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## G = C6×C32⋊2Q8order 432 = 24·33

### Direct product of C6 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6×C32⋊2Q8
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C32×Dic3 — C3×C32⋊2Q8 — C6×C32⋊2Q8
 Lower central C32 — C3×C6 — C6×C32⋊2Q8
 Upper central C1 — C2×C6

Generators and relations for C6×C322Q8
G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 576 in 210 conjugacy classes, 80 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, C2×Dic6, C6×Q8, C32×C6, C32×C6, C322Q8, C3×Dic6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C3×C62, C2×C322Q8, C6×Dic6, C3×C322Q8, Dic3×C3×C6, C6×C3⋊Dic3, C6×C322Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S32, S3×C6, C2×Dic6, C6×Q8, C322Q8, C3×Dic6, C2×S32, S3×C2×C6, C3×S32, C2×C322Q8, C6×Dic6, C3×C322Q8, S32×C6, C6×C322Q8

Smallest permutation representation of C6×C322Q8
On 48 points
Generators in S48
(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 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 28 16 24)(2 29 17 19)(3 30 18 20)(4 25 13 21)(5 26 14 22)(6 27 15 23)(7 35 47 39)(8 36 48 40)(9 31 43 41)(10 32 44 42)(11 33 45 37)(12 34 46 38)
(1 37 16 33)(2 38 17 34)(3 39 18 35)(4 40 13 36)(5 41 14 31)(6 42 15 32)(7 30 47 20)(8 25 48 21)(9 26 43 22)(10 27 44 23)(11 28 45 24)(12 29 46 19)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 12A ··· 12AF 12AG 12AH 12AI 12AJ order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 ··· 2 4 4 4 6 6 6 6 18 18 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - + + - + - + image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 C3×Dic6 S32 C32⋊2Q8 C2×S32 C3×S32 C3×C32⋊2Q8 S32×C6 kernel C6×C32⋊2Q8 C3×C32⋊2Q8 Dic3×C3×C6 C6×C3⋊Dic3 C2×C32⋊2Q8 C32⋊2Q8 C6×Dic3 C2×C3⋊Dic3 C6×Dic3 C32×C6 C3×Dic3 C62 C2×Dic3 C3×C6 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 2 1 2 8 4 2 2 2 4 2 4 8 4 8 4 16 1 2 1 2 4 2

Matrix representation of C6×C322Q8 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 0 3 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 0 12 0 0 0 0 1 12
,
 12 12 0 0 0 0 1 0 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 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 10 9 0 0 0 0 9 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C6×C322Q8 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C6xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(432,657);
// by ID

G=gap.SmallGroup(432,657);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,365,176,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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