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G = C33⋊9M4(2)  order 432 = 24·33

5th semidirect product of C33 and M4(2) acting via M4(2)/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊9M4(2), C12.72S32, C3⋊1(C24⋊S3), C32⋊4C8⋊14S3, (C3×C12).167D6, C33⋊5C4.4C4, C32⋊7(C8⋊S3), C6.8(C6.D6), C3⋊1(C12.31D6), (C32×C12).69C22, (C3×C3⋊C8)⋊8S3, C3⋊C8⋊5(C3⋊S3), C6.2(C4×C3⋊S3), C4.27(S3×C3⋊S3), C12.42(C2×C3⋊S3), (C32×C3⋊C8)⋊13C2, (C3×C6).47(C4×S3), C2.3(C33⋊8(C2×C4)), (C3×C32⋊4C8)⋊12C2, (C32×C6).36(C2×C4), (C2×C33⋊C2).4C4, (C4×C33⋊C2).4C2, SmallGroup(432,435)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊9M4(2)
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×C3⋊C8 — C33⋊9M4(2)
Lower central
C33 — C32×C6 — C33⋊9M4(2)
Upper central
C1 — C4

Generators and relations for C33⋊9M4(2)
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d5 >

Subgroups: 1160 in 184 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, M4(2), C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C33, C3⋊Dic3, C3×C12, C3×C12, C3×C12, C2×C3⋊S3, C8⋊S3, C33⋊C2, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C32⋊4C8, C3×C24, C4×C3⋊S3, C33⋊5C4, C32×C12, C2×C33⋊C2, C12.31D6, C24⋊S3, C32×C3⋊C8, C3×C32⋊4C8, C4×C33⋊C2, C33⋊9M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), C3⋊S3, C4×S3, S32, C2×C3⋊S3, C8⋊S3, C6.D6, C4×C3⋊S3, S3×C3⋊S3, C12.31D6, C24⋊S3, C33⋊8(C2×C4), C33⋊9M4(2)

Smallest permutation representation of C33⋊9M4(2)
►On 72 points
Generators in S72
(1 45 15)(2 46 16)(3 47 9)(4 48 10)(5 41 11)(6 42 12)(7 43 13)(8 44 14)(17 72 30)(18 65 31)(19 66 32)(20 67 25)(21 68 26)(22 69 27)(23 70 28)(24 71 29)(33 56 59)(34 49 60)(35 50 61)(36 51 62)(37 52 63)(38 53 64)(39 54 57)(40 55 58)

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

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

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

66 conjugacy classes

class 1 2A2B3A···3E3F3G3H3I4A4B4C6A···6E6F6G6H6I8A8B8C8D12A···12J12K···12R24A···24P24Q24R24S24T
order1223···333334446···66666888812···1212···1224···2424242424
size11542···2444411542···244446618182···24···46···618181818

66 irreducible representations

dim111111222222444
type+++++++++
imageC1C2C2C2C4C4S3S3D6M4(2)C4×S3C8⋊S3S32C6.D6C12.31D6
kernelC33⋊9M4(2)C32×C3⋊C8C3×C32⋊4C8C4×C33⋊C2C33⋊5C4C2×C33⋊C2C3×C3⋊C8C32⋊4C8C3×C12C33C3×C6C32C12C6C3
# reps11112241521020448

Matrix representation of C33⋊9M4(2) ►in GL8(𝔽73)

10000000
01000000
00100000
00010000
0000727200
00001000
00000010
00000001
,
10000000
01000000
007210000
007200000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000007272
00000010
,
641000000
199000000
007200000
000720000
000072000
000007200
00000010
0000007272
,
10000000
1872000000
00010000
00100000
00001000
0000727200
00000010
0000007272

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[64,19,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72],[1,18,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;
 

C33⋊9M4(2) in GAP, Magma, Sage, TeX 

C_3^3\rtimes_9M_4(2)
 
% in TeX
 
G:=Group("C3^3:9M4(2)");
 
// GroupNames label
 
G:=SmallGroup(432,435);
 
// by ID
 
G=gap.SmallGroup(432,435);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
 
// generators/relations
 

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