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G = 2+ 1+42C9order 288 = 25·32

2nd semidirect product of 2+ 1+4 and C9 acting via C9/C3=C3

non-abelian, soluble, monomial

Aliases: 2+ 1+42C9, (C3×Q8).2A4, C3.(C23⋊A4), Q82(C3.A4), (C22×C6).4A4, C232(C3.A4), C6.2(C22⋊A4), C2.2(C24⋊C9), (C3×2+ 1+4).2C3, SmallGroup(288,351)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4 — 2+ 1+42C9
C1C2C232+ 1+4C3×2+ 1+4 — 2+ 1+42C9
2+ 1+4 — 2+ 1+42C9
C1C6

Generators and relations for 2+ 1+42C9
 G = < a,b,c,d,e | a4=b2=d2=e9=1, c2=a2, bab=a-1, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, ebe-1=a2cd, dcd=a2c, ece-1=acd, ede-1=a-1b >

Subgroups: 285 in 81 conjugacy classes, 17 normal (7 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22 [×7], C6, C6 [×3], C2×C4 [×3], D4 [×6], Q8 [×2], C23 [×3], C23, C9, C12 [×2], C2×C6 [×7], C2×D4 [×3], C4○D4 [×2], C18, C2×C12 [×3], C3×D4 [×6], C3×Q8 [×2], C22×C6 [×3], C22×C6, 2+ 1+4, C3.A4 [×3], C6×D4 [×3], C3×C4○D4 [×2], Q8⋊C9 [×2], C2×C3.A4 [×3], C3×2+ 1+4, 2+ 1+42C9
Quotients: C1, C3, C9, A4 [×5], C3.A4 [×5], C22⋊A4, C23⋊A4, C24⋊C9, 2+ 1+42C9

Smallest permutation representation of 2+ 1+42C9
On 72 points
Generators in S72
(1 27 46 39)(2 10 47 29)(3 55 48 66)(4 21 49 42)(5 13 50 32)(6 58 51 69)(7 24 52 45)(8 16 53 35)(9 61 54 72)(11 20 30 41)(12 67 31 56)(14 23 33 44)(15 70 34 59)(17 26 36 38)(18 64 28 62)(19 63 40 65)(22 57 43 68)(25 60 37 71)
(1 39)(3 41)(4 42)(6 44)(7 45)(9 38)(10 29)(11 66)(12 56)(13 32)(14 69)(15 59)(16 35)(17 72)(18 62)(20 48)(21 49)(23 51)(24 52)(26 54)(27 46)(28 64)(30 55)(31 67)(33 58)(34 70)(36 61)(57 68)(60 71)(63 65)
(1 28 46 18)(2 40 47 19)(3 55 48 66)(4 31 49 12)(5 43 50 22)(6 58 51 69)(7 34 52 15)(8 37 53 25)(9 61 54 72)(10 65 29 63)(11 41 30 20)(13 68 32 57)(14 44 33 23)(16 71 35 60)(17 38 36 26)(21 56 42 67)(24 59 45 70)(27 62 39 64)
(1 18)(3 11)(4 12)(6 14)(7 15)(9 17)(19 40)(20 55)(21 67)(22 43)(23 58)(24 70)(25 37)(26 61)(27 64)(28 46)(30 48)(31 49)(33 51)(34 52)(36 54)(38 72)(39 62)(41 66)(42 56)(44 69)(45 59)(57 68)(60 71)(63 65)
(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)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D3A3B4A4B6A6B6C···6H9A···9F12A12B12C12D18A···18F
order122223344666···69···91212121218···18
size116661166116···616···16666616···16

33 irreducible representations

dim1113333444
type++++
imageC1C3C9A4A4C3.A4C3.A4C23⋊A4C23⋊A42+ 1+42C9
kernel2+ 1+42C9C3×2+ 1+42+ 1+4C3×Q8C22×C6Q8C23C3C3C1
# reps1262346126

Matrix representation of 2+ 1+42C9 in GL4(𝔽37) generated by

0010
361135
36000
361036
,
0010
361135
1000
00036
,
0100
36000
361135
360136
,
0100
1000
361135
00036
,
160016
016021
001621
2116165
G:=sub<GL(4,GF(37))| [0,36,36,36,0,1,0,1,1,1,0,0,0,35,0,36],[0,36,1,0,0,1,0,0,1,1,0,0,0,35,0,36],[0,36,36,36,1,0,1,0,0,0,1,1,0,0,35,36],[0,1,36,0,1,0,1,0,0,0,1,0,0,0,35,36],[16,0,0,21,0,16,0,16,0,0,16,16,16,21,21,5] >;

2+ 1+42C9 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_2C_9
% in TeX

G:=Group("ES+(2,2):2C9");
// GroupNames label

G:=SmallGroup(288,351);
// by ID

G=gap.SmallGroup(288,351);
# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,21,380,759,2524,375,4541,1027]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=d^2=e^9=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*c*d,d*c*d=a^2*c,e*c*e^-1=a*c*d,e*d*e^-1=a^-1*b>;
// generators/relations

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