direct product, non-abelian, soluble
Aliases: C2×C4.S4, CSU2(𝔽3)⋊2C22, SL2(𝔽3).3C23, C4.23(C2×S4), (C2×C4).17S4, C4○D4.15D6, (C2×Q8).21D6, C22.28(C2×S4), C2.13(C22×S4), Q8.3(C22×S3), C4.A4.10C22, (C2×CSU2(𝔽3))⋊5C2, (C2×SL2(𝔽3)).21C22, (C2×C4.A4).3C2, (C2×C4○D4).6S3, SmallGroup(192,1479)
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=f2=b2, 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: 459 in 141 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), Dic6, C2×Dic3, C2×C12, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, CSU2(𝔽3), C2×SL2(𝔽3), C4.A4, C2×Dic6, C2×C8.C22, C2×CSU2(𝔽3), C4.S4, C2×C4.A4, C2×C4.S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C4.S4, C22×S4, 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 | -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 | 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 | orthogonal lifted from D6 |
| 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 | orthogonal lifted from D6 |
| 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 C2×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 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 C2×S4 |
| ρ17 | 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 |
| ρ18 | 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 |
| ρ19 | 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 |
| ρ20 | 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 |
| 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 | symplectic lifted from C4.S4, Schur index 2 |
| ρ22 | 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 |
| 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 | symplectic lifted from C4.S4, Schur index 2 |
| ρ24 | 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 |
| 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 25)(2 26)(3 27)(4 28)(5 23)(6 24)(7 21)(8 22)(9 52)(10 49)(11 50)(12 51)(13 31)(14 32)(15 29)(16 30)(17 34)(18 35)(19 36)(20 33)(37 55)(38 56)(39 53)(40 54)(41 59)(42 60)(43 57)(44 58)(45 63)(46 64)(47 61)(48 62)
(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 13 3 15)(2 14 4 16)(5 20 7 18)(6 17 8 19)(9 63 11 61)(10 64 12 62)(21 35 23 33)(22 36 24 34)(25 31 27 29)(26 32 28 30)(37 41 39 43)(38 42 40 44)(45 50 47 52)(46 51 48 49)(53 57 55 59)(54 58 56 60)
(1 17 3 19)(2 18 4 20)(5 14 7 16)(6 15 8 13)(9 56 11 54)(10 53 12 55)(21 30 23 32)(22 31 24 29)(25 34 27 36)(26 35 28 33)(37 49 39 51)(38 50 40 52)(41 48 43 46)(42 45 44 47)(57 64 59 62)(58 61 60 63)
(5 14 18)(6 15 19)(7 16 20)(8 13 17)(9 63 58)(10 64 59)(11 61 60)(12 62 57)(21 30 33)(22 31 34)(23 32 35)(24 29 36)(41 49 46)(42 50 47)(43 51 48)(44 52 45)
(1 54 3 56)(2 53 4 55)(5 59 7 57)(6 58 8 60)(9 17 11 19)(10 20 12 18)(13 61 15 63)(14 64 16 62)(21 43 23 41)(22 42 24 44)(25 40 27 38)(26 39 28 37)(29 45 31 47)(30 48 32 46)(33 51 35 49)(34 50 36 52)
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,52)(10,49)(11,50)(12,51)(13,31)(14,32)(15,29)(16,30)(17,34)(18,35)(19,36)(20,33)(37,55)(38,56)(39,53)(40,54)(41,59)(42,60)(43,57)(44,58)(45,63)(46,64)(47,61)(48,62), (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), (1,13,3,15)(2,14,4,16)(5,20,7,18)(6,17,8,19)(9,63,11,61)(10,64,12,62)(21,35,23,33)(22,36,24,34)(25,31,27,29)(26,32,28,30)(37,41,39,43)(38,42,40,44)(45,50,47,52)(46,51,48,49)(53,57,55,59)(54,58,56,60), (1,17,3,19)(2,18,4,20)(5,14,7,16)(6,15,8,13)(9,56,11,54)(10,53,12,55)(21,30,23,32)(22,31,24,29)(25,34,27,36)(26,35,28,33)(37,49,39,51)(38,50,40,52)(41,48