34)(16 35)(17 46)(18 47)(19 48)(20 45)(25 29)(26 30)(27 31)(28 32)(37 41)(38 42)(39 43)(40 44)(49 56)(50 53)(51 54)(52 55)(57 61)(58 62)(59 63)(60 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 9 3 11)(2 10 4 12)(5 23 7 21)(6 24 8 22)(13 25 15 27)(14 26 16 28)(17 52 19 50)(18 49 20 51)(29 34 31 36)(30 35 32 33)(37 58 39 60)(38 59 40 57)(41 62 43 64)(42 63 44 61)(45 54 47 56)(46 55 48 53)
(1 15 3 13)(2 16 4 14)(5 36 7 34)(6 33 8 35)(9 25 11 27)(10 26 12 28)(17 44 19 42)(18 41 20 43)(21 29 23 31)(22 30 24 32)(37 45 39 47)(38 46 40 48)(49 64 51 62)(50 61 52 63)(53 57 55 59)(54 58 56 60)
(9 15 27)(10 16 28)(11 13 25)(12 14 26)(17 52 63)(18 49 64)(19 50 61)(20 51 62)(21 34 31)(22 35 32)(23 36 29)(24 33 30)(45 54 58)(46 55 59)(47 56 60)(48 53 57)
(1 42 7 38)(2 41 8 37)(3 44 5 40)(4 43 6 39)(9 50 21 53)(10 49 22 56)(11 52 23 55)(12 51 24 54)(13 17 36 46)(14 20 33 45)(15 19 34 48)(16 18 35 47)(25 63 29 59)(26 62 30 58)(27 61 31 57)(28 64 32 60)
G:=sub<Sym(64)| (1,7)(2,8)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,36)(14,33)(15,34)(16,35)(17,46)(18,47)(19,48)(20,45)(25,29)(26,30)(27,31)(28,32)(37,41)(38,42)(39,43)(40,44)(49,56)(50,53)(51,54)(52,55)(57,61)(58,62)(59,63)(60,64), (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), (1,9,3,11)(2,10,4,12)(5,23,7,21)(6,24,8,22)(13,25,15,27)(14,26,16,28)(17,52,19,50)(18,49,20,51)(29,34,31,36)(30,35,32,33)(37,58,39,60)(38,59,40,57)(41,62,43,64)(42,63,44,61)(45,54,47,56)(46,55,48,53), (1,15,3,13)(2,16,4,14)(5,36,7,34)(6,33,8,35)(9,25,11,27)(10,26,12,28)(17,44,19,42)(18,41,20,43)(21,29,23,31)(22,30,24,32)(37,45,39,47)(38,46,40,48)(49,64,51,62)(50,61,52,63)(53,57,55,59)(54,58,56,60), (9,15,27)(10,16,28)(11,13,25)(12,14,26)(17,52,63)(18,49,64)(19,50,61)(20,51,62)(21,34,31)(22,35,32)(23,36,29)(24,33,30)(45,54,58)(46,55,59)(47,56,60)(48,53,57), (1,42,7,38)(2,41,8,37)(3,44,5,40)(4,43,6,39)(9,50,21,53)(10,49,22,56)(11,52,23,55)(12,51,24,54)(13,17,36,46)(14,20,33,45)(15,19,34,48)(16,18,35,47)(25,63,29,59)(26,62,30,58)(27,61,31,57)(28,64,32,60)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,36)(14,33)(15,34)(16,35)(17,46)(18,47)(19,48)(20,45)(25,29)(26,30)(27,31)(28,32)(37,41)(38,42)(39,43)(40,44)(49,56)(50,53)(51,54)(52,55)(57,61)(58,62)(59,63)(60,64), (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), (1,9,3,11)(2,10,4,12)(5,23,7,21)(6,24,8,22)(13,25,15,27)(14,26,16,28)(17,52,19,50)(18,49,20,51)(29,34,31,36)(30,35,32,33)(37,58,39,60)(38,59,40,57)(41,62,43,64)(42,63,44,61)(45,54,47,56)(46,55,48,53), (1,15,3,13)(2,16,4,14)(5,36,7,34)(6,33,8,35)(9,25,11,27)(10,26,12,28)(17,44,19,42)(18,41,20,43)(21,29,23,31)(22,30,24,32)(37,45,39,47)(38,46,40,48)(49,64,51,62)(50,61,52,63)(53,57,55,59)(54,58,56,60), (9,15,27)(10,16,28)(11