non-abelian, soluble, monomial
Aliases: C24⋊1D9, C23.5D18, C6.S4⋊C2, C3.A4⋊2D4, C6.26(C2×S4), (C2×C6).11S4, C22⋊(C9⋊D4), C3.(A4⋊D4), (C23×C6).3S3, C22⋊2(C3.S4), (C22×C6).17D6, (C2×C3.S4)⋊2C2, (C2×C6).(C3⋊D4), C2.11(C2×C3.S4), (C22×C3.A4)⋊2C2, (C2×C3.A4).5C22, SmallGroup(288,342)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D18
G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=a, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ece-1=bc=cb, be=eb, dcd-1=b, ede-1=ad-1 >
Subgroups: 652 in 108 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22 [×2], C22 [×11], S3, C6, C6 [×4], C2×C4 [×3], D4 [×6], C23, C23 [×5], C9, Dic3 [×3], D6 [×3], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×3], C2×D4 [×3], C24, D9, C18 [×2], C2×Dic3 [×3], C3⋊D4 [×6], C22×S3, C22×C6, C22×C6 [×4], C22≀C2, Dic9, C3.A4, D18, C2×C18, C6.D4 [×3], C2×C3⋊D4 [×3], C23×C6, C9⋊D4, C3.S4, C2×C3.A4, C2×C3.A4, C24⋊4S3, C6.S4, C2×C3.S4, C22×C3.A4, C23.D18
Quotients: C1, C2 [×3], C22, S3, D4, D6, D9, C3⋊D4, S4, D18, C2×S4, C9⋊D4, C3.S4, A4⋊D4, C2×C3.S4, C23.D18
Character table of C23.D18
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 36 | 2 | 36 | 36 | 36 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | 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 | -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 | 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 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ13 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ14 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | -√-3 | 1 | √-3 | 1 | -1 | √-3 | -√-3 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ97-ζ92 | -ζ97+ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ97-ζ92 | ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ15 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | 1 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | √-3 | 1 | -√-3 | 1 | -1 | -√-3 | √-3 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ98-ζ9 | -ζ98+ζ9 | ζ95-ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95+ζ94 | complex lifted from C9⋊D4 |
| ρ17 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | -√-3 | √-3 | √-3 | √-3 | -√-3 | 1 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | √-3 | 1 | -√-3 | 1 | -1 | -√-3 | √-3 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ95-ζ94 | -ζ95+ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ19 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | -√-3 | 1 | √-3 | 1 | -1 | √-3 | -√-3 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ95-ζ94 | ζ95-ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97+ζ92 | complex lifted from C9⋊D4 |
| ρ20 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | √-3 | 1 | -√-3 | 1 | -1 | -√-3 | √-3 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ97-ζ92 | ζ97-ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ97+ζ92 | -ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98+ζ9 | complex lifted from C9⋊D4 |
| ρ21 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | -√-3 | 1 | √-3 | 1 | -1 | √-3 | -√-3 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ98-ζ9 | ζ98-ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ22 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | 3 | -1 | 1 | 1 | -3 | 3 | -3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 1 | 1 | -1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ24 | 3 | 3 | 3 | -1 | -1 | -1 | 1 | 3 | -1 | -1 | 1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ25 | 3 | 3 | -3 | -1 | -1 | 1 | 1 | 3 | 1 | -1 | -1 | -3 | 3 | -3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 6 | 6 | -6 | -2 | -2 | 2 | 0 | -3 | 0 | 0 | 0 | 3 | -3 | 3 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
| ρ27 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
| ρ28 | 6 | 6 | 6 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 0 | -3 | -3 | -3 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ29 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | -3√-3 | 3 | 3√-3 | -1 | 1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 3√-3 | 3 | -3√-3 | -1 | 1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)
(1 31)(3 33)(4 34)(6 36)(7 19)(9 21)(10 22)(12 24)(13 25)(15 27)(16 28)(18 30)
(1 31)(2 32)(4 34)(5 35)(7 19)(8 20)(10 22)(11 23)(13 25)(14 26)(16 28)(17 29)
(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)
(1 18 31 30)(2 29 32 17)(3 16 33 28)(4 27 34 15)(5 14 35 26)(6 25 36 13)(7 12 19 24)(8 23 20 11)(9 10 21 22)
G:=sub<Sym(36)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30), (1,31)(3,33)(4,34)(6,36)(7,19)(9,21)(10,22)(12,24)(13,25)(15,27)(16,28)(18,30), (1,31)(2,32)(4,34)(5,35)(7,19)(8,20)(10,22)(11,23)(13,25)(14,26)(16,28)(17,29), (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), (1,18,31,30)(2,29,32,17)(3,16,33,28)(4,27,34,15)(5,14,35,26)(6,25,36,13)(7,12,19,24)(8,23,20,11)(9,10,21,22)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30), (1,31)(3,33)(4,34)(6,36)(7,19)(9,21)(10,22)(12,24)(13,25)(15,27)(16,28)(18,30), (1,31)(2,32)(4,34)(5,35)(7,19)(8,20)(10,22)(11,23)(13,25)(14,26)(16,28)(17,29), (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), (1,18,31,30)(2,29,32,17)(3,16,33,28)(4,27,34,15)(5,14,35,26)(6,25,36,13)(7,12,19,24)(8,23,20,11)(9,10,21,22) );
G=PermutationGroup([(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30)], [(1,31),(3,33),(4,34),(6,36),(7,19),(9,21),(10,22),(12,24),(13,25),(15,27),(16,28),(18,30)], [(1,31),(2,32),(4,34),(5,35),(7,19),(8,20),(10,22),(11,23),(13,25),(14,26),(16,28),(17,29)], [(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)], [(1,18,31,30),(2,29,32,17),(3,16,33,28),(4,27,34,15),(5,14,35,26),(6,25,36,13),(7,12,19,24),(8,23,20,11),(9,10,21,22)])
Matrix representation of C23.D18 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 1 |
| 0 | 0 | 36 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 1 | 36 | 0 |
| 9 | 21 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 28 | 16 | 0 | 0 | 0 |
| 18 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[9,0,0,0,0,21,4,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[28,18,0,0,0,16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C23.D18 in GAP, Magma, Sage, TeX
C_2^3.D_{18} % in TeX
G:=Group("C2^3.D18"); // GroupNames label
G:=SmallGroup(288,342);
// by ID
G=gap.SmallGroup(288,342);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,1123,192,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^18=1,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*c*e^-1=b*c=c*b,b*e=e*b,d*c*d^-1=b,e*d*e^-1=a*d^-1>;
// generators/relations
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