C6^2.84D6 - GroupNames

G = C₆².₈₄D₆ order 432 = 2⁴·3³

32^nd non-split extension by C₆² of D₆ acting via D₆/C₃=C₂²

Aliases: C₆².₈₄D₆, (C₃xC₆).₄₇D₁₂, C₃:₂(D₆:Dic₃), C₆.₂₀(S₃xDic₃), (C₃²xC₆).₅₀D₄, C₃²:₁₀(D₆:C₄), C₃³:₁₃(C₂²:C₄), C₂.₁(C₃³:₉D₄), C₆.₄₀(C₃:D₁₂), C₆.₁₀(D₆:S₃), C₃:₂(C₆.D₁₂), C₆.₂₁(C₆.D₆), (C₃xC₆²).₁₄C₂², C₃²:₇(C₆.D₄), C₂².₃(C₃²:₄D₆), (C₆xC₃:S₃):₃C₄, (C₂xC₆).₆₀S₃², (C₃xC₆).₅₆(C₄xS₃), (C₆xC₃:Dic₃):₄C₂, (C₂xC₃:Dic₃):₉S₃, (C₂xC₃:S₃):₄Dic₃, (C₂²xC₃:S₃).₅S₃, C₂.₄(C₃³:₉(C₂xC₄)), (C₃xC₆).₆₈(C₃:D₄), (C₃²xC₆).₄₇(C₂xC₄), (C₃xC₆).₄₃(C₂xDic₃), (C₂xC₆xC₃:S₃).₃C₂, SmallGroup(432,461)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²xC₆ — C₆².₈₄D₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₆ — C₃xC₆² — C₂xC₆xC₃:S₃ — C₆².₈₄D₆

Lower central

C₃³ — C₃²xC₆ — C₆².₈₄D₆

Upper central

C₁ — C₂²

Generators and relations for C₆².₈₄D₆
G = < a,b,c,d | a⁶=b⁶=1, c⁶=a³, d²=a³b³, ab=ba, cac^-1=a^-1, ad=da, cbc^-1=dbd^-1=b^-1, dcd^-1=b³c⁵ >

Subgroups: 952 in 210 conjugacy classes, 51 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃:Dic₃, S₃xC₆, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₆², C₆², D₆:C₄, C₆.D₄, C₃xC₃:S₃, C₃²xC₆, C₃²xC₆, C₆xDic₃, C₂xC₃:Dic₃, S₃xC₂xC₆, C₂²xC₃:S₃, C₃xC₃:Dic₃, C₆xC₃:S₃, C₆xC₃:S₃, C₃xC₆², D₆:Dic₃, C₆.D₁₂, C₆xC₃:Dic₃, C₂xC₆xC₃:S₃, C₆².₈₄D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, Dic₃, D₆, C₂²:C₄, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, S₃², D₆:C₄, C₆.D₄, S₃xDic₃, C₆.D₆, D₆:S₃, C₃:D₁₂, C₃²:₄D₆, D₆:Dic₃, C₆.D₁₂, C₃³:₉(C₂xC₄), C₃³:₉D₄, C₆².₈₄D₆

Smallest permutation representation of C₆².₈₄D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	···	3H	4A	4B	4C	4D	6A	···	6I	6J	···	6X	6Y	6Z	6AA	6AB	12A	···	12H
order	1	2	2	2	2	2	3	3	3	3	···	3	4	4	4	4	6	···	6	6	···	6	6	6	6	6	12	···	12
size	1	1	1	1	18	18	2	2	2	4	···	4	18	18	18	18	2	···	2	4	···	4	18	18	18	18	18	···	18

54 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+		+	+	+	-	+		+		+	-	+	-	+
image	C₁	C₂	C₂	C₄	S₃	S₃	D₄	Dic₃	D₆	C₄xS₃	D₁₂	C₃:D₄	S₃²	S₃xDic₃	C₆.D₆	D₆:S₃	C₃:D₁₂	C₃²:₄D₆	C₃³:₉(C₂xC₄)	C₃³:₉D₄
kernel	C₆².₈₄D₆	C₆xC₃:Dic₃	C₂xC₆xC₃:S₃	C₆xC₃:S₃	C₂xC₃:Dic₃	C₂²xC₃:S₃	C₃²xC₆	C₂xC₃:S₃	C₆²	C₃xC₆	C₃xC₆	C₃xC₆	C₂xC₆	C₆	C₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	2	1	4	2	1	2	2	3	4	4	8	3	2	1	2	4	2	2	4

Matrix representation of C₆².₈₄D₆ ►in GL₆(F₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	1	0	0
0	0	12	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

12	1	0	0	0	0
12	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

0	8	0	0	0	0
8	0	0	0	0	0
0	0	12	0	0	0
0	0	1	1	0	0
0	0	0	0	11	11
0	0	0	0	2	9

0	5	0	0	0	0
5	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	2	9
0	0	0	0	11	11

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,11,2,0,0,0,0,11,9],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,11,0,0,0,0,9,11] >;

C₆².₈₄D₆ in GAP, Magma, Sage, TeX

C_6^2._{84}D_6

% in TeX

G:=Group("C6^2.84D6");

// GroupNames label

G:=SmallGroup(432,461);

// by ID

G=gap.SmallGroup(432,461);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,1124,571,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;

// generators/relations

G = C62.84D6 order 432 = 24·33

32nd non-split extension by C62 of D6 acting via D6/C3=C22

G = C₆².₈₄D₆ order 432 = 2⁴·3³

32^nd non-split extension by C₆² of D₆ acting via D₆/C₃=C₂²