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## G = C33⋊5(C2×Q8)  order 432 = 24·33

### 2nd semidirect product of C33 and C2×Q8 acting via C2×Q8/C2=C23

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊5(C2×Q8)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C3×C6.D6 — C33⋊5(C2×Q8)
 Lower central C33 — C32×C6 — C33⋊5(C2×Q8)
 Upper central C1 — C2

Generators and relations for C335(C2×Q8)
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, eae-1=a-1, af=fa, bc=cb, dbd=ebe-1=b-1, bf=fb, dcd=ece-1=fcf-1=c-1, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1020 in 198 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×Dic6, S3×Q8, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, C6.D6, C322Q8, C322Q8, C3×Dic6, S3×C12, C324Q8, C4×C3⋊S3, C32×Dic3, C32×Dic3, C3×C3⋊Dic3, C335C4, C6×C3⋊S3, S3×Dic6, Dic3.D6, C3×C6.D6, C3×C322Q8, Dic3×C3⋊S3, C334Q8, C339(C2×C4), C335(C2×Q8)
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, S3×Dic6, Dic3.D6, S33, C335(C2×Q8)

Smallest permutation representation of C335(C2×Q8)
On 48 points
Generators in S48
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 26 22)(6 23 27)(7 28 24)(8 21 25)(9 37 30)(10 31 38)(11 39 32)(12 29 40)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 22 26)(6 27 23)(7 24 28)(8 25 21)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 26 22)(6 23 27)(7 28 24)(8 21 25)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 8)(2 5)(3 6)(4 7)(9 19)(10 20)(11 17)(12 18)(13 40)(14 37)(15 38)(16 39)(21 45)(22 46)(23 47)(24 48)(25 44)(26 41)(27 42)(28 43)(29 35)(30 36)(31 33)(32 34)
(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 36 3 34)(2 35 4 33)(5 29 7 31)(6 32 8 30)(9 23 11 21)(10 22 12 24)(13 43 15 41)(14 42 16 44)(17 45 19 47)(18 48 20 46)(25 37 27 39)(26 40 28 38)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12N 12O 12P order 1 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 size 1 1 9 9 2 2 2 4 4 4 8 6 6 6 18 18 54 2 2 2 4 4 4 8 18 18 6 6 6 6 12 ··· 12 36 36

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + - + + + - + - + - + - image C1 C2 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 Dic6 S32 S3×Q8 C2×S32 S3×Dic6 Dic3.D6 S33 C33⋊5(C2×Q8) kernel C33⋊5(C2×Q8) C3×C6.D6 C3×C32⋊2Q8 Dic3×C3⋊S3 C33⋊4Q8 C33⋊9(C2×C4) C6.D6 C32⋊2Q8 C3×C3⋊S3 C3×Dic3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 Dic3 C32 C6 C3 C3 C2 C1 # reps 1 1 2 1 2 1 1 2 2 6 2 1 4 3 2 3 4 2 1 1

Matrix representation of C335(C2×Q8) in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1
,
 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12

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

C335(C2×Q8) in GAP, Magma, Sage, TeX

C_3^3\rtimes_5(C_2\times Q_8)
% in TeX

G:=Group("C3^3:5(C2xQ8)");
// GroupNames label

G:=SmallGroup(432,604);
// by ID

G=gap.SmallGroup(432,604);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,298,2028,14118]);
// 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,e*a*e^-1=a^-1,a*f=f*a,b*c=c*b,d*b*d=e*b*e^-1=b^-1,b*f=f*b,d*c*d=e*c*e^-1=f*c*f^-1=c^-1,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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