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G = D48D18order 288 = 25·32

4th semidirect product of D4 and D18 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D48D18, Q88D18, C922+ 1+4, D3611C22, C36.26C23, C18.12C24, D18.7C23, Dic9.7C23, Dic1812C22, C4○D45D9, (D4×D9)⋊5C2, (C2×C4)⋊4D18, C3.(D4○D12), (C2×D36)⋊13C2, (C2×C36)⋊5C22, Q83D95C2, (C3×D4).39D6, (C4×D9)⋊2C22, (D4×C9)⋊9C22, C9⋊D45C22, (C3×Q8).63D6, D365C28C2, (Q8×C9)⋊8C22, (C2×C12).105D6, (C2×C18).4C23, C6.49(S3×C23), C4.33(C22×D9), C2.13(C23×D9), (C22×D9)⋊4C22, C22.3(C22×D9), C12.187(C22×S3), (C9×C4○D4)⋊4C2, (C3×C4○D4).16S3, (C2×C6).10(C22×S3), SmallGroup(288,363)

Series: Derived Chief Lower central Upper central

C1C18 — D48D18
C1C3C9C18D18C22×D9D4×D9 — D48D18
C9C18 — D48D18
C1C2C4○D4

Generators and relations for D48D18
 G = < a,b,c,d | a4=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 1200 in 249 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C9, Dic3 [×2], C12, C12 [×3], D6 [×12], C2×C6 [×3], C2×D4 [×9], C4○D4, C4○D4 [×5], D9 [×6], C18, C18 [×3], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], 2+ 1+4, Dic9 [×2], C36, C36 [×3], D18 [×6], D18 [×6], C2×C18 [×3], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, Dic18, C4×D9 [×6], D36 [×9], C9⋊D4 [×6], C2×C36 [×3], D4×C9 [×3], Q8×C9, C22×D9 [×6], D4○D12, C2×D36 [×3], D365C2 [×3], D4×D9 [×6], Q83D9 [×2], C9×C4○D4, D48D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2+ 1+4, D18 [×7], S3×C23, C22×D9 [×7], D4○D12, C23×D9, D48D18

Smallest permutation representation of D48D18
On 72 points
Generators in S72
(1 34 50 57)(2 35 51 58)(3 36 52 59)(4 19 53 60)(5 20 54 61)(6 21 37 62)(7 22 38 63)(8 23 39 64)(9 24 40 65)(10 25 41 66)(11 26 42 67)(12 27 43 68)(13 28 44 69)(14 29 45 70)(15 30 46 71)(16 31 47 72)(17 32 48 55)(18 33 49 56)
(2 51)(4 53)(6 37)(8 39)(10 41)(12 43)(14 45)(16 47)(18 49)(20 61)(22 63)(24 65)(26 67)(28 69)(30 71)(32 55)(34 57)(36 59)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 36)(17 35)(18 34)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 72)(53 71)(54 70)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E···2J 3 4A4B4C4D4E4F6A6B6C6D9A9B9C12A12B12C12D12E18A18B18C18D···18L36A···36F36G···36O
order122222···234444446666999121212121218181818···1836···3636···36
size1122218···182222218182444222224442224···42···24···4

57 irreducible representations

dim11111122222222444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D6D6D6D9D18D18D182+ 1+4D4○D12D48D18
kernelD48D18C2×D36D365C2D4×D9Q83D9C9×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C9C3C1
# reps13362113313993126

Matrix representation of D48D18 in GL4(𝔽37) generated by

360514
0362328
63410
3901
,
1000
0100
313360
3428036
,
2533421
4291620
100124
010338
,
83300
252900
2702533
1010812
G:=sub<GL(4,GF(37))| [36,0,6,3,0,36,34,9,5,23,1,0,14,28,0,1],[1,0,31,34,0,1,3,28,0,0,36,0,0,0,0,36],[25,4,10,0,33,29,0,10,4,16,12,33,21,20,4,8],[8,25,27,10,33,29,0,10,0,0,25,8,0,0,33,12] >;

D48D18 in GAP, Magma, Sage, TeX

D_4\rtimes_8D_{18}
% in TeX

G:=Group("D4:8D18");
// GroupNames label

G:=SmallGroup(288,363);
// by ID

G=gap.SmallGroup(288,363);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^18=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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