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 |
(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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