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## G = C62.Dic3order 432 = 24·33

### 8th non-split extension by C62 of Dic3 acting via Dic3/C2=S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C62.Dic3
G = < a,b,c,d | a6=b6=1, c6=b2, d2=b2c3, cac-1=ab=ba, dad-1=a4b3, cbc-1=a3b4, dbd-1=a3b2, dcd-1=b4c5 >

Subgroups: 314 in 72 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C32, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, 3- 1+2, Dic9, C3.A4, C3.A4, C3×Dic3, C62, C62, C6.D4, C3×C22⋊C4, C2×3- 1+2, C2×C3.A4, C2×C3.A4, C6×Dic3, C2×C62, C9⋊C12, C32.A4, C6.S4, C3×C6.D4, C2×C32.A4, C62.Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, S4, C3×Dic3, A4⋊C4, C9⋊C6, C3×S4, C9⋊C12, C3×A4⋊C4, C32.S4, C62.Dic3

Smallest permutation representation of C62.Dic3
On 36 points
Generators in S36
(1 10)(2 14 8)(3 18 15 12 9 6)(4 13)(5 17 11)(7 16)(19 34 31 28 25 22)(20 29)(21 33 27)(23 32)(24 36 30)(26 35)
(1 4 7 10 13 16)(2 5 8 11 14 17)(3 15 9)(6 18 12)(19 31 25)(20 23 26 29 32 35)(21 24 27 30 33 36)(22 34 28)
(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 20 10 29)(2 19 11 28)(3 36 12 27)(4 35 13 26)(5 34 14 25)(6 33 15 24)(7 32 16 23)(8 31 17 22)(9 30 18 21)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B ··· 6G 6H ··· 6M 9A 9B 9C 12A ··· 12H 18A 18B 18C order 1 2 2 2 3 3 3 4 4 4 4 6 6 ··· 6 6 ··· 6 9 9 9 12 ··· 12 18 18 18 size 1 1 3 3 2 3 3 18 18 18 18 2 3 ··· 3 6 ··· 6 24 24 24 18 ··· 18 24 24 24

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 C9⋊C6 C9⋊C12 C32.S4 C32.S4 C62.Dic3 C62.Dic3 kernel C62.Dic3 C2×C32.A4 C6.S4 C32.A4 C2×C3.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 C62.Dic3 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 36 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 1 36
,
 0 1 0 0 0 0 36 1 0 0 0 0 0 0 0 36 0 0 0 0 1 36 0 0 0 0 0 0 0 1 0 0 0 0 36 1
,
 0 0 1 36 0 0 0 0 1 0 0 0 0 0 0 0 1 36 0 0 0 0 1 0 36 0 0 0 0 0 0 36 0 0 0 0
,
 35 13 0 0 0 0 11 2 0 0 0 0 0 0 0 0 26 35 0 0 0 0 24 11 0 0 26 35 0 0 0 0 24 11 0 0

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,1,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,36,36],[0,36,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,0,36,0,0,0,0,1,1],[0,0,0,0,36,0,0,0,0,0,0,36,1,1,0,0,0,0,36,0,0,0,0,0,0,0,1,1,0,0,0,0,36,0,0,0],[35,11,0,0,0,0,13,2,0,0,0,0,0,0,0,0,26,24,0,0,0,0,35,11,0,0,26,24,0,0,0,0,35,11,0,0] >;

C62.Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,249);
// by ID

G=gap.SmallGroup(432,249);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,1683,682,192,2524,9077,782,5298,1350]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^2,d^2=b^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=b^4*c^5>;
// generators/relations

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