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## G = C4○D4×D9order 288 = 25·32

### Direct product of C4○D4 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C4○D4×D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C4×D9 — C4○D4×D9
 Lower central C9 — C18 — C4○D4×D9
 Upper central C1 — C4 — C4○D4

Generators and relations for C4○D4×D9
G = < a,b,c,d,e | a4=c2=d9=e2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 992 in 246 conjugacy classes, 104 normal (21 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×5], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], C9, Dic3 [×4], C12, C12 [×3], D6 [×10], C2×C6 [×3], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], D9 [×2], D9 [×3], C18, C18 [×3], Dic6 [×3], C4×S3 [×10], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C2×C4○D4, Dic9, Dic9 [×3], C36, C36 [×3], D18, D18 [×3], D18 [×6], C2×C18 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, Dic18 [×3], C4×D9, C4×D9 [×9], D36 [×3], C2×Dic9 [×3], C9⋊D4 [×6], C2×C36 [×3], D4×C9 [×3], Q8×C9, C22×D9 [×3], S3×C4○D4, C2×C4×D9 [×3], D365C2 [×3], D4×D9 [×3], D42D9 [×3], Q8×D9, Q83D9, C9×C4○D4, C4○D4×D9
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, D9, C22×S3 [×7], C2×C4○D4, D18 [×7], S3×C23, C22×D9 [×7], S3×C4○D4, C23×D9, C4○D4×D9

Smallest permutation representation of C4○D4×D9
On 72 points
Generators in S72
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 64 46 55)(38 65 47 56)(39 66 48 57)(40 67 49 58)(41 68 50 59)(42 69 51 60)(43 70 52 61)(44 71 53 62)(45 72 54 63)
(1 41 14 50)(2 42 15 51)(3 43 16 52)(4 44 17 53)(5 45 18 54)(6 37 10 46)(7 38 11 47)(8 39 12 48)(9 40 13 49)(19 55 28 64)(20 56 29 65)(21 57 30 66)(22 58 31 67)(23 59 32 68)(24 60 33 69)(25 61 34 70)(26 62 35 71)(27 63 36 72)
(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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 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)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (1,41,14,50)(2,42,15,51)(3,43,16,52)(4,44,17,53)(5,45,18,54)(6,37,10,46)(7,38,11,47)(8,39,12,48)(9,40,13,49)(19,55,28,64)(20,56,29,65)(21,57,30,66)(22,58,31,67)(23,59,32,68)(24,60,33,69)(25,61,34,70)(26,62,35,71)(27,63,36,72), (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,9)(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( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (1,41,14,50)(2,42,15,51)(3,43,16,52)(4,44,17,53)(5,45,18,54)(6,37,10,46)(7,38,11,47)(8,39,12,48)(9,40,13,49)(19,55,28,64)(20,56,29,65)(21,57,30,66)(22,58,31,67)(23,59,32,68)(24,60,33,69)(25,61,34,70)(26,62,35,71)(27,63,36,72), (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,9)(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([(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,64,46,55),(38,65,47,56),(39,66,48,57),(40,67,49,58),(41,68,50,59),(42,69,51,60),(43,70,52,61),(44,71,53,62),(45,72,54,63)], [(1,41,14,50),(2,42,15,51),(3,43,16,52),(4,44,17,53),(5,45,18,54),(6,37,10,46),(7,38,11,47),(8,39,12,48),(9,40,13,49),(19,55,28,64),(20,56,29,65),(21,57,30,66),(22,58,31,67),(23,59,32,68),(24,60,33,69),(25,61,34,70),(26,62,35,71),(27,63,36,72)], [(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)], [(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)], [(1,9),(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 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18L 36A ··· 36F 36G ··· 36O order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 2 2 2 9 9 18 18 18 2 1 1 2 2 2 9 9 18 18 18 2 4 4 4 2 2 2 2 2 4 4 4 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 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 C2 S3 D6 D6 D6 C4○D4 D9 D18 D18 D18 S3×C4○D4 C4○D4×D9 kernel C4○D4×D9 C2×C4×D9 D36⋊5C2 D4×D9 D4⋊2D9 Q8×D9 Q8⋊3D9 C9×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 D9 C4○D4 C2×C4 D4 Q8 C3 C1 # reps 1 3 3 3 3 1 1 1 1 3 3 1 4 3 9 9 3 2 6

Matrix representation of C4○D4×D9 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 31 0 0 0 0 31
,
 36 0 0 0 0 36 0 0 0 0 36 13 0 0 34 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 3 36
,
 26 6 0 0 31 20 0 0 0 0 1 0 0 0 0 1
,
 31 11 0 0 17 6 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,31,0,0,0,0,31],[36,0,0,0,0,36,0,0,0,0,36,34,0,0,13,1],[1,0,0,0,0,1,0,0,0,0,1,3,0,0,0,36],[26,31,0,0,6,20,0,0,0,0,1,0,0,0,0,1],[31,17,0,0,11,6,0,0,0,0,1,0,0,0,0,1] >;

C4○D4×D9 in GAP, Magma, Sage, TeX

C_4\circ D_4\times D_9
% in TeX

G:=Group("C4oD4xD9");
// GroupNames label

G:=SmallGroup(288,362);
// by ID

G=gap.SmallGroup(288,362);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^9=e^2=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,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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