,13,25)(12,14,26)(17,52,63)(18,49,64)(19,50,61)(20,51,62)(21,34,31)(22,35,32)(23,36,29)(24,33,30)(45,54,58)(46,55,59)(47,56,60)(48,53,57), (1,42,7,38)(2,41,8,37)(3,44,5,40)(4,43,6,39)(9,50,21,53)(10,49,22,56)(11,52,23,55)(12,51,24,54)(13,17,36,46)(14,20,33,45)(15,19,34,48)(16,18,35,47)(25,63,29,59)(26,62,30,58)(27,61,31,57)(28,64,32,60) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,21),(10,22),(11,23),(12,24),(13,36),(14,33),(15,34),(16,35),(17,46),(18,47),(19,48),(20,45),(25,29),(26,30),(27,31),(28,32),(37,41),(38,42),(39,43),(40,44),(49,56),(50,53),(51,54),(52,55),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,9,3,11),(2,10,4,12),(5,23,7,21),(6,24,8,22),(13,25,15,27),(14,26,16,28),(17,52,19,50),(18,49,20,51),(29,34,31,36),(30,35,32,33),(37,58,39,60),(38,59,40,57),(41,62,43,64),(42,63,44,61),(45,54,47,56),(46,55,48,53)], [(1,15,3,13),(2,16,4,14),(5,36,7,34),(6,33,8,35),(9,25,11,27),(10,26,12,28),(17,44,19,42),(18,41,20,43),(21,29,23,31),(22,30,24,32),(37,45,39,47),(38,46,40,48),(49,64,51,62),(50,61,52,63),(53,57,55,59),(54,58,56,60)], [(9,15,27),(10,16,28),(11,13,25),(12,14,26),(17,52,63),(18,49,64),(19,50,61),(20,51,62),(21,34,31),(22,35,32),(23,36,29),(24,33,30),(45,54,58),(46,55,59),(47,56,60),(48,53,57)], [(1,42,7,38),(2,41,8,37),(3,44,5,40),(4,43,6,39),(9,50,21,53),(10,49,22,56),(11,52,23,55),(12,51,24,54),(13,17,36,46),(14,20,33,45),(15,19,34,48),(16,18,35,47),(25,63,29,59),(26,62,30,58),(27,61,31,57),(28,64,32,60)]])
Matrix representation of (C2×C4).S4 ►in GL7(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 7 | 66 |
| 0 | 0 | 0 | 7 | 0 | 66 | 66 |
| 0 | 0 | 0 | 66 | 7 | 0 | 66 |
| 0 | 0 | 0 | 7 | 7 | 7 | 0 |
| 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 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 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 43 | 69 | 4 | 69 |
| 0 | 0 | 0 | 69 | 4 | 43 | 69 |
| 0 | 0 | 0 | 4 | 43 | 69 | 69 |
| 0 | 0 | 0 | 69 | 69 | 69 | 30 |
G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,7,66,7,0,0,0,66,0,7,7,0,0,0,7,66,0,7,0,0,0,66,66,66,0],[0,0,1,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0],[0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,43,69,4,69,0,0,0,69,4,43,69,0,0,0,4,43,69,69,0,0,0,69,69,69,30] >;
(C2×C4).S4 in GAP, Magma, Sage, TeX
(C_2\times C_4).S_4
% in TeX
G:=Group("(C2xC4).S4"); // GroupNames label
G:=SmallGroup(192,985);
// by ID
G=gap.SmallGroup(192,985);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,1373,680,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=b^2,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b^-1,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;
// generators/relations
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