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G = C23.D18order 288 = 25·32

5th non-split extension by C23 of D18 acting via D18/C6=S3

non-abelian, soluble, monomial

Aliases: C241D9, C23.5D18, C6.S4⋊C2, C3.A42D4, C6.26(C2×S4), (C2×C6).11S4, C22⋊(C9⋊D4), C3.(A4⋊D4), (C23×C6).3S3, C222(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

C1C22C2×C3.A4 — C23.D18
C1C22C2×C6C3.A4C2×C3.A4C2×C3.S4 — C23.D18
C3.A4C2×C3.A4 — C23.D18
C1C2C22

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, C244S3, 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 12A2B2C2D2E2F34A4B4C6A6B6C6D6E6F6G9A9B9C18A18B18C18D18E18F18G18H18I
 size 1123363623636362226666888888888888
ρ1111111111111111111111111111111    trivial
ρ211-111-1-111-11-11-111-1-11111-1-1-1-1-111-1    linear of order 2
ρ3111111-11-1-1-11111111111111111111    linear of order 2
ρ411-111-111-11-1-11-111-1-11111-1-1-1-1-111-1    linear of order 2
ρ5222222020002222222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ622-222-202000-22-222-2-2-1-1-1-111111-1-11    orthogonal lifted from D6
ρ72-20-220020000-20-2200222-200000-2-20    orthogonal lifted from D4
ρ82222220-1000-1-1-1-1-1-1-1ζ989ζ9594ζ9792ζ989ζ989ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ92222220-1000-1-1-1-1-1-1-1ζ9594ζ9792ζ989ζ9594ζ9594ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ1022-222-20-10001-11-1-111ζ9594ζ9792ζ989ζ9594959497929899594989ζ989ζ97929792    orthogonal lifted from D18
ρ112222220-1000-1-1-1-1-1-1-1ζ9792ζ989ζ9594ζ9792ζ9792ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ989    orthogonal lifted from D9
ρ1222-222-20-10001-11-1-111ζ9792ζ989ζ9594ζ97929792989959497929594ζ9594ζ989989    orthogonal lifted from D18
ρ1322-222-20-10001-11-1-111ζ989ζ9594ζ9792ζ989989959497929899792ζ9792ζ95949594    orthogonal lifted from D18
ρ142-20-2200-1000--31-31-1-3--3ζ9792ζ989ζ9594979297929899594ζ9792ζ95949594989ζ989    complex lifted from C9⋊D4
ρ152-20-220020000-20-2200-1-1-11-3--3--3--3-311-3    complex lifted from C3⋊D4
ρ162-20-2200-1000-31--31-1--3-3ζ989ζ9594ζ9792989989ζ95949792ζ989ζ9792979295949594    complex lifted from C9⋊D4
ρ172-20-220020000-20-2200-1-1-11--3-3-3-3--311--3    complex lifted from C3⋊D4
ρ182-20-2200-1000-31--31-1--3-3ζ9594ζ9792ζ989959495949792ζ989ζ95949899899792ζ9792    complex lifted from C9⋊D4
ρ192-20-2200-1000--31-31-1-3--3ζ9594ζ9792ζ9899594ζ9594ζ97929899594ζ98998997929792    complex lifted from C9⋊D4
ρ202-20-2200-1000-31--31-1--3-3ζ9792ζ989ζ95949792ζ9792ζ989ζ9594979295949594989989    complex lifted from C9⋊D4
ρ212-20-2200-1000--31-31-1-3--3ζ989ζ9594ζ9792989ζ9899594ζ9792989979297929594ζ9594    complex lifted from C9⋊D4
ρ2233-3-1-11-13-111-33-3-1-111000000000000    orthogonal lifted from C2×S4
ρ23333-1-1-1-1311-1333-1-1-1-1000000000000    orthogonal lifted from S4
ρ24333-1-1-113-1-11333-1-1-1-1000000000000    orthogonal lifted from S4
ρ2533-3-1-11131-1-1-33-3-1-111000000000000    orthogonal lifted from C2×S4
ρ2666-6-2-220-30003-3311-1-1000000000000    orthogonal lifted from C2×C3.S4
ρ276-602-20060000-602-200000000000000    orthogonal lifted from A4⋊D4
ρ28666-2-2-20-3000-3-3-31111000000000000    orthogonal lifted from C3.S4
ρ296-602-200-3000-3-333-3-11--3-3000000000000    complex faithful
ρ306-602-200-30003-33-3-3-11-3--3000000000000    complex faithful

Smallest permutation representation of C23.D18
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)

360000
036000
00100
00010
00001
,
10000
01000
003600
003601
003610
,
10000
01000
000361
000360
001360
,
921000
04000
00001
00100
00010
,
2816000
189000
00100
00001
00010

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

Export

Character table of C23.D18 in TeX

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