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G = C9⋊He3⋊2C2  order 486 = 2·35

2nd semidirect product of C9⋊He3 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: C9⋊He3⋊2C2, C9⋊(C32⋊C6), C32⋊C9⋊9S3, C32⋊1(C9⋊C6), (C32×C9)⋊16C6, C32⋊4D9⋊7C3, C33.8(C3⋊S3), (C3×He3).12S3, C33.63(C3×S3), C3.4(He3⋊4S3), (C3×3- 1+2)⋊3S3, C3.4(He3.4S3), C3.3(C33.S3), (C3×C9).30(C3×S3), C32.38(C3×C3⋊S3), SmallGroup(486,148)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C9 — C9⋊He3⋊2C2
Chief series
C1 — C3 — C32 — C3×C9 — C32×C9 — C9⋊He3 — C9⋊He3⋊2C2
Lower central
C32×C9 — C9⋊He3⋊2C2
Upper central
C1

Generators and relations for C9⋊He3⋊2C2
 G = < a,b,c,d,e | a9=b3=c3=d3=e2=1, ab=ba, ac=ca, dad-1=a7, eae=a-1, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ece=c-1, ede=a3d >

Subgroups: 1124 in 109 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C3×3- 1+2, C32⋊D9, He3⋊4S3, C33.S3, C32⋊4D9, C9⋊He3, C9⋊He3⋊2C2
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C9⋊C6, C3×C3⋊S3, He3⋊4S3, C33.S3, He3.4S3, C9⋊He3⋊2C2

Character table of C9⋊He3⋊2C2

 class 123A3B3C3D3E3F3G3H3I3J3K6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O
 size 18122226669918188181666666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-111111111111-1-1111111111111111    linear of order 2
ρ31-11111111ζ32ζ3ζ32ζ3ζ6ζ65111111111ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 6
ρ4111111111ζ32ζ3ζ32ζ3ζ32ζ3111111111ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ51-11111111ζ3ζ32ζ3ζ32ζ65ζ6111111111ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 6
ρ6111111111ζ3ζ32ζ3ζ32ζ3ζ32111111111ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ7202222222222200-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ8202222-1-1-122-1-100-1-1222-1-1-1-1-1-1-1-122    orthogonal lifted from S3
ρ9202222-1-1-122-1-10022-1-1-1-1-1-12-1-122-1-1    orthogonal lifted from S3
ρ10202222-1-1-122-1-100-1-1-1-1-1222-122-1-1-1-1    orthogonal lifted from S3
ρ11202222-1-1-1-1-√-3-1+√-3ζ6ζ6500-1-1-1-1-1222-1-1+√-3-1-√-3ζ6ζ65ζ65ζ6    complex lifted from C3×S3
ρ12202222-1-1-1-1-√-3-1+√-3ζ6ζ6500-1-1222-1-1-1-1ζ65ζ6ζ6ζ65-1+√-3-1-√-3    complex lifted from C3×S3
ρ13202222-1-1-1-1+√-3-1-√-3ζ65ζ60022-1-1-1-1-1-12ζ6ζ65-1+√-3-1-√-3ζ6ζ65    complex lifted from C3×S3
ρ14202222-1-1-1-1-√-3-1+√-3ζ6ζ650022-1-1-1-1-1-12ζ65ζ6-1-√-3-1+√-3ζ65ζ6    complex lifted from C3×S3
ρ15202222-1-1-1-1+√-3-1-√-3ζ65ζ600-1-1222-1-1-1-1ζ6ζ65ζ65ζ6-1-√-3-1+√-3    complex lifted from C3×S3
ρ16202222222-1+√-3-1-√-3-1+√-3-1-√-300-1-1-1-1-1-1-1-1-1ζ6ζ65ζ65ζ6ζ6ζ65    complex lifted from C3×S3
ρ17202222222-1-√-3-1+√-3-1-√-3-1+√-300-1-1-1-1-1-1-1-1-1ζ65ζ6ζ6ζ65ζ65ζ6    complex lifted from C3×S3
ρ18202222-1-1-1-1+√-3-1-√-3ζ65ζ600-1-1-1-1-1222-1-1-√-3-1+√-3ζ65ζ6ζ6ζ65    complex lifted from C3×S3
ρ1960-3-3-36000000000006-3-30000000000    orthogonal lifted from C32⋊C6
ρ20606-3-3-36-3-3000000000000000000000    orthogonal lifted from C9⋊C6
ρ2160-3-3-3600000000000-36-30000000000    orthogonal lifted from C32⋊C6
ρ22606-3-3-3-36-3000000000000000000000    orthogonal lifted from C9⋊C6
ρ23606-3-3-3-3-36000000000000000000000    orthogonal lifted from C9⋊C6
ρ2460-3-3-3600000000000-3-360000000000    orthogonal lifted from C32⋊C6
ρ2560-36-3-3000000000000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000    orthogonal lifted from He3.4S3
ρ2660-3-36-30000000003ζ97+3ζ923ζ98+3ζ90000003ζ95+3ζ94000000    orthogonal lifted from He3.4S3
ρ2760-3-36-30000000003ζ95+3ζ943ζ97+3ζ920000003ζ98+3ζ9000000    orthogonal lifted from He3.4S3
ρ2860-36-3-3000000000000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000    orthogonal lifted from He3.4S3
ρ2960-36-3-3000000000000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000    orthogonal lifted from He3.4S3
ρ3060-3-36-30000000003ζ98+3ζ93ζ95+3ζ940000003ζ97+3ζ92000000    orthogonal lifted from He3.4S3

