C3xC12.D6 - GroupNames

G = C₃xC₁₂.D₆ order 432 = 2⁴·3³

Direct product of C₃ and C₁₂.D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₁₂.D₆, C₆².₉₉D₆, C₁₂.₃₉(S₃xC₆), (D₄xC₃³):₇C₂, C₃²:₇D₄:₈C₆, (C₃xC₁₂).₁₄₉D₆, (D₄xC₃²):₁₂C₆, (D₄xC₃²):₁₀S₃, C₃³:₃₁(C₄oD₄), C₆².₃₀(C₂xC₆), C₃²:₄Q₈:₁₂C₆, (C₃xC₆²).₃₈C₂², (C₃²xC₆).₉₀C₂³, C₃²:₂₆(D₄:₂S₃), (C₃²xC₁₂).₅₄C₂², (C₄xC₃:S₃):₉C₆, D₄:₂(C₃xC₃:S₃), C₄.₅(C₆xC₃:S₃), C₆.₅₇(S₃xC₂xC₆), (C₃xD₄):₃(C₃xS₃), (C₁₂xC₃:S₃):₁₁C₂, (C₃xD₄):₅(C₃:S₃), C₁₂.₅₆(C₂xC₃:S₃), C₃:₅(C₃xD₄:₂S₃), (C₂xC₆).₁₆(S₃xC₆), C₂².₁(C₆xC₃:S₃), (C₃xC₁₂).₅₈(C₂xC₆), (C₆xC₃:Dic₃):₁₇C₂, (C₂xC₃:Dic₃):₁₃C₆, C₃²:₁₄(C₃xC₄oD₄), C₆.₅₇(C₂²xC₃:S₃), (C₆xC₃:S₃).₆₂C₂², (C₃xC₃²:₇D₄):₁₀C₂, C₃:Dic₃.₂₃(C₂xC₆), (C₃xC₆).₆₄(C₂²xC₆), (C₃xC₃²:₄Q₈):₁₄C₂, (C₃xC₆).₁₇₉(C₂²xS₃), (C₃xC₃:Dic₃).₆₀C₂², C₂.₇(C₂xC₆xC₃:S₃), (C₂xC₆).₁₁(C₂xC₃:S₃), (C₂xC₃:S₃).₂₃(C₂xC₆), SmallGroup(432,715)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xC₁₂.D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — C₆xC₃:S₃ — C₁₂xC₃:S₃ — C₃xC₁₂.D₆

Lower central

C₃² — C₃xC₆ — C₃xC₁₂.D₆

Upper central

C₁ — C₆ — C₃xD₄

Generators and relations for C₃xC₁₂.D₆
G = < a,b,c,d | a³=b¹²=c⁶=1, d²=b⁶, ab=ba, ac=ca, ad=da, cbc^-1=b⁷, dbd^-1=b^-1, dcd^-1=b⁶c^-1 >

Subgroups: 852 in 304 conjugacy classes, 94 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, D₄, D₄, Q₈, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xD₄, C₃xQ₈, C₃³, C₃xDic₃, C₃:Dic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₆², C₆², D₄:₂S₃, C₃xC₄oD₄, C₃xC₃:S₃, C₃²xC₆, C₃²xC₆, C₃xDic₆, S₃xC₁₂, C₆xDic₃, C₃xC₃:D₄, C₃²:₄Q₈, C₄xC₃:S₃, C₂xC₃:Dic₃, C₃²:₇D₄, D₄xC₃², D₄xC₃², D₄xC₃², C₃xC₃:Dic₃, C₃xC₃:Dic₃, C₃²xC₁₂, C₆xC₃:S₃, C₃xC₆², C₃xD₄:₂S₃, C₁₂.D₆, C₃xC₃²:₄Q₈, C₁₂xC₃:S₃, C₆xC₃:Dic₃, C₃xC₃²:₇D₄, D₄xC₃³, C₃xC₁₂.D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃:S₃, C₂²xS₃, C₂²xC₆, S₃xC₆, C₂xC₃:S₃, D₄:₂S₃, C₃xC₄oD₄, C₃xC₃:S₃, S₃xC₂xC₆, C₂²xC₃:S₃, C₆xC₃:S₃, C₃xD₄:₂S₃, C₁₂.D₆, C₂xC₆xC₃:S₃, C₃xC₁₂.D₆

Smallest permutation representation of C₃xC₁₂.D₆
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

90 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	···	3N	4A	4B	4C	4D	4E	6A	6B	6C	···	6R	6S	···	6AP	6AQ	6AR	12A	12B	12C	···	12N	12O	12P	12Q	12R	12S	12T	12U	12V
order	1	2	2	2	2	3	3	3	···	3	4	4	4	4	4	6	6	6	···	6	6	···	6	6	6	12	12	12	···	12	12	12	12	12	12	12	12	12
size	1	1	2	2	18	1	1	2	···	2	2	9	9	18	18	1	1	2	···	2	4	···	4	18	18	2	2	4	···	4	9	9	9	9	18	18	18	18

90 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+						-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₆	D₆	C₄oD₄	C₃xS₃	S₃xC₆	S₃xC₆	C₃xC₄oD₄	D₄:₂S₃	C₃xD₄:₂S₃
kernel	C₃xC₁₂.D₆	C₃xC₃²:₄Q₈	C₁₂xC₃:S₃	C₆xC₃:Dic₃	C₃xC₃²:₇D₄	D₄xC₃³	C₁₂.D₆	C₃²:₄Q₈	C₄xC₃:S₃	C₂xC₃:Dic₃	C₃²:₇D₄	D₄xC₃²	D₄xC₃²	C₃xC₁₂	C₆²	C₃³	C₃xD₄	C₁₂	C₂xC₆	C₃²	C₃²	C₃
# reps	1	1	1	2	2	1	2	2	2	4	4	2	4	4	8	2	8	8	16	4	4	8

Matrix representation of C₃xC₁₂.D₆ ►in GL₆(F₁₃)

3	0	0	0	0	0
0	3	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

10	0	0	0	0	0
1	4	0	0	0	0
0	0	9	1	0	0
0	0	0	3	0	0
0	0	0	0	12	5
0	0	0	0	10	1

3	0	0	0	0	0
12	9	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	3	12

6	3	0	0	0	0
10	7	0	0	0	0
0	0	10	1	0	0
0	0	5	3	0	0
0	0	0	0	5	1
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,1,0,0,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,1,3,0,0,0,0,0,0,12,10,0,0,0,0,5,1],[3,12,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12],[6,10,0,0,0,0,3,7,0,0,0,0,0,0,10,5,0,0,0,0,1,3,0,0,0,0,0,0,5,0,0,0,0,0,1,8] >;

C₃xC₁₂.D₆ in GAP, Magma, Sage, TeX

C_3\times C_{12}.D_6

% in TeX

G:=Group("C3xC12.D6");

// GroupNames label

G:=SmallGroup(432,715);

// by ID

G=gap.SmallGroup(432,715);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,4037,14118]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;

// generators/relations

G = C3xC12.D6 order 432 = 24·33

Direct product of C3 and C12.D6

G = C₃xC₁₂.D₆ order 432 = 2⁴·3³

Direct product of C₃ and C₁₂.D₆