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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C9×C4≀C2
 Chief series C1 — C2 — C6 — C12 — C2×C12 — C2×C36 — C9×M4(2) — C9×C4≀C2
 Lower central C1 — C2 — C4 — C9×C4≀C2
 Upper central C1 — C36 — C2×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, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C9, C12, C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C18, C18, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4≀C2, C36, C36, C2×C18, C2×C18, C4×C12, C3×M4(2), C3×C4○D4, C72, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, C3×C4≀C2, C4×C36, C9×M4(2), C9×C4○D4, C9×C4≀C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C18, C2×C12, C3×D4, C4≀C2, C36, C2×C18, C3×C22⋊C4, C2×C36, D4×C9, 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 46 37)(2 60 47 38)(3 61 48 39)(4 62 49 40)(5 63 50 41)(6 55 51 42)(7 56 52 43)(8 57 53 44)(9 58 54 45)(10 67 24 29)(11 68 25 30)(12 69 26 31)(13 70 27 32)(14 71 19 33)(15 72 20 34)(16 64 21 35)(17 65 22 36)(18 66 23 28)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 37)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 64)(54 65)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 67 24 29)(11 68 25 30)(12 69 26 31)(13 70 27 32)(14 71 19 33)(15 72 20 34)(16 64 21 35)(17 65 22 36)(18 66 23 28)(37 59)(38 60)(39 61)(40 62)(41 63)(42 55)(43 56)(44 57)(45 58)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C ··· 4G 4H 6A 6B 6C 6D 6E 6F 8A 8B 9A ··· 9F 12A 12B 12C 12D 12E ··· 12N 12O 12P 18A ··· 18F 18G ··· 18L 18M ··· 18R 24A 24B 24C 24D 36A ··· 36L 36M ··· 36AP 36AQ ··· 36AV 72A ··· 72L order 1 2 2 2 3 3 4 4 4 ··· 4 4 6 6 6 6 6 6 8 8 9 ··· 9 12 12 12 12 12 ··· 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 4 1 1 1 1 2 ··· 2 4 1 1 2 2 4 4 4 4 1 ··· 1 1 1 1 1 2 ··· 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C9 C12 C12 C18 C18 C18 C36 C36 D4 D4 C3×D4 C3×D4 C4≀C2 D4×C9 D4×C9 C3×C4≀C2 C9×C4≀C2 kernel C9×C4≀C2 C4×C36 C9×M4(2) C9×C4○D4 C3×C4≀C2 D4×C9 Q8×C9 C4×C12 C3×M4(2) C3×C4○D4 C4≀C2 C3×D4 C3×Q8 C42 M4(2) C4○D4 D4 Q8 C36 C2×C18 C12 C2×C6 C9 C4 C22 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 6 4 4 6 6 6 12 12 1 1 2 2 4 6 6 8 24

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

 12 0 0 12
,
 31 0 0 6
,
 0 33 9 0
,
 6 0 0 1
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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