C6xC6.D6 - GroupNames

G = C₆xC₆.D₆ order 432 = 2⁴·3³

Direct product of C₆ and C₆.D₆

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₆xC₆.D₆, C₆².₁₀₉D₆, C₆:₁(S₃xC₁₂), (C₆xDic₃):₈C₆, Dic₃:₆(S₃xC₆), (C₃xDic₃):₂₀D₆, (C₆xDic₃):₁₁S₃, C₆².₂₅(C₂xC₆), C₃³:₁₁(C₂²xC₄), C₃²:₆(C₂²xC₁₂), (C₃²xC₆).₃₂C₂³, (C₃xC₆²).₁₉C₂², (C₃²xDic₃):₁₈C₂², C₂.₃(S₃²xC₆), C₃:₂(S₃xC₂xC₁₂), (C₆xC₃:S₃):₇C₄, (C₃xC₆):₈(C₄xS₃), (C₂xC₆).₇₂S₃², C₆.₁₃(S₃xC₂xC₆), (C₂xC₃:S₃):₆C₁₂, C₃:S₃:₃(C₂xC₁₂), C₆.₁₁₆(C₂xS₃²), (C₃xC₆):₅(C₂xC₁₂), C₃²:₁₆(S₃xC₂xC₄), C₂².₉(C₃xS₃²), (C₂xC₆).₂₇(S₃xC₆), (C₃²xC₆):₅(C₂xC₄), (Dic₃xC₃xC₆):₁₂C₂, (C₃xDic₃):₇(C₂xC₆), (C₂xDic₃):₅(C₃xS₃), (C₂²xC₃:S₃).₇C₆, (C₆xC₃:S₃).₄₉C₂², (C₃xC₆).₂₃(C₂²xC₆), (C₃xC₆).₁₃₇(C₂²xS₃), (C₂xC₆xC₃:S₃).₇C₂, (C₃xC₃:S₃):₈(C₂xC₄), (C₂xC₃:S₃).₂₁(C₂xC₆), SmallGroup(432,654)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₆xC₆.D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — C₃²xDic₃ — C₃xC₆.D₆ — C₆xC₆.D₆

Lower central

C₃² — C₆xC₆.D₆

Upper central

C₁ — C₂xC₆

Generators and relations for C₆xC₆.D₆
G = < a,b,c,d | a⁶=b⁶=d²=1, c⁶=b³, ab=ba, ac=ca, ad=da, cbc^-1=dbd=b^-1, dcd=c⁵ >

Subgroups: 928 in 290 conjugacy classes, 96 normal (16 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, C₂²xC₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₆², C₆², C₆², S₃xC₂xC₄, C₂²xC₁₂, C₃xC₃:S₃, C₃²xC₆, C₃²xC₆, C₆.D₆, S₃xC₁₂, C₆xDic₃, C₆xDic₃, C₆xC₁₂, S₃xC₂xC₆, C₂²xC₃:S₃, C₃²xDic₃, C₆xC₃:S₃, C₃xC₆², C₂xC₆.D₆, S₃xC₂xC₁₂, C₃xC₆.D₆, Dic₃xC₃xC₆, C₂xC₆xC₃:S₃, C₆xC₆.D₆
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, C₂³, C₁₂, D₆, C₂xC₆, C₂²xC₄, C₃xS₃, C₄xS₃, C₂xC₁₂, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, S₃xC₂xC₄, C₂²xC₁₂, C₆.D₆, S₃xC₁₂, C₂xS₃², S₃xC₂xC₆, C₃xS₃², C₂xC₆.D₆, S₃xC₂xC₁₂, C₃xC₆.D₆, S₃²xC₆, C₆xC₆.D₆

Smallest permutation representation of C₆xC₆.D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

108 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3H	3I	3J	3K	4A	···	4H	6A	···	6F	6G	···	6X	6Y	···	6AG	6AH	···	6AO	12A	···	12P	12Q	···	12AN
order	1	2	2	2	2	2	2	2	3	3	3	···	3	3	3	3	4	···	4	6	···	6	6	···	6	6	···	6	6	···	6	12	···	12	12	···	12
size	1	1	1	1	9	9	9	9	1	1	2	···	2	4	4	4	3	···	3	1	···	1	2	···	2	4	···	4	9	···	9	3	···	3	6	···	6

108 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+							+	+	+						+	+	+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₁₂	S₃	D₆	D₆	C₃xS₃	C₄xS₃	S₃xC₆	S₃xC₆	S₃xC₁₂	S₃²	C₆.D₆	C₂xS₃²	C₃xS₃²	C₃xC₆.D₆	S₃²xC₆
kernel	C₆xC₆.D₆	C₃xC₆.D₆	Dic₃xC₃xC₆	C₂xC₆xC₃:S₃	C₂xC₆.D₆	C₆xC₃:S₃	C₆.D₆	C₆xDic₃	C₂²xC₃:S₃	C₂xC₃:S₃	C₆xDic₃	C₃xDic₃	C₆²	C₂xDic₃	C₃xC₆	Dic₃	C₂xC₆	C₆	C₂xC₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	4	2	1	2	8	8	4	2	16	2	4	2	4	8	8	4	16	1	2	1	2	4	2

Matrix representation of C₆xC₆.D₆ ►in GL₆(F₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	3	0	0	0
0	0	0	3	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	12	1	0	0
0	0	12	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	5
0	0	0	0	8	5

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

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;

C₆xC₆.D₆ in GAP, Magma, Sage, TeX

C_6\times C_6.D_6

% in TeX

G:=Group("C6xC6.D6");

// GroupNames label

G:=SmallGroup(432,654);

// by ID

G=gap.SmallGroup(432,654);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,176,2028,14118]);

// Polycyclic

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

// generators/relations

G = C6xC6.D6 order 432 = 24·33

Direct product of C6 and C6.D6

G = C₆xC₆.D₆ order 432 = 2⁴·3³

Direct product of C₆ and C₆.D₆