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## G = C22×C33⋊C4order 432 = 24·33

### Direct product of C22 and C33⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C22×C33⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C33⋊C4 — C2×C33⋊C4 — C22×C33⋊C4
 Lower central C33 — C22×C33⋊C4
 Upper central C1 — C22

Generators and relations for C22×C33⋊C4
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cd-1, de=ed, fdf-1=c-1d-1, fef-1=e-1 >

Subgroups: 1064 in 192 conjugacy classes, 53 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C22×S3, C22×C6, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C62, C62, C22×Dic3, C3×C3⋊S3, C3×C3⋊S3, C32×C6, C2×C32⋊C4, S3×C2×C6, C22×C3⋊S3, C33⋊C4, C6×C3⋊S3, C3×C62, C22×C32⋊C4, C2×C33⋊C4, C2×C6×C3⋊S3, C22×C33⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C2×Dic3, C22×S3, C32⋊C4, C22×Dic3, C2×C32⋊C4, C33⋊C4, C22×C32⋊C4, C2×C33⋊C4, C22×C33⋊C4

Smallest permutation representation of C22×C33⋊C4
On 48 points
Generators in S48
(1 8)(2 5)(3 6)(4 7)(9 21)(10 22)(11 23)(12 24)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(25 37)(26 38)(27 39)(28 40)(41 45)(42 46)(43 47)(44 48)
(1 43)(2 44)(3 41)(4 42)(5 48)(6 45)(7 46)(8 47)(9 15)(10 16)(11 13)(12 14)(17 26)(18 27)(19 28)(20 25)(21 31)(22 32)(23 29)(24 30)(33 38)(34 39)(35 40)(36 37)
(1 17 16)(2 13 18)(3 14 19)(4 20 15)(5 29 34)(6 30 35)(7 36 31)(8 33 32)(9 42 25)(10 43 26)(11 27 44)(12 28 41)(21 46 37)(22 47 38)(23 39 48)(24 40 45)
(1 16 17)(3 19 14)(6 35 30)(8 32 33)(10 26 43)(12 41 28)(22 38 47)(24 45 40)
(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 34 29)(6 30 35)(7 36 31)(8 32 33)(9 42 25)(10 26 43)(11 44 27)(12 28 41)(21 46 37)(22 38 47)(23 48 39)(24 40 45)
(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)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B ··· 3G 4A ··· 4H 6A 6B 6C 6D ··· 6U 6V 6W 6X 6Y order 1 2 2 2 2 2 2 2 3 3 ··· 3 4 ··· 4 6 6 6 6 ··· 6 6 6 6 6 size 1 1 1 1 9 9 9 9 2 4 ··· 4 27 ··· 27 2 2 2 4 ··· 4 18 18 18 18

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 C32⋊C4 C2×C32⋊C4 C33⋊C4 C2×C33⋊C4 kernel C22×C33⋊C4 C2×C33⋊C4 C2×C6×C3⋊S3 C6×C3⋊S3 C3×C62 C22×C3⋊S3 C2×C3⋊S3 C2×C3⋊S3 C62 C2×C6 C6 C22 C2 # reps 1 6 1 6 2 1 3 3 1 2 6 4 12

Matrix representation of C22×C33⋊C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 10 0 0 0 3 6 0 0 0 0 0 9 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 2 12 0 0 0 9 5 12 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 3 0 7 3 0 0 0 3 6 4 0 0 0 0 9 0 0 0 0 0 0 9
,
 8 0 0 0 0 0 5 5 0 0 0 0 0 0 12 2 3 10 0 0 3 11 5 8 0 0 0 11 2 10 0 0 11 0 11 1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,6,9,0,0,0,10,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,2,5,1,0,0,0,12,12,0,1],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,7,6,9,0,0,0,3,4,0,9],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,12,3,0,11,0,0,2,11,11,0,0,0,3,5,2,11,0,0,10,8,10,1] >;

C22×C33⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_3^3\rtimes C_4
% in TeX

G:=Group("C2^2xC3^3:C4");
// GroupNames label

G:=SmallGroup(432,766);
// by ID

G=gap.SmallGroup(432,766);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,2804,165,2693,530,14118]);
// Polycyclic

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

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