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## G = M4(2)×D9order 288 = 25·32

### Direct product of M4(2) and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — M4(2)×D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — C2×C4×D9 — M4(2)×D9
 Lower central C9 — C18 — M4(2)×D9
 Upper central C1 — C4 — M4(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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, D9, D9, C18, C18, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), Dic9, C36, D18, D18, C2×C18, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C9⋊C8, C72, C4×D9, C2×Dic9, C2×C36, C22×D9, S3×M4(2), C8×D9, C8⋊D9, C4.Dic9, C9×M4(2), C2×C4×D9, M4(2)×D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, D9, C4×S3, C22×S3, C2×M4(2), D18, S3×C2×C4, C4×D9, 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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 9A 9B 9C 12A 12B 12C 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G 36H 36I 72A ··· 72L order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 8 8 8 8 8 8 8 8 9 9 9 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 36 36 72 ··· 72 size 1 1 2 9 9 18 2 1 1 2 9 9 18 2 4 2 2 2 2 18 18 18 18 2 2 2 2 2 4 2 2 2 4 4 4 4 4 4 4 2 ··· 2 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) D9 C4×S3 C4×S3 D18 D18 C4×D9 C4×D9 S3×M4(2) M4(2)×D9 kernel M4(2)×D9 C8×D9 C8⋊D9 C4.Dic9 C9×M4(2) C2×C4×D9 C4×D9 C2×Dic9 C22×D9 C3×M4(2) C24 C2×C12 D9 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 1 4 3 2 2 6 3 6 6 2 6

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

 46 0 0 0 0 46 0 0 0 0 1 3 0 0 15 72
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 48 72
,
 70 45 0 0 28 42 0 0 0 0 1 0 0 0 0 1
,
 3 42 0 0 45 70 0 0 0 0 72 0 0 0 0 72
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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