Copied to
clipboard

## G = C3×C3⋊S3⋊3C8order 432 = 24·33

### Direct product of C3 and C3⋊S3⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3⋊3C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×C32⋊2C8 — C3×C3⋊S3⋊3C8
 Lower central C32 — C3×C3⋊S3⋊3C8
 Upper central C1 — C12

Generators and relations for C3×C3⋊S33C8
G = < a,b,c,d,e | a3=b3=c3=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc-1, dcd=c-1, ece-1=b-1c-1, de=ed >

Subgroups: 364 in 88 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C24, C3×C3⋊S3, C32×C6, C322C8, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3⋊S33C8, C3×C322C8, C12×C3⋊S3, C3×C3⋊S33C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C2×C8, C24, C2×C12, C32⋊C4, C2×C24, C2×C32⋊C4, C3×C32⋊C4, C3⋊S33C8, C6×C32⋊C4, C3×C3⋊S33C8

Smallest permutation representation of C3×C3⋊S33C8
On 48 points
Generators in S48
(1 13 47)(2 14 48)(3 15 41)(4 16 42)(5 9 43)(6 10 44)(7 11 45)(8 12 46)(17 40 27)(18 33 28)(19 34 29)(20 35 30)(21 36 31)(22 37 32)(23 38 25)(24 39 26)
(2 14 48)(4 42 16)(6 10 44)(8 46 12)(17 27 40)(19 34 29)(21 31 36)(23 38 25)
(1 47 13)(2 14 48)(3 15 41)(4 42 16)(5 43 9)(6 10 44)(7 11 45)(8 46 12)(17 27 40)(18 28 33)(19 34 29)(20 35 30)(21 31 36)(22 32 37)(23 38 25)(24 39 26)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)| (1,13,47)(2,14,48)(3,15,41)(4,16,42)(5,9,43)(6,10,44)(7,11,45)(8,12,46)(17,40,27)(18,33,28)(19,34,29)(20,35,30)(21,36,31)(22,37,32)(23,38,25)(24,39,26), (2,14,48)(4,42,16)(6,10,44)(8,46,12)(17,27,40)(19,34,29)(21,31,36)(23,38,25), (1,47,13)(2,14,48)(3,15,41)(4,42,16)(5,43,9)(6,10,44)(7,11,45)(8,46,12)(17,27,40)(18,28,33)(19,34,29)(20,35,30)(21,31,36)(22,32,37)(23,38,25)(24,39,26), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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( (1,13,47)(2,14,48)(3,15,41)(4,16,42)(5,9,43)(6,10,44)(7,11,45)(8,12,46)(17,40,27)(18,33,28)(19,34,29)(20,35,30)(21,36,31)(22,37,32)(23,38,25)(24,39,26), (2,14,48)(4,42,16)(6,10,44)(8,46,12)(17,27,40)(19,34,29)(21,31,36)(23,38,25), (1,47,13)(2,14,48)(3,15,41)(4,42,16)(5,43,9)(6,10,44)(7,11,45)(8,46,12)(17,27,40)(18,28,33)(19,34,29)(20,35,30)(21,31,36)(22,32,37)(23,38,25)(24,39,26), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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([[(1,13,47),(2,14,48),(3,15,41),(4,16,42),(5,9,43),(6,10,44),(7,11,45),(8,12,46),(17,40,27),(18,33,28),(19,34,29),(20,35,30),(21,36,31),(22,37,32),(23,38,25),(24,39,26)], [(2,14,48),(4,42,16),(6,10,44),(8,46,12),(17,27,40),(19,34,29),(21,31,36),(23,38,25)], [(1,47,13),(2,14,48),(3,15,41),(4,42,16),(5,43,9),(6,10,44),(7,11,45),(8,46,12),(17,27,40),(18,28,33),(19,34,29),(20,35,30),(21,31,36),(22,32,37),(23,38,25),(24,39,26)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)]])

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 8A ··· 8H 12A 12B 12C 12D 12E ··· 12P 12Q 12R 12S 12T 24A ··· 24P order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 9 9 1 1 4 ··· 4 1 1 9 9 1 1 4 ··· 4 9 9 9 9 9 ··· 9 1 1 1 1 4 ··· 4 9 9 9 9 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C3⋊S3⋊3C8 C6×C32⋊C4 C3×C3⋊S3⋊3C8 kernel C3×C3⋊S3⋊3C8 C3×C32⋊2C8 C12×C3⋊S3 C3⋊S3⋊3C8 C32×C12 C6×C3⋊S3 C32⋊2C8 C4×C3⋊S3 C3×C3⋊S3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 4 4 4 8

Matrix representation of C3×C3⋊S33C8 in GL5(𝔽73)

 8 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 64
,
 1 0 0 0 0 0 8 0 0 0 0 0 64 0 0 0 0 0 8 0 0 0 0 0 64
,
 1 0 0 0 0 0 0 72 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 72 0
,
 22 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0

G:=sub<GL(5,GF(73))| [8,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,72,0],[22,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0] >;

C3×C3⋊S33C8 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes S_3\rtimes_3C_8
% in TeX

G:=Group("C3xC3:S3:3C8");
// GroupNames label

G:=SmallGroup(432,628);
// by ID

G=gap.SmallGroup(432,628);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,176,80,14117,362,18822,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1,e*b*e^-1=b*c^-1,d*c*d=c^-1,e*c*e^-1=b^-1*c^-1,d*e=e*d>;
// generators/relations

׿
×
𝔽