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## G = M4(2)×3- 1+2order 432 = 24·33

### Direct product of M4(2) and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — M4(2)×3- 1+2
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×3- 1+2 — C8×3- 1+2 — M4(2)×3- 1+2
 Lower central C1 — C6 — M4(2)×3- 1+2
 Upper central C1 — C12 — M4(2)×3- 1+2

Generators and relations for M4(2)×3- 1+2
G = < a,b,c,d | a8=b2=c9=d3=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 110 in 80 conjugacy classes, 63 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C24, C24, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C3×M4(2), C3×M4(2), C2×3- 1+2, C2×3- 1+2, C72, C2×C36, C3×C24, C6×C12, C4×3- 1+2, C22×3- 1+2, C9×M4(2), C32×M4(2), C8×3- 1+2, C2×C4×3- 1+2, M4(2)×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, 3- 1+2, C3×C12, C62, C3×M4(2), C2×3- 1+2, C6×C12, C4×3- 1+2, C22×3- 1+2, C32×M4(2), C2×C4×3- 1+2, M4(2)×3- 1+2

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

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

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

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

110 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 18A ··· 18F 18G ··· 18L 24A ··· 24H 24I ··· 24P 36A ··· 36L 36M ··· 36R 72A ··· 72X order 1 2 2 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 9 ··· 9 12 12 12 12 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 24 ··· 24 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 1 1 3 3 1 1 2 1 1 2 2 3 3 6 6 2 2 2 2 3 ··· 3 1 1 1 1 2 2 3 3 3 3 6 6 3 ··· 3 6 ··· 6 2 ··· 2 6 ··· 6 3 ··· 3 6 ··· 6 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 6 type + + + image C1 C2 C2 C3 C3 C4 C4 C6 C6 C6 C6 C12 C12 C12 C12 M4(2) C3×M4(2) C3×M4(2) 3- 1+2 C2×3- 1+2 C2×3- 1+2 C4×3- 1+2 C4×3- 1+2 M4(2)×3- 1+2 kernel M4(2)×3- 1+2 C8×3- 1+2 C2×C4×3- 1+2 C9×M4(2) C32×M4(2) C4×3- 1+2 C22×3- 1+2 C72 C2×C36 C3×C24 C6×C12 C36 C2×C18 C3×C12 C62 3- 1+2 C9 C32 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 1 6 2 2 2 12 6 4 2 12 12 4 4 2 12 4 2 4 2 4 4 4

Matrix representation of M4(2)×3- 1+2 in GL5(𝔽73)

 46 19 0 0 0 14 27 0 0 0 0 0 46 0 0 0 0 0 46 0 0 0 0 0 46
,
 1 0 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 8 0 0 0 0 0 8 0 0 0 0 0 64 0 72 0 0 0 0 64 0 0 7 65 9
,
 64 0 0 0 0 0 64 0 0 0 0 0 1 72 64 0 0 0 64 0 0 0 0 0 8

G:=sub<GL(5,GF(73))| [46,14,0,0,0,19,27,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[1,72,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,8,0,0,0,0,0,64,0,7,0,0,0,0,65,0,0,72,64,9],[64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,72,64,0,0,0,64,0,8] >;

M4(2)×3- 1+2 in GAP, Magma, Sage, TeX

M_4(2)\times 3_-^{1+2}
% in TeX

G:=Group("M4(2)xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,214);
// by ID

G=gap.SmallGroup(432,214);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,3053,394,605,242]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^3=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^-1=c^4>;
// generators/relations

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