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G = C424D9order 288 = 25·32

3rd semidirect product of C42 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D361C4, C424D9, C4.17D36, C36.33D4, Dic181C4, C12.52D12, C91C4≀C2, (C4×C36)⋊6C2, C4.6(C4×D9), C12.54(C4×S3), C36.16(C2×C4), (C4×C12).11S3, C6.8(D6⋊C4), (C2×C4).71D18, (C2×C18).27D4, C4.Dic91C2, C2.3(D18⋊C4), C3.(C424S3), (C2×C12).391D6, D365C2.1C2, C18.1(C22⋊C4), (C2×C36).94C22, C22.7(C9⋊D4), (C2×C6).65(C3⋊D4), SmallGroup(288,12)

Series: Derived Chief Lower central Upper central

C1C36 — C424D9
C1C3C9C18C2×C18C2×C36D365C2 — C424D9
C9C18C36 — C424D9
C1C4C2×C4C42

Generators and relations for C424D9
 G = < a,b,c,d | a4=b4=c9=d2=1, dad=ab=ba, ac=ca, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 292 in 66 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C9, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C42, M4(2), C4○D4, D9, C18, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C4≀C2, Dic9, C36 [×2], C36 [×2], D18, C2×C18, C4.Dic3, C4×C12, C4○D12, C9⋊C8, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, C424S3, C4.Dic9, C4×C36, D365C2, C424D9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4≀C2, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C424S3, D18⋊C4, C424D9

Smallest permutation representation of C424D9
On 72 points
Generators in S72
(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 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 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 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 39)(2 38)(3 37)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 54)(18 53)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 63)(27 62)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 72)(36 71)

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

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

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

78 conjugacy classes

class 1 2A2B2C 3 4A4B4C···4G4H6A6B6C8A8B9A9B9C12A···12L18A···18I36A···36AJ
order12223444···446668899912···1218···1836···36
size112362112···23622236362222···22···22···2

78 irreducible representations

dim111111222222222222222
type++++++++++++
imageC1C2C2C2C4C4S3D4D4D6D9C4×S3D12C3⋊D4C4≀C2D18C4×D9D36C9⋊D4C424S3C424D9
kernelC424D9C4.Dic9C4×C36D365C2Dic18D36C4×C12C36C2×C18C2×C12C42C12C12C2×C6C9C2×C4C4C4C22C3C1
# reps1111221111322243666824

Matrix representation of C424D9 in GL2(𝔽37) generated by

2825
22
,
923
2728
,
251
621
,
257
612
G:=sub<GL(2,GF(37))| [28,2,25,2],[9,27,23,28],[25,6,1,21],[25,6,7,12] >;

C424D9 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4D_9
% in TeX

G:=Group("C4^2:4D9");
// GroupNames label

G:=SmallGroup(288,12);
// by ID

G=gap.SmallGroup(288,12);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,675,80,6725,292,9414]);
// Polycyclic

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

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