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## G = C2×D4×D9order 288 = 25·32

### Direct product of C2, D4 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×D4×D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C23×D9 — C2×D4×D9
 Lower central C9 — C18 — C2×D4×D9
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×D9
G = < a,b,c,d,e | a2=b4=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1704 in 354 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], C9, Dic3 [×2], C12 [×2], D6 [×30], C2×C6, C2×C6 [×4], C2×C6 [×4], C22×C4, C2×D4, C2×D4 [×11], C24 [×2], D9 [×4], D9 [×4], C18, C18 [×2], C18 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×19], C22×C6 [×2], C22×D4, Dic9 [×2], C36 [×2], D18 [×10], D18 [×20], C2×C18, C2×C18 [×4], C2×C18 [×4], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C4×D9 [×4], D36 [×4], C2×Dic9, C9⋊D4 [×8], C2×C36, D4×C9 [×4], C22×D9, C22×D9 [×10], C22×D9 [×8], C22×C18 [×2], C2×S3×D4, C2×C4×D9, C2×D36, D4×D9 [×8], C2×C9⋊D4 [×2], D4×C18, C23×D9 [×2], C2×D4×D9
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D9, C22×S3 [×7], C22×D4, D18 [×7], S3×D4 [×2], S3×C23, C22×D9 [×7], C2×S3×D4, D4×D9 [×2], C23×D9, C2×D4×D9

Smallest permutation representation of C2×D4×D9
On 72 points
Generators in S72
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(1 68 14 59)(2 69 15 60)(3 70 16 61)(4 71 17 62)(5 72 18 63)(6 64 10 55)(7 65 11 56)(8 66 12 57)(9 67 13 58)(19 46 28 37)(20 47 29 38)(21 48 30 39)(22 49 31 40)(23 50 32 41)(24 51 33 42)(25 52 34 43)(26 53 35 44)(27 54 36 45)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 62)(56 61)(57 60)(58 59)(64 71)(65 70)(66 69)(67 68)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 9 9 9 9 18 18 18 18 2 2 2 18 18 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D9 D18 D18 D18 S3×D4 D4×D9 kernel C2×D4×D9 C2×C4×D9 C2×D36 D4×D9 C2×C9⋊D4 D4×C18 C23×D9 C6×D4 D18 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C6 C2 # reps 1 1 1 8 2 1 2 1 4 1 4 2 3 3 12 6 2 6

Matrix representation of C2×D4×D9 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 13 0 0 0 0 34 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 13 0 0 0 0 0 36
,
 0 1 0 0 0 0 36 36 0 0 0 0 0 0 20 6 0 0 0 0 31 26 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 31 26 0 0 0 0 20 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,34,0,0,0,0,13,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,13,36],[0,36,0,0,0,0,1,36,0,0,0,0,0,0,20,31,0,0,0,0,6,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,20,0,0,0,0,26,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×D4×D9 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_9
% in TeX

G:=Group("C2xD4xD9");
// GroupNames label

G:=SmallGroup(288,356);
// by ID

G=gap.SmallGroup(288,356);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,185,6725,292,9414]);
// Polycyclic

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

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