Smallest permutation representation of C9⋊He3⋊2C2
►On 81 points
Generators in S81
(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 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)

(1 7 4)(2 8 5)(3 9 6)(10 79 20)(11 80 21)(12 81 22)(13 73 23)(14 74 24)(15 75 25)(16 76 26)(17 77 27)(18 78 19)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 71 56)(47 72 57)(48 64 58)(49 65 59)(50 66 60)(51 67 61)(52 68 62)(53 69 63)(54 70 55)

(1 33 44)(2 34 45)(3 35 37)(4 36 38)(5 28 39)(6 29 40)(7 30 41)(8 31 42)(9 32 43)(10 73 26)(11 74 27)(12 75 19)(13 76 20)(14 77 21)(15 78 22)(16 79 23)(17 80 24)(18 81 25)(46 62 65)(47 63 66)(48 55 67)(49 56 68)(50 57 69)(51 58 70)(52 59 71)(53 60 72)(54 61 64)

(1 51 13)(2 46 11)(3 50 18)(4 54 16)(5 49 14)(6 53 12)(7 48 10)(8 52 17)(9 47 15)(19 40 72)(20 44 70)(21 39 68)(22 43 66)(23 38 64)(24 42 71)(25 37 69)(26 41 67)(27 45 65)(28 56 77)(29 60 75)(30 55 73)(31 59 80)(32 63 78)(33 58 76)(34 62 74)(35 57 81)(36 61 79)

(2 9)(3 8)(4 7)(5 6)(10 13)(11 12)(14 18)(15 17)(19 74)(20 73)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 40)(29 39)(30 38)(31 37)(32 45)(33 44)(34 43)(35 42)(36 41)(46 50)(47 49)(51 54)(52 53)(55 67)(56 66)(57 65)(58 64)(59 72)(60 71)(61 70)(62 69)(63 68)
 
G:=sub<Sym(81)| (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)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,7,4)(2,8,5)(3,9,6)(10,79,20)(11,80,21)(12,81,22)(13,73,23)(14,74,24)(15,75,25)(16,76,26)(17,77,27)(18,78,19)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,71,56)(47,72,57)(48,64,58)(49,65,59)(50,66,60)(51,67,61)(52,68,62)(53,69,63)(54,70,55), (1,33,44)(2,34,45)(3,35,37)(4,36,38)(5,28,39)(6,29,40)(7,30,41)(8,31,42)(9,32,43)(10,73,26)(11,74,27)(12,75,19)(13,76,20)(14,77,21)(15,78,22)(16,79,23)(17,80,24)(18,81,25)(46,62,65)(47,63,66)(48,55,67)(49,56,68)(50,57,69)(51,58,70)(52,59,71)(53,60,72)(54,61,64), (1,51,13)(2,46,11)(3,50,18)(4,54,16)(5,49,14)(6,53,12)(7,48,10)(8,52,17)(9,47,15)(19,40,72)(20,44,70)(21,39,68)(22,43,66)(23,38,64)(24,42,71)(25,37,69)(26,41,67)(27,45,65)(28,56,77)(29,60,75)(30,55,73)(31,59,80)(32,63,78)(33,58,76)(34,62,74)(35,57,81)(36,61,79), (2,9)(3,8)(4,7)(5,6)(10,13)(11,12)(14,18)(15,17)(19,74)(20,73)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,40)(29,39)(30,38)(31,37)(32,45)(33,44)(34,43)(35,42)(36,41)(46,50)(47,49)(51,54)(52,53)(55,67)(56,66)(57,65)(58,64)(59,72)(60,71)(61,70)(62,69)(63,68)>;
 
