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G = 2+ 1+4⋊C9order 288 = 25·32

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

non-abelian, soluble

Aliases: 2+ 1+41C9, C4○D4.C18, C12.9(C2×A4), Q8⋊C94C22, (C3×Q8).4A4, Q8.(C3.A4), C3.(Q8.A4), Q8.C183C2, Q8.2(C2×C18), C6.17(C22×A4), (C3×2+ 1+4).1C3, C4.2(C2×C3.A4), (C3×C4○D4).3C6, (C3×Q8).13(C2×C6), C2.6(C22×C3.A4), SmallGroup(288,348)

Series: Derived Chief Lower central Upper central

C1C2Q8 — 2+ 1+4⋊C9
C1C2Q8C3×Q8Q8⋊C9Q8.C18 — 2+ 1+4⋊C9
Q8 — 2+ 1+4⋊C9
C1C6C3×Q8

Generators and relations for 2+ 1+4⋊C9
 G = < a,b,c,d,e | a4=b2=d2=e9=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a-1bcd, dcd=a2c, ece-1=a-1d, ede-1=cd >

Subgroups: 249 in 81 conjugacy classes, 27 normal (12 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×5], C6, C6 [×3], C2×C4 [×3], D4 [×6], Q8 [×2], C23 [×2], C9, C12 [×3], C12, C2×C6 [×5], C2×D4 [×3], C4○D4 [×3], C4○D4, C18, C2×C12 [×3], C3×D4 [×6], C3×Q8 [×2], C22×C6 [×2], 2+ 1+4, C36 [×3], C6×D4 [×3], C3×C4○D4 [×3], C3×C4○D4, Q8⋊C9, Q8×C9, C3×2+ 1+4, Q8.C18 [×3], 2+ 1+4⋊C9
Quotients: C1, C2 [×3], C3, C22, C6 [×3], C9, A4, C2×C6, C18 [×3], C2×A4 [×3], C3.A4, C2×C18, C22×A4, C2×C3.A4 [×3], Q8.A4, C22×C3.A4, 2+ 1+4⋊C9

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

57 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D6A6B6C···6H9A···9F12A···12F12G12H18A···18F36A···36R
order12222334444666···69···912···12121218···1836···36
size11666112226116···64···42···2664···48···8

57 irreducible representations

dim1111113333444
type+++++
imageC1C2C3C6C9C18A4C2×A4C3.A4C2×C3.A4Q8.A4Q8.A42+ 1+4⋊C9
kernel2+ 1+4⋊C9Q8.C18C3×2+ 1+4C3×C4○D42+ 1+4C4○D4C3×Q8C12Q8C4C3C3C1
# reps13266181326126

Matrix representation of 2+ 1+4⋊C9 in GL7(𝔽37)

1000000
0100000
0010000
00003620
00010035
00000036
0000010
,
1030000
03600000
00360000
0000100
0001000
0000001
0000010
,
1030000
03600000
00360000
00003620
00036002
00036001
00003610
,
360340000
036210000
0010000
0000100
0001000
00010036
00001360
,
016320000
303600000
01410000
0000102727
0000271027
000532027
00055027

G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,2,0,0,1,0,0,0,0,35,36,0],[1,0,0,0,0,0,0,0,36,0,0,0,0,0,3,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,36,0,0,0,0,0,3,0,36,0,0,0,0,0,0,0,0,36,36,0,0,0,0,36,0,0,36,0,0,0,2,0,0,1,0,0,0,0,2,1,0],[36,0,0,0,0,0,0,0,36,0,0,0,0,0,34,21,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,36,0],[0,30,0,0,0,0,0,16,36,14,0,0,0,0,32,0,1,0,0,0,0,0,0,0,0,0,5,5,0,0,0,10,27,32,5,0,0,0,27,10,0,0,0,0,0,27,27,27,27] >;

2+ 1+4⋊C9 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes C_9
% in TeX

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

G:=SmallGroup(288,348);
// by ID

G=gap.SmallGroup(288,348);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,79,648,172,1153,285,124]);
// 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,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b*c*d,d*c*d=a^2*c,e*c*e^-1=a^-1*d,e*d*e^-1=c*d>;
// generators/relations

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