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G = C33⋊9(C4⋊C4)  order 432 = 24·33

2nd semidirect product of C33 and C4⋊C4 acting via C4⋊C4/C4=C4

metabelian, soluble, monomial

Aliases: C33⋊9(C4⋊C4), C4⋊(C33⋊C4), C3⋊S3.4D12, C12⋊1(C32⋊C4), (C32×C12)⋊1C4, (C3×C12)⋊6Dic3, C3⋊S3.4Dic6, C3⋊Dic3⋊7Dic3, C32⋊6(C4⋊Dic3), (C3×C3⋊S3).6Q8, (C12×C3⋊S3).4C2, (C4×C3⋊S3).12S3, (C3×C3⋊Dic3)⋊5C4, (C3×C3⋊S3).15D4, (C2×C3⋊S3).40D6, C6.11(C2×C32⋊C4), C3⋊1(C4⋊(C32⋊C4)), C2.5(C2×C33⋊C4), (C6×C3⋊S3).42C22, (C32×C6).18(C2×C4), (C2×C33⋊C4).6C2, (C3×C6).25(C2×Dic3), SmallGroup(432,638)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊9(C4⋊C4)
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C33⋊C4 — C33⋊9(C4⋊C4)
Lower central
C33 — C32×C6 — C33⋊9(C4⋊C4)
Upper central
C1 — C2 — C4

Generators and relations for C33⋊9(C4⋊C4)
 G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, ac=ca, dad-1=a-1, eae-1=ab-1, bc=cb, dbd-1=b-1, ebe-1=a-1b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 584 in 96 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C3⋊S3, C32×C6, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C3×C3⋊Dic3, C32×C12, C33⋊C4, C6×C3⋊S3, C4⋊(C32⋊C4), C12×C3⋊S3, C2×C33⋊C4, C33⋊9(C4⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, D12, C2×Dic3, C32⋊C4, C4⋊Dic3, C2×C32⋊C4, C33⋊C4, C4⋊(C32⋊C4), C2×C33⋊C4, C33⋊9(C4⋊C4)

Smallest permutation representation of C33⋊9(C4⋊C4)
►On 48 points
Generators in S48
(17 41 22)(18 23 42)(19 43 24)(20 21 44)(25 30 45)(26 46 31)(27 32 47)(28 48 29)

(1 35 39)(2 40 36)(3 33 37)(4 38 34)(5 14 10)(6 11 15)(7 16 12)(8 9 13)(17 41 22)(18 23 42)(19 43 24)(20 21 44)(25 30 45)(26 46 31)(27 32 47)(28 48 29)

(1 39 35)(2 40 36)(3 37 33)(4 38 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 22 41)(18 23 42)(19 24 43)(20 21 44)(25 45 30)(26 46 31)(27 47 32)(28 48 29)

(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)

(1 45 5 42)(2 48 6 41)(3 47 7 44)(4 46 8 43)(9 19 34 31)(10 18 35 30)(11 17 36 29)(12 20 33 32)(13 24 38 26)(14 23 39 25)(15 22 40 28)(16 21 37 27)
 
G:=sub<Sym(48)| (17,41,22)(18,23,42)(19,43,24)(20,21,44)(25,30,45)(26,46,31)(27,32,47)(28,48,29), (1,35,39)(2,40,36)(3,33,37)(4,38,34)(5,14,10)(6,11,15)(7,16,12)(8,9,13)(17,41,22)(18,23,42)(19,43,24)(20,21,44)(25,30,45)(26,46,31)(27,32,47)(28,48,29), (1,39,35)(2,40,36)(3,37,33)(4,38,34)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,22,41)(18,23,42)(19,24,43)(20,21,44)(25,45,30)(26,46,31)(27,47,32)(28,48,29), (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), (1,45,5,42)(2,48,6,41)(3,47,7,44)(4,46,8,43)(9,19,34,31)(10,18,35,30)(11,17,36,29)(12,20,33,32)(13,24,38,26)(14,23,39,25)(15,22,40,28)(16,21,37,27)>;
 
