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G = C33⋊C4⋊C4order 432 = 24·33

2nd semidirect product of C33⋊C4 and C4 acting via C4/C2=C2

non-abelian, soluble, monomial

Aliases: C6.14S3≀C2, C333(C4⋊C4), C33⋊C42C4, C3⋊S3.1Dic6, (C32×C6).8D4, C6.D6.2S3, C2.3(C33⋊D4), C323(Dic3⋊C4), C3⋊S3.3(C4×S3), (C3×C3⋊S3).3Q8, (C2×C3⋊S3).12D6, C31(C3⋊S3.Q8), C339(C2×C4).3C2, (C6×C3⋊S3).8C22, (C3×C6).14(C3⋊D4), (C2×C33⋊C4).3C2, (C3×C6.D6).4C2, (C3×C3⋊S3).10(C2×C4), SmallGroup(432,581)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — C33⋊C4⋊C4
C1C3C33C3×C3⋊S3C6×C3⋊S3C339(C2×C4) — C33⋊C4⋊C4
C33C3×C3⋊S3 — C33⋊C4⋊C4
C1C2

Generators and relations for C33⋊C4⋊C4
 G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, ac=ca, dad-1=ab-1, ae=ea, bc=cb, dbd-1=ebe-1=a-1b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 580 in 96 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×4], C22, S3 [×4], C6, C6 [×6], C2×C4 [×3], C32, C32 [×4], Dic3 [×6], C12 [×4], D6 [×2], C2×C6, C4⋊C4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×6], C3⋊Dic3, C3×C12, C32⋊C4 [×2], S3×C6 [×2], C2×C3⋊S3, Dic3⋊C4, C3×C3⋊S3 [×2], C32×C6, S3×Dic3, C6.D6, C6.D6, S3×C12, C2×C32⋊C4, C32×Dic3, C3×C3⋊Dic3, C33⋊C4 [×2], C6×C3⋊S3, C3⋊S3.Q8, C3×C6.D6, C339(C2×C4), C2×C33⋊C4, C33⋊C4⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, Dic3⋊C4, S3≀C2, C3⋊S3.Q8, C33⋊D4, C33⋊C4⋊C4

Smallest permutation representation of C33⋊C4⋊C4
On 48 points
Generators in S48
(1 16 45)(2 13 46)(3 47 14)(4 48 15)(5 28 38)(6 39 25)(7 40 26)(8 27 37)(9 17 34)(10 18 35)(11 36 19)(12 33 20)(21 29 43)(22 44 30)(23 41 31)(24 32 42)
(2 46 13)(4 15 48)(5 38 28)(7 26 40)(10 35 18)(12 20 33)(21 43 29)(23 31 41)
(1 45 16)(2 13 46)(3 47 14)(4 15 48)(5 38 28)(6 25 39)(7 40 26)(8 27 37)(9 34 17)(10 18 35)(11 36 19)(12 20 33)(21 29 43)(22 44 30)(23 31 41)(24 42 32)
(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 33 25 23)(2 36 26 22)(3 35 27 21)(4 34 28 24)(5 42 15 17)(6 41 16 20)(7 44 13 19)(8 43 14 18)(9 38 32 48)(10 37 29 47)(11 40 30 46)(12 39 31 45)

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

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

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

36 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D4E4F6A6B6C6D6E6F6G6H12A12B12C12D12E···12J12K12L
order1222333333444444666666661212121212···121212
size119924444866181854542444481818666612···123636

36 irreducible representations

dim111112222222444488
type+++++-++-++-
imageC1C2C2C2C4S3Q8D4D6Dic6C4×S3C3⋊D4S3≀C2C3⋊S3.Q8C33⋊D4C33⋊C4⋊C4C33⋊D4C33⋊C4⋊C4
kernelC33⋊C4⋊C4C3×C6.D6C339(C2×C4)C2×C33⋊C4C33⋊C4C6.D6C3×C3⋊S3C32×C6C2×C3⋊S3C3⋊S3C3⋊S3C3×C6C6C3C2C1C2C1
# reps111141111222444411

Matrix representation of C33⋊C4⋊C4 in GL6(𝔽13)

100000
010000
00120120
000001
001000
00012012
,
100000
010000
001000
00012012
000010
000100
,
0120000
1120000
001000
000100
000010
000001
,
920000
1140000
0001200
0012000
0000012
001010
,
290000
4110000
000500
005000
000808
008080

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,1,0,0,0,0,12,0,0],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,11,0,0,0,0,2,4,0,0,0,0,0,0,0,12,0,1,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,5,0,8,0,0,5,0,8,0,0,0,0,0,0,8,0,0,0,0,8,0] >;

C33⋊C4⋊C4 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_4\rtimes C_4
% in TeX

G:=Group("C3^3:C4:C4");
// GroupNames label

G:=SmallGroup(432,581);
// by ID

G=gap.SmallGroup(432,581);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,1684,571,298,677,1027,14118]);
// 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*b^-1,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a^-1*b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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