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## G = C3×C3⋊S3.Q8order 432 = 24·33

### Direct product of C3 and C3⋊S3.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C3×C3⋊S3.Q8
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C6×C3⋊S3 — C3×C6.D6 — C3×C3⋊S3.Q8
 Lower central C32 — C3⋊S3 — C3×C3⋊S3.Q8
 Upper central C1 — C6

Generators and relations for C3×C3⋊S3.Q8
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=ece-1=b-1, ebe-1=dcd=fcf-1=c-1, bf=fb, de=ed, df=fd, fef-1=de-1 >

Subgroups: 428 in 96 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×4], C22, S3 [×4], C6, C6 [×6], C2×C4 [×3], C32, C32 [×4], Dic3 [×2], C12 [×8], D6 [×2], C2×C6, C4⋊C4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3 [×2], C2×C12 [×3], C33, C3×Dic3 [×6], C3×C12 [×2], C32⋊C4 [×2], S3×C6 [×2], C2×C3⋊S3, C3×C4⋊C4, C3×C3⋊S3 [×2], C32×C6, C6.D6 [×2], S3×C12 [×2], C2×C32⋊C4, C32×Dic3 [×2], C3×C32⋊C4 [×2], C6×C3⋊S3, C3⋊S3.Q8, C3×C6.D6 [×2], C6×C32⋊C4, C3×C3⋊S3.Q8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4, Q8, C12 [×2], C2×C6, C4⋊C4, C2×C12, C3×D4, C3×Q8, C3×C4⋊C4, S3≀C2, C3⋊S3.Q8, C3×S3≀C2, C3×C3⋊S3.Q8

Smallest permutation representation of C3×C3⋊S3.Q8
On 48 points
Generators in S48
(1 17 5)(2 18 6)(3 19 7)(4 20 8)(9 23 35)(10 24 36)(11 21 33)(12 22 34)(13 40 47)(14 37 48)(15 38 45)(16 39 46)(25 29 44)(26 30 41)(27 31 42)(28 32 43)
(1 5 17)(3 7 19)(9 23 35)(11 21 33)(13 47 40)(15 45 38)(25 29 44)(27 31 42)
(2 18 6)(4 20 8)(10 36 24)(12 34 22)(14 37 48)(16 39 46)(26 41 30)(28 43 32)
(1 9)(2 10)(3 11)(4 12)(5 35)(6 36)(7 33)(8 34)(13 25)(14 26)(15 27)(16 28)(17 23)(18 24)(19 21)(20 22)(29 40)(30 37)(31 38)(32 39)(41 48)(42 45)(43 46)(44 47)
(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 40 3 38)(2 32 4 30)(5 13 7 15)(6 28 8 26)(9 29 11 31)(10 39 12 37)(14 36 16 34)(17 47 19 45)(18 43 20 41)(21 42 23 44)(22 48 24 46)(25 33 27 35)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6H 6I 6J 6K 6L 12A ··· 12H 12I ··· 12T 12U 12V 12W 12X order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 9 9 1 1 4 ··· 4 6 6 6 6 18 18 1 1 4 ··· 4 9 9 9 9 6 ··· 6 12 ··· 12 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + + image C1 C2 C2 C3 C4 C6 C6 C12 Q8 D4 C3×Q8 C3×D4 S3≀C2 C3⋊S3.Q8 C3×S3≀C2 C3×C3⋊S3.Q8 kernel C3×C3⋊S3.Q8 C3×C6.D6 C6×C32⋊C4 C3⋊S3.Q8 C3×C32⋊C4 C6.D6 C2×C32⋊C4 C32⋊C4 C3×C3⋊S3 C32×C6 C3⋊S3 C3×C6 C6 C3 C2 C1 # reps 1 2 1 2 4 4 2 8 1 1 2 2 4 4 8 8

Matrix representation of C3×C3⋊S3.Q8 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 3 9 0 0 0 0 0 0 1 0 0 0 2 0 0 1
,
 1 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 9 0 0 0 11 0 3 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 12 0 0 0 0 11 0 12 2 0 0 0 2 0 1
,
 5 0 0 0 0 0 1 8 0 0 0 0 0 0 0 0 1 0 0 0 12 0 12 1 0 0 1 0 0 0 0 0 1 1 1 0
,
 12 10 0 0 0 0 5 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 11 0 12 2 0 0 0 0 0 1

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

C3×C3⋊S3.Q8 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes S_3.Q_8
% in TeX

G:=Group("C3xC3:S3.Q8");
// GroupNames label

G:=SmallGroup(432,575);
// by ID

G=gap.SmallGroup(432,575);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,168,197,176,4037,3036,362,1189,1203]);
// Polycyclic

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

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