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G = C9×C4≀C2order 288 = 25·32

Direct product of C9 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×C4≀C2, D42C36, Q83C36, C426C18, C36.66D4, M4(2)⋊4C18, (D4×C9)⋊5C4, (Q8×C9)⋊5C4, (C4×C36)⋊10C2, C4.3(C2×C36), C4.17(D4×C9), C36.32(C2×C4), (C4×C12).14C6, C4○D4.3C18, (C3×D4).3C12, C12.83(C3×D4), (C2×C18).23D4, (C3×Q8).7C12, C22.3(D4×C9), C12.32(C2×C12), (C9×M4(2))⋊10C2, (C3×M4(2)).5C6, C18.26(C22⋊C4), (C2×C36).115C22, C3.(C3×C4≀C2), (C3×C4≀C2).C3, (C9×C4○D4).4C2, (C2×C6).26(C3×D4), C2.8(C9×C22⋊C4), (C2×C4).14(C2×C18), (C3×C4○D4).12C6, C6.26(C3×C22⋊C4), (C2×C12).135(C2×C6), SmallGroup(288,54)

Series: Derived Chief Lower central Upper central

C1C4 — C9×C4≀C2
C1C2C6C12C2×C12C2×C36C9×M4(2) — C9×C4≀C2
C1C2C4 — C9×C4≀C2
C1C36C2×C36 — C9×C4≀C2

Generators and relations for C9×C4≀C2
 G = < a,b,c,d | a9=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 102 in 66 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C9, C12 [×2], C12 [×3], C2×C6, C2×C6, C42, M4(2), C4○D4, C18, C18 [×2], C24, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C4≀C2, C36 [×2], C36 [×3], C2×C18, C2×C18, C4×C12, C3×M4(2), C3×C4○D4, C72, C2×C36, C2×C36 [×2], D4×C9, D4×C9, Q8×C9, C3×C4≀C2, C4×C36, C9×M4(2), C9×C4○D4, C9×C4≀C2
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, C18 [×3], C2×C12, C3×D4 [×2], C4≀C2, C36 [×2], C2×C18, C3×C22⋊C4, C2×C36, D4×C9 [×2], C3×C4≀C2, C9×C22⋊C4, C9×C4≀C2

Smallest permutation representation of C9×C4≀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 59 50 43)(2 60 51 44)(3 61 52 45)(4 62 53 37)(5 63 54 38)(6 55 46 39)(7 56 47 40)(8 57 48 41)(9 58 49 42)(10 32 27 70)(11 33 19 71)(12 34 20 72)(13 35 21 64)(14 36 22 65)(15 28 23 66)(16 29 24 67)(17 30 25 68)(18 31 26 69)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 63)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 37)(27 38)(46 71)(47 72)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 46)(7 47)(8 48)(9 49)(10 32 27 70)(11 33 19 71)(12 34 20 72)(13 35 21 64)(14 36 22 65)(15 28 23 66)(16 29 24 67)(17 30 25 68)(18 31 26 69)(37 62)(38 63)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)

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

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

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

126 conjugacy classes

class 1 2A2B2C3A3B4A4B4C···4G4H6A6B6C6D6E6F8A8B9A···9F12A12B12C12D12E···12N12O12P18A···18F18G···18L18M···18R24A24B24C24D36A···36L36M···36AP36AQ···36AV72A···72L
order122233444···44666666889···91212121212···12121218···1818···1818···182424242436···3636···3636···3672···72
size112411112···24112244441···111112···2441···12···24···444441···12···24···44···4

126 irreducible representations

dim111111111111111111222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C9C12C12C18C18C18C36C36D4D4C3×D4C3×D4C4≀C2D4×C9D4×C9C3×C4≀C2C9×C4≀C2
kernelC9×C4≀C2C4×C36C9×M4(2)C9×C4○D4C3×C4≀C2D4×C9Q8×C9C4×C12C3×M4(2)C3×C4○D4C4≀C2C3×D4C3×Q8C42M4(2)C4○D4D4Q8C36C2×C18C12C2×C6C9C4C22C3C1
# reps111122222264466612121122466824

Matrix representation of C9×C4≀C2 in GL2(𝔽37) generated by

120
012
,
310
06
,
033
90
,
60
01
G:=sub<GL(2,GF(37))| [12,0,0,12],[31,0,0,6],[0,9,33,0],[6,0,0,1] >;

C9×C4≀C2 in GAP, Magma, Sage, TeX

C_9\times C_4\wr C_2
% in TeX

G:=Group("C9xC4wrC2");
// GroupNames label

G:=SmallGroup(288,54);
// by ID

G=gap.SmallGroup(288,54);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2194,360,242]);
// Polycyclic

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

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