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G = C3×C3⋊S3.Q8order 432 = 24·33

Direct product of C3 and C3⋊S3.Q8

direct product, non-abelian, soluble, monomial

Aliases: C3×C3⋊S3.Q8, C6.20S3≀C2, C32⋊C41C12, C332(C4⋊C4), (C32×C6).2D4, C6.D6.2C6, C3⋊S3.(C3×Q8), C2.2(C3×S3≀C2), (C3×C32⋊C4)⋊4C4, C321(C3×C4⋊C4), (C3×C6).2(C3×D4), (C3×C3⋊S3).1Q8, C3⋊S3.3(C2×C12), (C2×C32⋊C4).1C6, (C6×C3⋊S3).5C22, (C6×C32⋊C4).10C2, (C3×C6.D6).3C2, (C2×C3⋊S3).5(C2×C6), (C3×C3⋊S3).5(C2×C4), SmallGroup(432,575)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C3×C3⋊S3.Q8
C1C32C3×C6C2×C3⋊S3C6×C3⋊S3C3×C6.D6 — C3×C3⋊S3.Q8
C32C3⋊S3 — C3×C3⋊S3.Q8
C1C6

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 2A2B2C3A3B3C···3H4A4B4C4D4E4F6A6B6C···6H6I6J6K6L12A···12H12I···12T12U12V12W12X
order1222333···3444444666···6666612···1212···1212121212
size1199114···466661818114···499996···612···1218181818

54 irreducible representations

dim1111111122224444
type+++-++
imageC1C2C2C3C4C6C6C12Q8D4C3×Q8C3×D4S3≀C2C3⋊S3.Q8C3×S3≀C2C3×C3⋊S3.Q8
kernelC3×C3⋊S3.Q8C3×C6.D6C6×C32⋊C4C3⋊S3.Q8C3×C32⋊C4C6.D6C2×C32⋊C4C32⋊C4C3×C3⋊S3C32×C6C3⋊S3C3×C6C6C3C2C1
# reps1212442811224488

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

300000
030000
009000
000900
000090
000009
,
100000
010000
003000
003900
000010
002001
,
100000
010000
001000
000100
000090
0011033
,
100000
010000
001200
0001200
00110122
000201
,
500000
180000
000010
00120121
001000
001110
,
12100000
510000
001000
000100
00110122
000001

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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