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G = Q83Dic9order 288 = 25·32

2nd semidirect product of Q8 and Dic9 acting via Dic9/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q83Dic9, D42Dic9, C36.56D4, C93C4≀C2, (D4×C9)⋊2C4, (Q8×C9)⋊2C4, C36.9(C2×C4), C4○D4.3D9, (C2×C18).3D4, (C4×Dic9)⋊2C2, (C2×C12).49D6, (C2×C4).43D18, C4.Dic94C2, C4.3(C2×Dic9), C4.31(C9⋊D4), (C3×Q8).6Dic3, (C3×D4).2Dic3, C12.3(C2×Dic3), C3.(Q83Dic3), (C2×C36).27C22, C22.3(C9⋊D4), C12.126(C3⋊D4), C18.18(C22⋊C4), C2.8(C18.D4), C6.19(C6.D4), (C3×C4○D4).9S3, (C9×C4○D4).1C2, (C2×C6).3(C3⋊D4), SmallGroup(288,44)

Series: Derived Chief Lower central Upper central

C1C36 — Q83Dic9
C1C3C9C18C36C2×C36C4.Dic9 — Q83Dic9
C9C18C36 — Q83Dic9
C1C4C2×C4C4○D4

Generators and relations for Q83Dic9
 G = < a,b,c,d | a4=c18=1, b2=a2, d2=c9, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 212 in 66 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C9, Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C18, C18 [×2], C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4≀C2, Dic9 [×2], C36 [×2], C36, C2×C18, C2×C18, C4.Dic3, C4×Dic3, C3×C4○D4, C9⋊C8, C2×Dic9, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q83Dic3, C4.Dic9, C4×Dic9, C9×C4○D4, Q83Dic9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D9, C2×Dic3, C3⋊D4 [×2], C4≀C2, Dic9 [×2], D18, C6.D4, C2×Dic9, C9⋊D4 [×2], Q83Dic3, C18.D4, Q83Dic9

Smallest permutation representation of Q83Dic9
On 72 points
Generators in S72
(1 16 36 21)(2 17 28 22)(3 18 29 23)(4 10 30 24)(5 11 31 25)(6 12 32 26)(7 13 33 27)(8 14 34 19)(9 15 35 20)(37 66 46 57)(38 67 47 58)(39 68 48 59)(40 69 49 60)(41 70 50 61)(42 71 51 62)(43 72 52 63)(44 55 53 64)(45 56 54 65)
(1 55 36 64)(2 65 28 56)(3 57 29 66)(4 67 30 58)(5 59 31 68)(6 69 32 60)(7 61 33 70)(8 71 34 62)(9 63 35 72)(10 38 24 47)(11 48 25 39)(12 40 26 49)(13 50 27 41)(14 42 19 51)(15 52 20 43)(16 44 21 53)(17 54 22 45)(18 46 23 37)
(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 4)(2 3)(5 9)(6 8)(10 16)(11 15)(12 14)(17 18)(19 26)(20 25)(21 24)(22 23)(28 29)(30 36)(31 35)(32 34)(37 65 46 56)(38 64 47 55)(39 63 48 72)(40 62 49 71)(41 61 50 70)(42 60 51 69)(43 59 52 68)(44 58 53 67)(45 57 54 66)

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

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

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

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H6A6B6C6D8A8B9A9B9C12A12B12C12D12E18A18B18C18D···18L36A···36F36G···36O
order1222344444444666688999121212121218181818···1836···3636···36
size1124211241818181824443636222224442224···42···24···4

54 irreducible representations

dim11111122222222222222244
type++++++++--++--
imageC1C2C2C2C4C4S3D4D4D6Dic3Dic3D9C3⋊D4C3⋊D4C4≀C2D18Dic9Dic9C9⋊D4C9⋊D4Q83Dic3Q83Dic9
kernelQ83Dic9C4.Dic9C4×Dic9C9×C4○D4D4×C9Q8×C9C3×C4○D4C36C2×C18C2×C12C3×D4C3×Q8C4○D4C12C2×C6C9C2×C4D4Q8C4C22C3C1
# reps11112211111132243336626

Matrix representation of Q83Dic9 in GL4(𝔽73) generated by

27000
04600
0010
0001
,
02700
27000
00720
00072
,
72000
0100
004531
00423
,
46000
0100
0001
0010
G:=sub<GL(4,GF(73))| [27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,27,0,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,45,42,0,0,31,3],[46,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

Q83Dic9 in GAP, Magma, Sage, TeX

Q_8\rtimes_3{\rm Dic}_9
% in TeX

G:=Group("Q8:3Dic9");
// GroupNames label

G:=SmallGroup(288,44);
// by ID

G=gap.SmallGroup(288,44);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,100,675,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^18=1,b^2=a^2,d^2=c^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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