C6xC3:Dic3 - GroupNames

G = C₆xC₃:Dic₃ order 216 = 2³·3³

Direct product of C₆ and C₃:Dic₃

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

Aliases: C₆xC₃:Dic₃, C₆².₁₄C₆, C₆².₁₅S₃, C₆:(C₃xDic₃), (C₃xC₆):₅C₁₂, C₆.₂₇(S₃xC₆), (C₃²xC₆):₄C₄, C₃³:₁₄(C₂xC₄), (C₃xC₆):₄Dic₃, C₃:₂(C₆xDic₃), (C₃xC₆).₆₀D₆, (C₃xC₆²).₂C₂, C₃²:₁₀(C₂xC₁₂), C₃²:₉(C₂xDic₃), (C₃²xC₆).₂₄C₂², C₂.₂(C₆xC₃:S₃), C₂².(C₃xC₃:S₃), C₆.₂₅(C₂xC₃:S₃), (C₃xC₆).₃₂(C₂xC₆), (C₂xC₆).₁₉(C₃xS₃), (C₂xC₆).₁₁(C₃:S₃), SmallGroup(216,143)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₆xC₃:Dic₃

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — C₃xC₃:Dic₃ — C₆xC₃:Dic₃

Lower central

C₃² — C₆xC₃:Dic₃

Upper central

C₁ — C₂xC₆

Generators and relations for C₆xC₃:Dic₃
G = < a,b,c,d | a⁶=b³=c⁶=1, d²=c³, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c^-1 >

Subgroups: 272 in 136 conjugacy classes, 66 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₂xC₄, C₃², C₃², C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₃³, C₃xDic₃, C₃:Dic₃, C₆², C₆², C₆², C₃²xC₆, C₃²xC₆, C₆xDic₃, C₂xC₃:Dic₃, C₃xC₃:Dic₃, C₃xC₆², C₆xC₃:Dic₃
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, Dic₃, C₁₂, D₆, C₂xC₆, C₃xS₃, C₃:S₃, C₂xDic₃, C₂xC₁₂, C₃xDic₃, C₃:Dic₃, S₃xC₆, C₂xC₃:S₃, C₃xC₃:S₃, C₆xDic₃, C₂xC₃:Dic₃, C₃xC₃:Dic₃, C₆xC₃:S₃, C₆xC₃:Dic₃

Smallest permutation representation of C₆xC₃:Dic₃
►On 72 points

Generators in S₇₂

(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)

(1 57 44)(2 58 45)(3 59 46)(4 60 47)(5 55 48)(6 56 43)(7 36 63)(8 31 64)(9 32 65)(10 33 66)(11 34 61)(12 35 62)(13 50 23)(14 51 24)(15 52 19)(16 53 20)(17 54 21)(18 49 22)(25 37 70)(26 38 71)(27 39 72)(28 40 67)(29 41 68)(30 42 69)

(1 51 59 20 48 18)(2 52 60 21 43 13)(3 53 55 22 44 14)(4 54 56 23 45 15)(5 49 57 24 46 16)(6 50 58 19 47 17)(7 26 34 42 65 67)(8 27 35 37 66 68)(9 28 36 38 61 69)(10 29 31 39 62 70)(11 30 32 40 63 71)(12 25 33 41 64 72)

(1 32 20 71)(2 33 21 72)(3 34 22 67)(4 35 23 68)(5 36 24 69)(6 31 19 70)(7 14 42 55)(8 15 37 56)(9 16 38 57)(10 17 39 58)(11 18 40 59)(12 13 41 60)(25 43 64 52)(26 44 65 53)(27 45 66 54)(28 46 61 49)(29 47 62 50)(30 48 63 51)

G:=sub<Sym(72)| (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,57,44)(2,58,45)(3,59,46)(4,60,47)(5,55,48)(6,56,43)(7,36,63)(8,31,64)(9,32,65)(10,33,66)(11,34,61)(12,35,62)(13,50,23)(14,51,24)(15,52,19)(16,53,20)(17,54,21)(18,49,22)(25,37,70)(26,38,71)(27,39,72)(28,40,67)(29,41,68)(30,42,69), (1,51,59,20,48,18)(2,52,60,21,43,13)(3,53,55,22,44,14)(4,54,56,23,45,15)(5,49,57,24,46,16)(6,50,58,19,47,17)(7,26,34,42,65,67)(8,27,35,37,66,68)(9,28,36,38,61,69)(10,29,31,39,62,70)(11,30,32,40,63,71)(12,25,33,41,64,72), (1,32,20,71)(2,33,21,72)(3,34,22,67)(4,35,23,68)(5,36,24,69)(6,31,19,70)(7,14,42,55)(8,15,37,56)(9,16,38,57)(10,17,39,58)(11,18,40,59)(12,13,41,60)(25,43,64,52)(26,44,65,53)(27,45,66,54)(28,46,61,49)(29,47,62,50)(30,48,63,51)>;

