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G = M4(2)×D9order 288 = 25·32

Direct product of M4(2) and D9

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)×D9, C86D18, C726C22, C24.44D6, C36.38C23, (C8×D9)⋊7C2, C8⋊D95C2, C9⋊C811C22, (C4×D9).1C4, C4.15(C4×D9), C92(C2×M4(2)), C3.(S3×M4(2)), C12.11(C4×S3), C36.12(C2×C4), D18.6(C2×C4), (C2×C12).51D6, (C2×C4).46D18, C4.Dic95C2, C22.7(C4×D9), (C9×M4(2))⋊3C2, (C2×Dic9).6C4, Dic9.8(C2×C4), (C22×D9).4C4, C4.38(C22×D9), C18.15(C22×C4), (C2×C36).29C22, (C4×D9).18C22, (C3×M4(2)).3S3, C12.199(C22×S3), C6.54(S3×C2×C4), (C2×C4×D9).3C2, C2.16(C2×C4×D9), (C2×C6).8(C4×S3), (C2×C18).5(C2×C4), SmallGroup(288,116)

Series: Derived Chief Lower central Upper central

C1C18 — M4(2)×D9
C1C3C9C18C36C4×D9C2×C4×D9 — M4(2)×D9
C9C18 — M4(2)×D9
C1C4M4(2)

Generators and relations for M4(2)×D9
 G = < a,b,c,d | a8=b2=c9=d2=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 384 in 102 conjugacy classes, 48 normal (34 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×3], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C9, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, D9 [×2], D9, C18, C18, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C2×M4(2), Dic9 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, C9⋊C8 [×2], C72 [×2], C4×D9 [×4], C2×Dic9, C2×C36, C22×D9, S3×M4(2), C8×D9 [×2], C8⋊D9 [×2], C4.Dic9, C9×M4(2), C2×C4×D9, M4(2)×D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, D9, C4×S3 [×2], C22×S3, C2×M4(2), D18 [×3], S3×C2×C4, C4×D9 [×2], C22×D9, S3×M4(2), C2×C4×D9, M4(2)×D9

Smallest permutation representation of M4(2)×D9
On 72 points
Generators in S72
(1 59 23 50 14 68 32 41)(2 60 24 51 15 69 33 42)(3 61 25 52 16 70 34 43)(4 62 26 53 17 71 35 44)(5 63 27 54 18 72 36 45)(6 55 19 46 10 64 28 37)(7 56 20 47 11 65 29 38)(8 57 21 48 12 66 30 39)(9 58 22 49 13 67 31 40)
(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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B8A8B8C8D8E8F8G8H9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122222344444466888888889991212121818181818182424242436···3636363672···72
size1129918211299182422221818181822222422244444442···24444···4

60 irreducible representations

dim1111111112222222222244
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6M4(2)D9C4×S3C4×S3D18D18C4×D9C4×D9S3×M4(2)M4(2)×D9
kernelM4(2)×D9C8×D9C8⋊D9C4.Dic9C9×M4(2)C2×C4×D9C4×D9C2×Dic9C22×D9C3×M4(2)C24C2×C12D9M4(2)C12C2×C6C8C2×C4C4C22C3C1
# reps1221114221214322636626

Matrix representation of M4(2)×D9 in GL4(𝔽73) generated by

46000
04600
0013
001572
,
1000
0100
0010
004872
,
704500
284200
0010
0001
,
34200
457000
00720
00072
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,1,15,0,0,3,72],[1,0,0,0,0,1,0,0,0,0,1,48,0,0,0,72],[70,28,0,0,45,42,0,0,0,0,1,0,0,0,0,1],[3,45,0,0,42,70,0,0,0,0,72,0,0,0,0,72] >;

M4(2)×D9 in GAP, Magma, Sage, TeX

M_4(2)\times D_9
% in TeX

G:=Group("M4(2)xD9");
// GroupNames label

G:=SmallGroup(288,116);
// by ID

G=gap.SmallGroup(288,116);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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