G:=Group( (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)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,7,4)(2,8,5)(3,9,6)(10,79,20)(11,80,21)(12,81,22)(13,73,23)(14,74,24)(15,75,25)(16,76,26)(17,77,27)(18,78,19)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,71,56)(47,72,57)(48,64,58)(49,65,59)(50,66,60)(51,67,61)(52,68,62)(53,69,63)(54,70,55), (1,33,44)(2,34,45)(3,35,37)(4,36,38)(5,28,39)(6,29,40)(7,30,41)(8,31,42)(9,32,43)(10,73,26)(11,74,27)(12,75,19)(13,76,20)(14,77,21)(15,78,22)(16,79,23)(17,80,24)(18,81,25)(46,62,65)(47,63,66)(48,55,67)(49,56,68)(50,57,69)(51,58,70)(52,59,71)(53,60,72)(54,61,64), (1,51,13)(2,46,11)(3,50,18)(4,54,16)(5,49,14)(6,53,12)(7,48,10)(8,52,17)(9,47,15)(19,40,72)(20,44,70)(21,39,68)(22,43,66)(23,38,64)(24,42,71)(25,37,69)(26,41,67)(27,45,65)(28,56,77)(29,60,75)(30,55,73)(31,59,80)(32,63,78)(33,58,76)(34,62,74)(35,57,81)(36,61,79), (2,9)(3,8)(4,7)(5,6)(10,13)(11,12)(14,18)(15,17)(19,74)(20,73)(21,81)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,40)(29,39)(30,38)(31,37)(32,45)(33,44)(34,43)(35,42)(36,41)(46,50)(47,49)(51,54)(52,53)(55,67)(56,66)(57,65)(58,64)(59,72)(60,71)(61,70)(62,69)(63,68) );
 
G=PermutationGroup([[(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),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(1,7,4),(2,8,5),(3,9,6),(10,79,20),(11,80,21),(12,81,22),(13,73,23),(14,74,24),(15,75,25),(16,76,26),(17,77,27),(18,78,19),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,71,56),(47,72,57),(48,64,58),(49,65,59),(50,66,60),(51,67,61),(52,68,62),(53,69,63),(54,70,55)], [(1,33,44),(2,34,45),(3,35,37),(4,36,38),(5,28,39),(6,29,40),(7,30,41),(8,31,42),(9,32,43),(10,73,26),(11,74,27),(12,75,19),(13,76,20),(14,77,21),(15,78,22),(16,79,23),(17,80,24),(18,81,25),(46,62,65),(47,63,66),(48,55,67),(49,56,68),(50,57,69),(51,58,70),(52,59,71),(53,60,72),(54,61,64)], [(1,51,13),(2,46,11),(3,50,18),(4,54,16),(5,49,14),(6,53,12),(7,48,10),(8,52,17),(9,47,15),(19,40,72),(20,44,70),(21,39,68),(22,43,66),(23,38,64),(24,42,71),(25,37,69),(26,41,67),(27,45,65),(28,56,77),(29,60,75),(30,55,73),(31,59,80),(32,63,78),(33,58,76),(34,62,74),(35,57,81),(36,61,79)], [(2,9),(3,8),(4,7),(5,6),(10,13),(11,12),(14,18),(15,17),(19,74),(20,73),(21,81),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,40),(29,39),(30,38),(31,37),(32,45),(33,44),(34,43),(35,42),(36,41),(46,50),(47,49),(51,54),(52,53),(55,67),(56,66),(57,65),(58,64),(59,72),(60,71),(61,70),(62,69),(63,68)]])
 

Matrix representation of C9⋊He3⋊2C2 ►in GL12(𝔽19)

00181800000000
001000000000
00001818000000
000010000000
010000000000
18180000000000
000000570000
00000012170000
0000001713121700
0000006421400
00000013300214
00000016100057
,
010000000000
18180000000000
000100000000
00181800000000
000001000000
00001818000000
000000010000
00000018180000
0000000161000
000000330100
0000001414001818
000000500010
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000010000
00000018180000
000000000100
00000000181800
000000000001
00000000001818
,
222121710000000
1707997000000
171017079000000
970171017000000
7997017000000
101712222000000
000000550012
000000140001718
000000168001616
0000001180030
000000131618181414
0000003161050
,
100000000000
18180000000000
000001000000
000010000000
000100000000
001000000000
0000000180000
0000001800000
000000001000
00000000181800
000000000010
00000000001818

G:=sub<GL(12,GF(19))| [0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,12,17,6,13,16,0,0,0,0,0,0,7,17,13,4,3,10,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,14,7],[0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,3,14,5,0,0,0,0,0,0,1,18,16,3,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18],[2,17,17,9,7,10,0,0,0,0,0,0,2,0,10,7,9,17,0,0,0,0,0,0,2,7,17,0,9,12,0,0,0,0,0,0,12,9,0,17,7,2,0,0,0,0,0,0,17,9,7,10,0,2,0,0,0,0,0,0,10,7,9,17,17,2,0,0,0,0,0,0,0,0,0,0,0,0,5,14,16,11,13,3,0,0,0,0,0,0,5,0,8,8,16,16,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,17,16,3,14,5,0,0,0,0,0,0,2,18,16,0,14,0],[1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18] >;
 

C9⋊He3⋊2C2 in GAP, Magma, Sage, TeX 

C_9\rtimes {\rm He}_3\rtimes_2C_2
 
% in TeX
 
G:=Group("C9:He3:2C2");
 
// GroupNames label
 
G:=SmallGroup(486,148);
 
// by ID
 
G=gap.SmallGroup(486,148);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,4755,2169,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^9=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=a^3*d>;
 
// generators/relations
 

Export

Character table of C9⋊He3⋊2C2 in TeX

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