Aliases: C23.15S4, Q8⋊Dic3⋊6C2, (C2×Q8)⋊2Dic3, (C2×Q8).14D6, C22.18(C2×S4), Q8.1(C2×Dic3), (C22×Q8).4S3, C22.5(A4⋊C4), (C2×SL2(𝔽3))⋊3C4, C2.3(Q8.D6), SL2(𝔽3).5(C2×C4), (C22×SL2(𝔽3)).4C2, (C2×SL2(𝔽3)).14C22, C2.6(C2×A4⋊C4), SmallGroup(192,979)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C23.15S4 |
Generators and relations for C23.15S4
G = < a,b,c,d,e,f,g | a2=b2=c2=f3=1, d2=e2=c, g2=b, ab=ba, gag-1=ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, bg=gb, ede-1=cd=dc, geg-1=ce=ec, cf=fc, cg=gc, fdf-1=cde, gdg-1=de, fef-1=d, gfg-1=f-1 >
Subgroups: 283 in 81 conjugacy classes, 21 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C8, C2×C4, Q8, Q8, C23, Dic3, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C22×C6, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C6.D4, C2×SL2(𝔽3), C2×SL2(𝔽3), C23.38D4, Q8⋊Dic3, C22×SL2(𝔽3), C23.15S4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, Q8.D6, C2×A4⋊C4, C23.15S4
Character table of C23.15S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | |
| 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 | 1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -1 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 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 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | -3 | -3 | -3 | 3 | 0 | -1 | -1 | 1 | 1 | -i | i | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from A4⋊C4 |
| ρ18 | 3 | 3 | -3 | -3 | 3 | -3 | 0 | -1 | 1 | 1 | -1 | -i | -i | i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | -i | i | complex lifted from A4⋊C4 |
| ρ19 | 3 | 3 | -3 | -3 | -3 | 3 | 0 | -1 | -1 | 1 | 1 | i | -i | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from A4⋊C4 |
| ρ20 | 3 | 3 | -3 | -3 | 3 | -3 | 0 | -1 | 1 | 1 | -1 | i | i | -i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | i | -i | complex lifted from A4⋊C4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | √-3 | -√-3 | -√-3 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -√-3 | √-3 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | -√-3 | √-3 | -√-3 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | -√-3 | √-3 | √-3 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
►On 32 points
Generators in S32
(1 16)(2 13)(3 14)(4 15)(5 25)(6 26)(7 27)(8 28)(9 17)(10 18)(11 19)(12 20)(21 31)(22 32)(23 29)(24 30)
(1 16)(2 13)(3 14)(4 15)(5 27)(6 28)(7 25)(8 26)(9 17)(10 18)(11 19)(12 20)(21 29)(22 30)(23 31)(24 32)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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)
(1 10 3 12)(2 9 4 11)(5 30 7 32)(6 29 8 31)(13 17 15 19)(14 20 16 18)(21 26 23 28)(22 25 24 27)
(2 10 11)(4 12 9)(5 30 8)(6 7 32)(13 18 19)(15 20 17)(22 26 27)(24 28 25)
(1 31 16 23)(2 5 13 27)(3 29 14 21)(4 7 15 25)(6 20 28 12)(8 18 26 10)(9 32 17 24)(11 30 19 22)
G:=sub<Sym(32)| (1,16)(2,13)(3,14)(4,15)(5,25)(6,26)(7,27)(8,28)(9,17)(10,18)(11,19)(12,20)(21,31)(22,32)(23,29)(24,30), (1,16)(2,13)(3,14)(4,15)(5,27)(6,28)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,10,3,12)(2,9,4,11)(5,30,7,32)(6,29,8,31)(13,17,15,19)(14,20,16,18)(21,26,23,28)(22,25,24,27), (2,10,11)(4,12,9)(5,30,8)(6,7,32)(13,18,19)(15,20,17)(22,26,27)(24,28,25), (1,31,16,23)(2,5,13,27)(3,29,14,21)(4,7,15,25)(6,20,28,12)(8,18,26,10)(9,32,17,24)(11,30,19,22)>;
G:=Group( (1,16)(2,13)(3,14)(4,15)(5,25)(6,26)(7,27)(8,28)(9,17)(10,18)(11,19)(12,20)(21,31)(22,32)(23,29)(24,30), (1,16)(2,13)(3,14)(4,15)(5,27)(6,28)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,10,3,12)(2,9,4,11)(5,30,7,32)(6,29,8,31)(13,17,15,19)(14,20,16,18)(21,26,23,28)(22,25,24,27), (2,10,11)(4,12,9)(5,30,8)(6,7,32)(13,18,19)(15,20,17)(22,26,27)(24,28,25), (1,31,16,23)(2,5,13,27)(3,29,14,21)(4,7,15,25)(6,20,28,12)(8,18,26,10)(9,32,17,24)(11,30,19,22) );
G=PermutationGroup([[(1,16),(2,13),(3,14),(4,15),(5,25),(6,26),(7,27),(8,28),(9,17),(10,18),(11,19),(12,20),(21,31),(22,32),(23,29),(24,30)], [(1,16),(2,13),(3,14),(4,15),(5,27),(6,28),(7,25),(8,26),(9,17),(10,18),(11,19),(12,20),(21,29),(22,30),(23,31),(24,32)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,10,3,12),(2,9,4,11),(5,30,7,32),(6,29,8,31),(13,17,15,19),(14,20,16,18),(21,26,23,28),(22,25,24,27)], [(2,10,11),(4,12,9),(5,30,8),(6,7,32),(13,18,19),(15,20,17),(22,26,27),(24,28,25)], [(1,31,16,23),(2,5,13,27),(3,29,14,21),(4,7,15,25),(6,20,28,12),(8,18,26,10),(9,32,17,24),(11,30,19,22)]])
Matrix representation of C23.15S4 ►in GL7(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 | 64 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 65 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 | 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 | 64 | 65 | 0 | 0 |
| 0 | 0 | 0 | 65 | 9 | 0 | 0 |
| 0 | 0 | 0 | 23 | 33 | 9 | 8 |
| 0 | 0 | 0 | 33 | 40 | 8 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 33 | 40 | 8 | 64 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 71 | 0 |
| 0 | 0 | 0 | 65 | 9 | 0 | 71 |
| 0 | 0 | 0 | 1 | 0 | 72 | 0 |
| 0 | 0 | 0 | 33 | 41 | 8 | 64 |
G:=sub<GL(7,GF(73))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,72,8,0,0,0,0,72,0,64,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72],[0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,65,9,0,0,0,0,0,9,8,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,0,64,65,23,33,0,0,0,65,9,33,40,0,0,0,0,0,9,8,0,0,0,0,0,8,64],[1,72,0,0,0,0,0,0,72,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,9,0,33,0,0,0,0,8,0,40,0,0,0,0,0,1,8,0,0,0,0,0,0,64],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,65,1,33,0,0,0,0,9,0,41,0,0,0,71,0,72,8,0,0,0,0,71,0,64] >;
C23.15S4 in GAP, Magma, Sage, TeX
C_2^3._{15}S_4 % in TeX
G:=Group("C2^3.15S4"); // GroupNames label
G:=SmallGroup(192,979);
// by ID
G=gap.SmallGroup(192,979);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,1373,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^3=1,d^2=e^2=c,g^2=b,a*b=b*a,g*a*g^-1=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,b*f=f*b,b*g=g*b,e*d*e^-1=c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=c*d*e,g*d*g^-1=d*e,f*e*f^-1=d,g*f*g^-1=f^-1>;
// generators/relations
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