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## G = C2×C6×C32⋊C4order 432 = 24·33

### Direct product of C2×C6 and C32⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C6×C32⋊C4
G = < a,b,c,d,e | a2=b6=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 844 in 192 conjugacy classes, 64 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×4], C4 [×4], C22, C22 [×6], S3 [×8], C6 [×3], C6 [×16], C2×C4 [×6], C23, C32, C32 [×4], C12 [×4], D6 [×12], C2×C6, C2×C6 [×10], C22×C4, C3×S3 [×8], C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C3×C6 [×12], C2×C12 [×6], C22×S3 [×2], C22×C6, C33, C32⋊C4 [×4], S3×C6 [×12], C2×C3⋊S3 [×6], C62, C62 [×4], C22×C12, C3×C3⋊S3, C3×C3⋊S3 [×3], C32×C6 [×3], C2×C32⋊C4 [×6], S3×C2×C6 [×2], C22×C3⋊S3, C3×C32⋊C4 [×4], C6×C3⋊S3 [×6], C3×C62, C22×C32⋊C4, C6×C32⋊C4 [×6], C2×C6×C3⋊S3, C2×C6×C32⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×C12 [×6], C22×C6, C32⋊C4, C22×C12, C2×C32⋊C4 [×3], C3×C32⋊C4, C22×C32⋊C4, C6×C32⋊C4 [×3], C2×C6×C32⋊C4

Smallest permutation representation of C2×C6×C32⋊C4
On 48 points
Generators in S48
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(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)
(7 9 11)(8 10 12)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 8 16 48)(2 9 17 43)(3 10 18 44)(4 11 13 45)(5 12 14 46)(6 7 15 47)(19 41 29 31)(20 42 30 32)(21 37 25 33)(22 38 26 34)(23 39 27 35)(24 40 28 36)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 4A ··· 4H 6A ··· 6F 6G ··· 6X 6Y ··· 6AF 12A ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 9 9 9 9 1 1 4 ··· 4 9 ··· 9 1 ··· 1 4 ··· 4 9 ··· 9 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C6×C32⋊C4 kernel C2×C6×C32⋊C4 C6×C32⋊C4 C2×C6×C3⋊S3 C22×C32⋊C4 C6×C3⋊S3 C3×C62 C2×C32⋊C4 C22×C3⋊S3 C2×C3⋊S3 C62 C2×C6 C6 C22 C2 # reps 1 6 1 2 6 2 12 2 12 4 2 6 4 12

Matrix representation of C2×C6×C32⋊C4 in GL5(𝔽13)

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

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[10,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,12,0,0,0,0,0,12,0,0] >;

C2×C6×C32⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_3^2\rtimes C_4
% in TeX

G:=Group("C2xC6xC3^2:C4");
// GroupNames label

G:=SmallGroup(432,765);
// by ID

G=gap.SmallGroup(432,765);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,3,168,14117,201,18822,622]);
// Polycyclic

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

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