G:=Group( (17,41,22)(18,23,42)(19,43,24)(20,21,44)(25,30,45)(26,46,31)(27,32,47)(28,48,29), (1,35,39)(2,40,36)(3,33,37)(4,38,34)(5,14,10)(6,11,15)(7,16,12)(8,9,13)(17,41,22)(18,23,42)(19,43,24)(20,21,44)(25,30,45)(26,46,31)(27,32,47)(28,48,29), (1,39,35)(2,40,36)(3,37,33)(4,38,34)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,22,41)(18,23,42)(19,24,43)(20,21,44)(25,45,30)(26,46,31)(27,47,32)(28,48,29), (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), (1,45,5,42)(2,48,6,41)(3,47,7,44)(4,46,8,43)(9,19,34,31)(10,18,35,30)(11,17,36,29)(12,20,33,32)(13,24,38,26)(14,23,39,25)(15,22,40,28)(16,21,37,27) );
 
G=PermutationGroup([[(17,41,22),(18,23,42),(19,43,24),(20,21,44),(25,30,45),(26,46,31),(27,32,47),(28,48,29)], [(1,35,39),(2,40,36),(3,33,37),(4,38,34),(5,14,10),(6,11,15),(7,16,12),(8,9,13),(17,41,22),(18,23,42),(19,43,24),(20,21,44),(25,30,45),(26,46,31),(27,32,47),(28,48,29)], [(1,39,35),(2,40,36),(3,37,33),(4,38,34),(5,14,10),(6,15,11),(7,16,12),(8,13,9),(17,22,41),(18,23,42),(19,24,43),(20,21,44),(25,45,30),(26,46,31),(27,47,32),(28,48,29)], [(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)], [(1,45,5,42),(2,48,6,41),(3,47,7,44),(4,46,8,43),(9,19,34,31),(10,18,35,30),(11,17,36,29),(12,20,33,32),(13,24,38,26),(14,23,39,25),(15,22,40,28),(16,21,37,27)]])
 

42 conjugacy classes

class 1 2A2B2C3A3B···3G4A4B4C4D4E4F6A6B···6G6H6I12A12B12C···12N12O12P
order122233···344444466···666121212···121212
size119924···42185454545424···41818224···41818

42 irreducible representations

dim1111122222222444444
type+++++---+-+++
imageC1C2C2C4C4S3D4Q8Dic3Dic3D6Dic6D12C32⋊C4C2×C32⋊C4C33⋊C4C4⋊(C32⋊C4)C2×C33⋊C4C33⋊9(C4⋊C4)
kernelC33⋊9(C4⋊C4)C12×C3⋊S3C2×C33⋊C4C3×C3⋊Dic3C32×C12C4×C3⋊S3C3×C3⋊S3C3×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C3⋊S3C12C6C4C3C2C1
# reps1122211111122224448

Matrix representation of C33⋊9(C4⋊C4) ►in GL4(𝔽13) generated by

10123
0100
0090
0003
,
9904
0300
0090
0003
,
3099
0300
0090
0009
,
501111
1855
0008
0080
,
12000
0001
0100
81211
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,12,0,9,0,3,0,0,3],[9,0,0,0,9,3,0,0,0,0,9,0,4,0,0,3],[3,0,0,0,0,3,0,0,9,0,9,0,9,0,0,9],[5,1,0,0,0,8,0,0,11,5,0,8,11,5,8,0],[12,0,0,8,0,0,1,12,0,0,0,1,0,1,0,1] >;
 

C33⋊9(C4⋊C4) in GAP, Magma, Sage, TeX 

C_3^3\rtimes_9(C_4\rtimes C_4)
 
% in TeX
 
G:=Group("C3^3:9(C4:C4)");
 
// GroupNames label
 
G:=SmallGroup(432,638);
 
// by ID
 
G=gap.SmallGroup(432,638);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,64,2804,298,2693,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,e*a*e^-1=a*b^-1,b*c=c*b,d*b*d^-1=b^-1,e*b*e^-1=a^-1*b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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