C6^2.Dic3 - GroupNames

G = C₆².Dic₃ order 432 = 2⁴·3³

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

Aliases: C₆².₈Dic₃, C₃.A₄⋊C₁₂, C₆.S₄⋊C₃, C₆.₇(C₃×S₄), (C₃×C₆).₅S₄, C₃².A₄⋊C₄, C₂²⋊(C₉⋊C₁₂), C₂³.(C₉⋊C₆), C₃².(A₄⋊C₄), (C₂×C₆²).₅S₃, C₂.₁(C₃².S₄), (C₂×C₃.A₄).C₆, C₃.₁(C₃×A₄⋊C₄), (C₂×C₃².A₄).C₂, (C₂²×C₆).₄(C₃×S₃), (C₂×C₆).₂(C₃×Dic₃), SmallGroup(432,249)

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

G = C62.Dic3 order 432 = 24·33

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

G = C₆².Dic₃ order 432 = 2⁴·3³

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