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)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,57,44)(2,58,45)(3,59,46)(4,60,47)(5,55,48)(6,56,43)(7,36,63)(8,31,64)(9,32,65)(10,33,66)(11,34,61)(12,35,62)(13,50,23)(14,51,24)(15,52,19)(16,53,20)(17,54,21)(18,49,22)(25,37,70)(26,38,71)(27,39,72)(28,40,67)(29,41,68)(30,42,69), (1,51,59,20,48,18)(2,52,60,21,43,13)(3,53,55,22,44,14)(4,54,56,23,45,15)(5,49,57,24,46,16)(6,50,58,19,47,17)(7,26,34,42,65,67)(8,27,35,37,66,68)(9,28,36,38,61,69)(10,29,31,39,62,70)(11,30,32,40,63,71)(12,25,33,41,64,72), (1,32,20,71)(2,33,21,72)(3,34,22,67)(4,35,23,68)(5,36,24,69)(6,31,19,70)(7,14,42,55)(8,15,37,56)(9,16,38,57)(10,17,39,58)(11,18,40,59)(12,13,41,60)(25,43,64,52)(26,44,65,53)(27,45,66,54)(28,46,61,49)(29,47,62,50)(30,48,63,51) );

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),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,57,44),(2,58,45),(3,59,46),(4,60,47),(5,55,48),(6,56,43),(7,36,63),(8,31,64),(9,32,65),(10,33,66),(11,34,61),(12,35,62),(13,50,23),(14,51,24),(15,52,19),(16,53,20),(17,54,21),(18,49,22),(25,37,70),(26,38,71),(27,39,72),(28,40,67),(29,41,68),(30,42,69)], [(1,51,59,20,48,18),(2,52,60,21,43,13),(3,53,55,22,44,14),(4,54,56,23,45,15),(5,49,57,24,46,16),(6,50,58,19,47,17),(7,26,34,42,65,67),(8,27,35,37,66,68),(9,28,36,38,61,69),(10,29,31,39,62,70),(11,30,32,40,63,71),(12,25,33,41,64,72)], [(1,32,20,71),(2,33,21,72),(3,34,22,67),(4,35,23,68),(5,36,24,69),(6,31,19,70),(7,14,42,55),(8,15,37,56),(9,16,38,57),(10,17,39,58),(11,18,40,59),(12,13,41,60),(25,43,64,52),(26,44,65,53),(27,45,66,54),(28,46,61,49),(29,47,62,50),(30,48,63,51)]])

C₆xC₃:Dic₃ is a maximal subgroup of
C₃xDic₃² C₆².₇₇D₆ C₆².₇₉D₆ C₆².₈₀D₆ C₆².₈₁D₆ C₆².₈₂D₆ C₃³:₆C₄² C₆².₈₄D₆ C₆².₈₅D₆ C₃³:₁₂M₄(2) S₃xC₆xDic₃ C₆².₉₀D₆ C₆².₉₆D₆ C₃:S₃xC₂xC₁₂

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3N	4A	4B	4C	4D	6A	···	6F	6G	···	6AP	12A	···	12H
order	1	2	2	2	3	3	3	···	3	4	4	4	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	1	1	2	···	2	9	9	9	9	1	···	1	2	···	2	9	···	9

72 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+						+	-	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	S₃	Dic₃	D₆	C₃xS₃	C₃xDic₃	S₃xC₆
kernel	C₆xC₃:Dic₃	C₃xC₃:Dic₃	C₃xC₆²	C₂xC₃:Dic₃	C₃²xC₆	C₃:Dic₃	C₆²	C₃xC₆	C₆²	C₃xC₆	C₃xC₆	C₂xC₆	C₆	C₆
# reps	1	2	1	2	4	4	2	8	4	8	4	8	16	8

Matrix representation of C₆xC₃:Dic₃ ►in GL₅(F₁₃)

9	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	10	0
0	0	0	0	10

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	3	0
0	0	0	0	9

12	0	0	0	0
0	4	11	0	0
0	0	10	0	0
0	0	0	12	0
0	0	0	0	12

5	0	0	0	0
0	5	0	0	0
0	11	8	0	0
0	0	0	0	1
0	0	0	12	0

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[12,0,0,0,0,0,4,0,0,0,0,11,10,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,0,5,11,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,1,0] >;

C₆xC₃:Dic₃ in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes {\rm Dic}_3

% in TeX

G:=Group("C6xC3:Dic3");

// GroupNames label

G:=SmallGroup(216,143);

// by ID

G=gap.SmallGroup(216,143);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,1444,5189]);

// Polycyclic

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

// generators/relations

G = C6xC3:Dic3 order 216 = 23·33

Direct product of C6 and C3:Dic3

G = C₆xC₃:Dic₃ order 216 = 2³·3³

Direct product of C₆ and C₃:Dic₃