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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

Aliases: M4(2)×3- 1+2, C727C6, C36.4C12, C62.3C12, C12.42C62, (C3×C24).7C6, (C2×C36).6C6, (C9×M4(2))⋊C3, C24.15(C3×C6), (C6×C12).11C6, C36.29(C2×C6), C12.9(C3×C12), (C2×C18).3C12, C6.27(C6×C12), (C3×C12).6C12, C93(C3×M4(2)), C18.14(C2×C12), C32.(C3×M4(2)), C4.(C4×3- 1+2), C83(C2×3- 1+2), (C32×M4(2)).C3, (C8×3- 1+2)⋊7C2, C3.3(C32×M4(2)), C22.(C4×3- 1+2), (C3×M4(2)).3C32, (C4×3- 1+2).4C4, C4.6(C22×3- 1+2), (C22×3- 1+2).3C4, (C4×3- 1+2).22C22, (C2×C12).15(C3×C6), (C2×C6).11(C3×C12), (C3×C6).35(C2×C12), (C3×C12).69(C2×C6), C2.5(C2×C4×3- 1+2), (C2×C4×3- 1+2).6C2, (C2×C4).2(C2×3- 1+2), (C2×3- 1+2).14(C2×C4), SmallGroup(432,214)

Series: Derived Chief Lower central Upper central

C1C6 — M4(2)×3- 1+2
C1C2C6C12C3×C12C4×3- 1+2C8×3- 1+2 — M4(2)×3- 1+2
C1C6 — M4(2)×3- 1+2
C1C12 — 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 [×2], C22, C6, C6 [×3], C8 [×2], C2×C4, C9 [×3], C32, C12 [×2], C12 [×2], C2×C6, C2×C6, M4(2), C18 [×3], C18 [×3], C3×C6, C3×C6, C24 [×2], C24 [×2], C2×C12, C2×C12, 3- 1+2, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C3×M4(2), C3×M4(2), C2×3- 1+2, C2×3- 1+2, C72 [×6], C2×C36 [×3], C3×C24 [×2], C6×C12, C4×3- 1+2 [×2], C22×3- 1+2, C9×M4(2) [×3], C32×M4(2), C8×3- 1+2 [×2], C2×C4×3- 1+2, M4(2)×3- 1+2
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, C32, C12 [×8], C2×C6 [×4], M4(2), C3×C6 [×3], C2×C12 [×4], 3- 1+2, C3×C12 [×2], C62, C3×M4(2) [×4], C2×3- 1+2 [×3], C6×C12, C4×3- 1+2 [×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 54 15 42 21 63 31)(2 67 46 16 43 22 55 32)(3 68 47 17 44 23 56 33)(4 69 48 18 45 24 57 34)(5 70 49 10 37 25 58 35)(6 71 50 11 38 26 59 36)(7 72 51 12 39 27 60 28)(8 64 52 13 40 19 61 29)(9 65 53 14 41 20 62 30)
(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(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 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(46 52 49)(47 50 53)(55 61 58)(56 59 62)(64 70 67)(65 68 71)

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

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

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

110 conjugacy classes

class 1 2A2B3A3B3C3D4A4B4C6A6B6C6D6E6F6G6H8A8B8C8D9A···9F12A12B12C12D12E12F12G12H12I12J12K12L18A···18F18G···18L24A···24H24I···24P36A···36L36M···36R72A···72X
order12233334446666666688889···912121212121212121212121218···1818···1824···2424···2436···3636···3672···72
size11211331121122336622223···31111223333663···36···62···26···63···36···66···6

110 irreducible representations

dim111111111111111222333336
type+++
imageC1C2C2C3C3C4C4C6C6C6C6C12C12C12C12M4(2)C3×M4(2)C3×M4(2)3- 1+2C2×3- 1+2C2×3- 1+2C4×3- 1+2C4×3- 1+2M4(2)×3- 1+2
kernelM4(2)×3- 1+2C8×3- 1+2C2×C4×3- 1+2C9×M4(2)C32×M4(2)C4×3- 1+2C22×3- 1+2C72C2×C36C3×C24C6×C12C36C2×C18C3×C12C623- 1+2C9C32M4(2)C8C2×C4C4C22C1
# reps1216222126421212442124242444

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

4619000
1427000
004600
000460
000046
,
10000
7272000
00100
00010
00001
,
80000
08000
0064072
000064
007659
,
640000
064000
0017264
000640
00008

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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