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G = C9×C22≀C2order 288 = 25·32

Direct product of C9 and C22≀C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C9×C22≀C2, C245C18, (C2×C18)⋊7D4, (C2×D4)⋊1C18, (C6×D4).9C6, C2.4(D4×C18), C6.67(C6×D4), C223(D4×C9), (D4×C18)⋊10C2, C22⋊C42C18, C231(C2×C18), (C23×C18)⋊1C2, (C2×C36)⋊8C22, C18.67(C2×D4), (C23×C6).9C6, (C2×C18).75C23, (C22×C18)⋊1C22, C22.10(C22×C18), (C2×C4)⋊1(C2×C18), C3.(C3×C22≀C2), (C2×C6).27(C3×D4), (C9×C22⋊C4)⋊10C2, (C2×C12).62(C2×C6), (C3×C22⋊C4).5C6, (C3×C22≀C2).2C3, (C2×C6).80(C22×C6), (C22×C6).45(C2×C6), SmallGroup(288,170)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C22≀C2
C1C3C6C2×C6C2×C18C22×C18D4×C18 — C9×C22≀C2
C1C22 — C9×C22≀C2
C1C2×C18 — C9×C22≀C2

Generators and relations for C9×C22≀C2
 G = < a,b,c,d,e,f | a9=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 318 in 195 conjugacy classes, 78 normal (15 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×6], C22 [×17], C6 [×3], C6 [×7], C2×C4 [×3], D4 [×6], C23, C23 [×3], C23 [×6], C9, C12 [×3], C2×C6, C2×C6 [×6], C2×C6 [×17], C22⋊C4 [×3], C2×D4 [×3], C24, C18 [×3], C18 [×7], C2×C12 [×3], C3×D4 [×6], C22×C6, C22×C6 [×3], C22×C6 [×6], C22≀C2, C36 [×3], C2×C18, C2×C18 [×6], C2×C18 [×17], C3×C22⋊C4 [×3], C6×D4 [×3], C23×C6, C2×C36 [×3], D4×C9 [×6], C22×C18, C22×C18 [×3], C22×C18 [×6], C3×C22≀C2, C9×C22⋊C4 [×3], D4×C18 [×3], C23×C18, C9×C22≀C2
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, C9, C2×C6 [×7], C2×D4 [×3], C18 [×7], C3×D4 [×6], C22×C6, C22≀C2, C2×C18 [×7], C6×D4 [×3], D4×C9 [×6], C22×C18, C3×C22≀C2, D4×C18 [×3], C9×C22≀C2

Smallest permutation representation of C9×C22≀C2
On 72 points
Generators in S72
(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)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 68)(56 69)(57 70)(58 71)(59 72)(60 64)(61 65)(62 66)(63 67)
(1 54)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 37)(36 38)
(1 45)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 46)(36 47)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 68)(11 69)(12 70)(13 71)(14 72)(15 64)(16 65)(17 66)(18 67)(19 59)(20 60)(21 61)(22 62)(23 63)(24 55)(25 56)(26 57)(27 58)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 46)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 55)(44 56)(45 57)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D···2I2J3A3B4A4B4C6A···6F6G···6R6S6T9A···9F12A···12F18A···18R18S···18BB18BC···18BH36A···36R
order12222···22334446···66···6669···912···1218···1818···1818···1836···36
size11112···24114441···12···2441···14···41···12···24···44···4

126 irreducible representations

dim111111111111222
type+++++
imageC1C2C2C2C3C6C6C6C9C18C18C18D4C3×D4D4×C9
kernelC9×C22≀C2C9×C22⋊C4D4×C18C23×C18C3×C22≀C2C3×C22⋊C4C6×D4C23×C6C22≀C2C22⋊C4C2×D4C24C2×C18C2×C6C22
# reps1331266261818661236

Matrix representation of C9×C22≀C2 in GL4(𝔽37) generated by

12000
01200
0010
0001
,
36000
0100
0010
00036
,
1000
03600
00360
00036
,
36000
03600
00360
00036
,
36000
03600
0010
0001
,
0100
1000
0001
0010
G:=sub<GL(4,GF(37))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C9×C22≀C2 in GAP, Magma, Sage, TeX

C_9\times C_2^2\wr C_2
% in TeX

G:=Group("C9xC2^2wrC2");
// GroupNames label

G:=SmallGroup(288,170);
// by ID

G=gap.SmallGroup(288,170);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,1094,360]);
// Polycyclic

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

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