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## G = C2×C33⋊5Q8order 432 = 24·33

### Direct product of C2 and C33⋊5Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C2×C33⋊5Q8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊5Q8 — C2×C33⋊5Q8
 Lower central C33 — C32×C6 — C2×C33⋊5Q8
 Upper central C1 — C22

Generators and relations for C2×C335Q8
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf-1=b-1, cd=dc, ece-1=c-1, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 856 in 210 conjugacy classes, 63 normal (7 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C2×Dic6, C32×C6, C32×C6, C322Q8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, C2×C322Q8, C335Q8, C6×C3⋊Dic3, C2×C335Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, C322Q8, C2×S32, C324D6, C2×C322Q8, C335Q8, C2×C324D6, C2×C335Q8

Smallest permutation representation of C2×C335Q8
On 48 points
Generators in S48
(1 5)(2 6)(3 7)(4 8)(9 34)(10 35)(11 36)(12 33)(13 38)(14 39)(15 40)(16 37)(17 29)(18 30)(19 31)(20 32)(21 27)(22 28)(23 25)(24 26)(41 48)(42 45)(43 46)(44 47)
(1 35 39)(2 40 36)(3 33 37)(4 38 34)(5 10 14)(6 15 11)(7 12 16)(8 13 9)(17 22 41)(18 42 23)(19 24 43)(20 44 21)(25 30 45)(26 46 31)(27 32 47)(28 48 29)
(1 39 35)(2 36 40)(3 37 33)(4 34 38)(5 14 10)(6 11 15)(7 16 12)(8 9 13)(17 22 41)(18 42 23)(19 24 43)(20 44 21)(25 30 45)(26 46 31)(27 32 47)(28 48 29)
(1 39 35)(2 40 36)(3 37 33)(4 38 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 22 41)(18 23 42)(19 24 43)(20 21 44)(25 45 30)(26 46 31)(27 47 32)(28 48 29)
(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 44 3 42)(2 43 4 41)(5 47 7 45)(6 46 8 48)(9 29 11 31)(10 32 12 30)(13 28 15 26)(14 27 16 25)(17 36 19 34)(18 35 20 33)(21 37 23 39)(22 40 24 38)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D ··· 3H 4A ··· 4F 6A ··· 6I 6J ··· 6X 12A ··· 12L order 1 2 2 2 3 3 3 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 4 ··· 4 18 ··· 18 2 ··· 2 4 ··· 4 18 ··· 18

54 irreducible representations

 dim 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + + - + - + image C1 C2 C2 S3 Q8 D6 D6 Dic6 S32 C32⋊2Q8 C2×S32 C32⋊4D6 C33⋊5Q8 C2×C32⋊4D6 kernel C2×C33⋊5Q8 C33⋊5Q8 C6×C3⋊Dic3 C2×C3⋊Dic3 C32×C6 C3⋊Dic3 C62 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 3 3 2 6 3 12 3 6 3 2 4 2

Matrix representation of C2×C335Q8 in GL6(𝔽13)

 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 0 12 0 0 0 0 1 12 0 0 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 1 0 0 0 0 0 0 1 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 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 6 0 0 0 0 7 10
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 5 0 0 0 0 0 8

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

C2×C335Q8 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_5Q_8
% in TeX

G:=Group("C2xC3^3:5Q8");
// GroupNames label

G:=SmallGroup(432,695);
// by ID

G=gap.SmallGroup(432,695);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,64,1124,571,2028,14118]);
// Polycyclic

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

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