,43,46)(42,45,44,47)(57,64,59,62)(58,61,60,63), (5,14,18)(6,15,19)(7,16,20)(8,13,17)(9,63,58)(10,64,59)(11,61,60)(12,62,57)(21,30,33)(22,31,34)(23,32,35)(24,29,36)(41,49,46)(42,50,47)(43,51,48)(44,52,45), (1,54,3,56)(2,53,4,55)(5,59,7,57)(6,58,8,60)(9,17,11,19)(10,20,12,18)(13,61,15,63)(14,64,16,62)(21,43,23,41)(22,42,24,44)(25,40,27,38)(26,39,28,37)(29,45,31,47)(30,48,32,46)(33,51,35,49)(34,50,36,52)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,52)(10,49)(11,50)(12,51)(13,31)(14,32)(15,29)(16,30)(17,34)(18,35)(19,36)(20,33)(37,55)(38,56)(39,53)(40,54)(41,59)(42,60)(43,57)(44,58)(45,63)(46,64)(47,61)(48,62), (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), (1,13,3,15)(2,14,4,16)(5,20,7,18)(6,17,8,19)(9,63,11,61)(10,64,12,62)(21,35,23,33)(22,36,24,34)(25,31,27,29)(26,32,28,30)(37,41,39,43)(38,42,40,44)(45,50,47,52)(46,51,48,49)(53,57,55,59)(54,58,56,60), (1,17,3,19)(2,18,4,20)(5,14,7,16)(6,15,8,13)(9,56,11,54)(10,53,12,55)(21,30,23,32)(22,31,24,29)(25,34,27,36)(26,35,28,33)(37,49,39,51)(38,50,40,52)(41,48,43,46)(42,45,44,47)(57,64,59,62)(58,61,60,63), (5,14,18)(6,15,19)(7,16,20)(8,13,17)(9,63,58)(10,64,59)(11,61,60)(12,62,57)(21,30,33)(22,31,34)(23,32,35)(24,29,36)(41,49,46)(42,50,47)(43,51,48)(44,52,45), (1,54,3,56)(2,53,4,55)(5,59,7,57)(6,58,8,60)(9,17,11,19)(10,20,12,18)(13,61,15,63)(14,64,16,62)(21,43,23,41)(22,42,24,44)(25,40,27,38)(26,39,28,37)(29,45,31,47)(30,48,32,46)(33,51,35,49)(34,50,36,52) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,23),(6,24),(7,21),(8,22),(9,52),(10,49),(11,50),(12,51),(13,31),(14,32),(15,29),(16,30),(17,34),(18,35),(19,36),(20,33),(37,55),(38,56),(39,53),(40,54),(41,59),(42,60),(43,57),(44,58),(45,63),(46,64),(47,61),(48,62)], [(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)], [(1,13,3,15),(2,14,4,16),(5,20,7,18),(6,17,8,19),(9,63,11,61),(10,64,12,62),(21,35,23,33),(22,36,24,34),(25,31,27,29),(26,32,28,30),(37,41,39,43),(38,42,40,44),(45,50,47,52),(46,51,48,49),(53,57,55,59),(54,58,56,60)], [(1,17,3,19),(2,18,4,20),(5,14,7,16),(6,15,8,13),(9,56,11,54),(10,53,12,55),(21,30,23,32),(22,31,24,29),(25,34,27,36),(26,35,28,33),(37,49,39,51),(38,50,40,52),(41,48,43,46),(42,45,44,47),(57,64,59,62),(58,61,60,63)], [(5,14,18),(6,15,19),(7,16,20),(8,13,17),(9,63,58),(10,64,59),(11,61,60),(12,62,57),(21,30,33),(22,31,34),(23,32,35),(24,29,36),(41,49,46),(42,50,47),(43,51,48),(44,52,45)], [(1,54,3,56),(2,53,4,55),(5,59,7,57),(6,58,8,60),(9,17,11,19),(10,20,12,18),(13,61,15,63),(14,64,16,62),(21,43,23,41),(22,42,24,44),(25,40,27,38),(26,39,28,37),(29,45,31,47),(30,48,32,46),(33,51,35,49),(34,50,36,52)]])
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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 | 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 | 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 | 72 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 |
G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,66,7,66,0,0,0,7,0,66,66,0,0,0,66,7,0,66,0,0,0,7,7,7,0],[72,72,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0],[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],[0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,0,0,0,27] >;
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(192,1479);
// by ID
G=gap.SmallGroup(192,1479);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,2102,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=f^2=b^2,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
Export