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G = C4×C32⋊4D6order 432 = 24·33

Direct product of C4 and C32⋊4D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C4×C32⋊4D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C2×C32⋊4D6 — C4×C32⋊4D6
 Lower central C33 — C4×C32⋊4D6
 Upper central C1 — C4

Generators and relations for C4×C324D6
G = < a,b,c,d,e | a4=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1320 in 270 conjugacy classes, 63 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, C12, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C3×C3⋊Dic3, C32×C12, C324D6, C6×C3⋊S3, C4×S32, C339(C2×C4), C12×C3⋊S3, C2×C324D6, C4×C324D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C2×S32, C324D6, C4×S32, C2×C324D6, C4×C324D6

Smallest permutation representation of C4×C324D6
On 48 points
Generators in S48
(1 30 19 7)(2 25 20 8)(3 26 21 9)(4 27 22 10)(5 28 23 11)(6 29 24 12)(13 47 38 34)(14 48 39 35)(15 43 40 36)(16 44 41 31)(17 45 42 32)(18 46 37 33)
(1 3 5)(2 6 4)(7 9 11)(8 12 10)(13 15 17)(14 18 16)(19 21 23)(20 24 22)(25 29 27)(26 28 30)(31 35 33)(32 34 36)(37 41 39)(38 40 42)(43 45 47)(44 48 46)
(1 3 5)(2 6 4)(7 9 11)(8 12 10)(13 17 15)(14 16 18)(19 21 23)(20 24 22)(25 29 27)(26 28 30)(31 33 35)(32 36 34)(37 39 41)(38 42 40)(43 47 45)(44 46 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 33)(2 32)(3 31)(4 36)(5 35)(6 34)(7 37)(8 42)(9 41)(10 40)(11 39)(12 38)(13 29)(14 28)(15 27)(16 26)(17 25)(18 30)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)

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

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

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

60 conjugacy classes

 class 1 2A 2B ··· 2G 3A 3B 3C 3D ··· 3H 4A 4B 4C ··· 4H 6A 6B 6C 6D ··· 6H 6I ··· 6N 12A ··· 12F 12G ··· 12P 12Q ··· 12V order 1 2 2 ··· 2 3 3 3 3 ··· 3 4 4 4 ··· 4 6 6 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 9 ··· 9 2 2 2 4 ··· 4 1 1 9 ··· 9 2 2 2 4 ··· 4 18 ··· 18 2 ··· 2 4 ··· 4 18 ··· 18

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D6 D6 D6 C4×S3 S32 C2×S32 C32⋊4D6 C4×S32 C2×C32⋊4D6 C4×C32⋊4D6 kernel C4×C32⋊4D6 C33⋊9(C2×C4) C12×C3⋊S3 C2×C32⋊4D6 C32⋊4D6 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C6 C4 C3 C2 C1 # reps 1 3 3 1 8 3 3 3 3 12 3 3 2 6 2 4

Matrix representation of C4×C324D6 in GL6(𝔽13)

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

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

C4×C324D6 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_4D_6
% in TeX

G:=Group("C4xC3^2:4D6");
// GroupNames label

G:=SmallGroup(432,690);
// by ID

G=gap.SmallGroup(432,690);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,1124,571,2028,14118]);
// Polycyclic

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

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