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## G = C62⋊5Dic3order 432 = 24·33

### 4th semidirect product of C62 and Dic3 acting via Dic3/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C62⋊5Dic3
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C2×C32⋊A4 — C62⋊5Dic3
 Lower central C3×A4 — C62⋊5Dic3
 Upper central C1 — C2

Generators and relations for C625Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, cac-1=ab=ba, dad-1=a4b3, cbc-1=a3b4, dbd-1=a3b2, dcd-1=c-1 >

Subgroups: 449 in 82 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, A4, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×A4, C22×C6, C22×C6, He3, C3×Dic3, C3⋊Dic3, C3×A4, C3×A4, C62, C62, C6.D4, C3×C22⋊C4, A4⋊C4, C2×He3, C6×Dic3, C6×A4, C6×A4, C2×C62, C32⋊C12, C32⋊A4, C3×C6.D4, C6.7S4, C2×C32⋊A4, C625Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, S4, C3×Dic3, A4⋊C4, C32⋊C6, C3×S4, C32⋊C12, C3×A4⋊C4, C62⋊S3, C625Dic3

Smallest permutation representation of C625Dic3
On 36 points
Generators in S36
(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)
(1 4 5 2 3 6)(7 10 12 8 9 11)(13 17 15 16 14 18)(19 24 21 23 20 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)
(1 19 32 2 23 35)(3 20 34 4 24 31)(5 21 36 6 22 33)(7 16 25 8 13 28)(9 17 27 10 14 30)(11 15 26 12 18 29)
(1 7 2 8)(3 12 4 11)(5 9 6 10)(13 35 16 32)(14 33 17 36)(15 31 18 34)(19 28 23 25)(20 26 24 29)(21 30 22 27)

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

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), (1,4,5,2,3,6)(7,10,12,8,9,11)(13,17,15,16,14,18)(19,24,21,23,20,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36), (1,19,32,2,23,35)(3,20,34,4,24,31)(5,21,36,6,22,33)(7,16,25,8,13,28)(9,17,27,10,14,30)(11,15,26,12,18,29), (1,7,2,8)(3,12,4,11)(5,9,6,10)(13,35,16,32)(14,33,17,36)(15,31,18,34)(19,28,23,25)(20,26,24,29)(21,30,22,27) );

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)], [(1,4,5,2,3,6),(7,10,12,8,9,11),(13,17,15,16,14,18),(19,24,21,23,20,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36)], [(1,19,32,2,23,35),(3,20,34,4,24,31),(5,21,36,6,22,33),(7,16,25,8,13,28),(9,17,27,10,14,30),(11,15,26,12,18,29)], [(1,7,2,8),(3,12,4,11),(5,9,6,10),(13,35,16,32),(14,33,17,36),(15,31,18,34),(19,28,23,25),(20,26,24,29),(21,30,22,27)]])

38 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B ··· 6G 6H ··· 6M 6N 6O 6P 12A ··· 12H order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 6 ··· 6 6 ··· 6 6 6 6 12 ··· 12 size 1 1 3 3 2 3 3 24 24 24 18 18 18 18 2 3 ··· 3 6 ··· 6 24 24 24 18 ··· 18

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 6 6 6 6 type + + + - + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 S4 A4⋊C4 C3×S4 C3×A4⋊C4 C32⋊C6 C32⋊C12 C62⋊S3 C62⋊S3 C62⋊5Dic3 C62⋊5Dic3 kernel C62⋊5Dic3 C2×C32⋊A4 C6.7S4 C32⋊A4 C6×A4 C3×A4 C2×C62 C62 C22×C6 C2×C6 C3×C6 C32 C6 C3 C23 C22 C2 C2 C1 C1 # reps 1 1 2 2 2 4 1 1 2 2 2 2 4 4 1 1 1 2 1 2

Matrix representation of C625Dic3 in GL9(𝔽13)

 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 4 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 4 0 9
,
 12 0 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 0 0 0 9 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3
,
 0 12 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 0 1 2 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 12
,
 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 0 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 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(9,GF(13))| [4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,9],[12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3],[0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,12],[8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C625Dic3 in GAP, Magma, Sage, TeX

C_6^2\rtimes_5{\rm Dic}_3
% in TeX

G:=Group("C6^2:5Dic3");
// GroupNames label

G:=SmallGroup(432,251);
// by ID

G=gap.SmallGroup(432,251);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,675,682,2524,9077,782,5298,1350]);
// Polycyclic

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

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