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G = C9×2+ 1+4order 288 = 25·32

Direct product of C9 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×2+ 1+4
 Chief series C1 — C3 — C6 — C18 — C2×C18 — D4×C9 — D4×C18 — C9×2+ 1+4
 Lower central C1 — C2 — C9×2+ 1+4
 Upper central C1 — C18 — C9×2+ 1+4

Generators and relations for C9×2+ 1+4
G = < a,b,c,d,e | a9=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 330 in 249 conjugacy classes, 204 normal (9 characteristic)
C1, C2, C2 [×9], C3, C4 [×6], C22 [×9], C22 [×6], C6, C6 [×9], C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C9, C12 [×6], C2×C6 [×9], C2×C6 [×6], C2×D4 [×9], C4○D4 [×6], C18, C18 [×9], C2×C12 [×9], C3×D4 [×18], C3×Q8 [×2], C22×C6 [×6], 2+ 1+4, C36 [×6], C2×C18 [×9], C2×C18 [×6], C6×D4 [×9], C3×C4○D4 [×6], C2×C36 [×9], D4×C9 [×18], Q8×C9 [×2], C22×C18 [×6], C3×2+ 1+4, D4×C18 [×9], C9×C4○D4 [×6], C9×2+ 1+4
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C24, C18 [×15], C22×C6 [×15], 2+ 1+4, C2×C18 [×35], C23×C6, C22×C18 [×15], C3×2+ 1+4, C23×C18, C9×2+ 1+4

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

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

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

153 conjugacy classes

 class 1 2A 2B ··· 2J 3A 3B 4A ··· 4F 6A 6B 6C ··· 6T 9A ··· 9F 12A ··· 12L 18A ··· 18F 18G ··· 18BH 36A ··· 36AJ order 1 2 2 ··· 2 3 3 4 ··· 4 6 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

153 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 type + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 2+ 1+4 C3×2+ 1+4 C9×2+ 1+4 kernel C9×2+ 1+4 D4×C18 C9×C4○D4 C3×2+ 1+4 C6×D4 C3×C4○D4 2+ 1+4 C2×D4 C4○D4 C9 C3 C1 # reps 1 9 6 2 18 12 6 54 36 1 2 6

Matrix representation of C9×2+ 1+4 in GL4(𝔽37) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 21 2 0 0 1 16 0 0 21 1 0 1 2 16 36 0
,
 16 35 0 0 35 21 0 0 0 36 0 1 35 21 1 0
,
 21 0 0 2 2 0 36 16 21 1 0 1 1 0 0 16
,
 1 0 0 0 0 1 0 0 1 0 36 0 16 0 0 36
G:=sub<GL(4,GF(37))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[21,1,21,2,2,16,1,16,0,0,0,36,0,0,1,0],[16,35,0,35,35,21,36,21,0,0,0,1,0,0,1,0],[21,2,21,1,0,0,1,0,0,36,0,0,2,16,1,16],[1,0,1,16,0,1,0,0,0,0,36,0,0,0,0,36] >;

C9×2+ 1+4 in GAP, Magma, Sage, TeX

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

G:=Group("C9xES+(2,2)");
// GroupNames label

G:=SmallGroup(288,371);
// by ID

G=gap.SmallGroup(288,371);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,555,1571,242]);
// Polycyclic

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

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