C3xC12.59D6 - GroupNames

G = C₃xC₁₂.₅₉D₆ order 432 = 2⁴·3³

Direct product of C₃ and C₁₂.₅₉D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₁₂.₅₉D₆, C₆².₁₅₅D₆, (C₆xC₁₂):₁₃C₆, (C₆xC₁₂):₁₄S₃, C₁₂:S₃:₁₄C₆, C₃²:₇D₄:₉C₆, C₁₂.₁₀₅(S₃xC₆), (C₃xC₁₂).₂₁₁D₆, C₃³:₂₈(C₄oD₄), C₆².₇₈(C₂xC₆), C₃²:₄Q₈:₁₄C₆, C₃²:₂₆(C₄oD₁₂), (C₃xC₆²).₆₄C₂², (C₃²xC₆).₈₈C₂³, (C₃²xC₁₂).₁₀₀C₂², (C₃xC₆xC₁₂):₈C₂, C₆.₅₅(S₃xC₂xC₆), (C₄xC₃:S₃):₁₃C₆, (C₂xC₁₂):₄(C₃xS₃), C₄.₁₆(C₆xC₃:S₃), C₃:₅(C₃xC₄oD₁₂), (C₁₂xC₃:S₃):₁₇C₂, (C₂xC₁₂):₇(C₃:S₃), C₁₂.₉₈(C₂xC₃:S₃), (C₂xC₆).₇₈(S₃xC₆), C₂².₂(C₆xC₃:S₃), (C₃xC₁₂).₇₆(C₂xC₆), (C₃xC₁₂:S₃):₁₇C₂, C₃²:₁₂(C₃xC₄oD₄), C₆.₅₅(C₂²xC₃:S₃), (C₆xC₃:S₃).₆₁C₂², (C₃xC₃²:₇D₄):₁₁C₂, C₃:Dic₃.₂₂(C₂xC₆), (C₃xC₆).₆₂(C₂²xC₆), (C₃xC₃²:₄Q₈):₁₇C₂, (C₃xC₆).₁₇₇(C₂²xS₃), (C₃xC₃:Dic₃).₅₉C₂², C₂.₅(C₂xC₆xC₃:S₃), (C₂xC₄):₃(C₃xC₃:S₃), (C₂xC₆).₂₈(C₂xC₃:S₃), (C₂xC₃:S₃).₂₂(C₂xC₆), SmallGroup(432,713)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xC₁₂.₅₉D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — C₆xC₃:S₃ — C₁₂xC₃:S₃ — C₃xC₁₂.₅₉D₆

Lower central

C₃² — C₃xC₆ — C₃xC₁₂.₅₉D₆

Upper central

C₁ — C₁₂ — C₂xC₁₂

Generators and relations for C₃xC₁₂.₅₉D₆
G = < a,b,c,d | a³=b¹²=c⁶=1, d²=b⁶, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b⁵, dcd^-1=b⁶c^-1 >

Subgroups: 884 in 304 conjugacy classes, 94 normal (30 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃³, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₆², C₆², C₆², C₄oD₁₂, C₃xC₄oD₄, C₃xC₃:S₃, C₃²xC₆, C₃²xC₆, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₃²:₄Q₈, C₄xC₃:S₃, C₁₂:S₃, C₃²:₇D₄, C₆xC₁₂, C₆xC₁₂, C₆xC₁₂, C₃xC₃:Dic₃, C₃²xC₁₂, C₆xC₃:S₃, C₃xC₆², C₃xC₄oD₁₂, C₁₂.₅₉D₆, C₃xC₃²:₄Q₈, C₁₂xC₃:S₃, C₃xC₁₂:S₃, C₃xC₃²:₇D₄, C₃xC₆xC₁₂, C₃xC₁₂.₅₉D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃:S₃, C₂²xS₃, C₂²xC₆, S₃xC₆, C₂xC₃:S₃, C₄oD₁₂, C₃xC₄oD₄, C₃xC₃:S₃, S₃xC₂xC₆, C₂²xC₃:S₃, C₆xC₃:S₃, C₃xC₄oD₁₂, C₁₂.₅₉D₆, C₂xC₆xC₃:S₃, C₃xC₁₂.₅₉D₆

Smallest permutation representation of C₃xC₁₂.₅₉D₆
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

126 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	···	3N	4A	4B	4C	4D	4E	6A	6B	6C	···	6AN	6AO	6AP	6AQ	6AR	12A	12B	12C	12D	12E	···	12BB	12BC	12BD	12BE	12BF
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
size	1	1	2	18	18	1	1	2	···	2	1	1	2	18	18	1	1	2	···	2	18	18	18	18	1	1	1	1	2	···	2	18	18	18	18

126 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+							+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₆	D₆	C₄oD₄	C₃xS₃	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂
kernel	C₃xC₁₂.₅₉D₆	C₃xC₃²:₄Q₈	C₁₂xC₃:S₃	C₃xC₁₂:S₃	C₃xC₃²:₇D₄	C₃xC₆xC₁₂	C₁₂.₅₉D₆	C₃²:₄Q₈	C₄xC₃:S₃	C₁₂:S₃	C₃²:₇D₄	C₆xC₁₂	C₆xC₁₂	C₃xC₁₂	C₆²	C₃³	C₂xC₁₂	C₁₂	C₂xC₆	C₃²	C₃²	C₃
# reps	1	1	2	1	2	1	2	2	4	2	4	2	4	8	4	2	8	16	8	16	4	32

Matrix representation of C₃xC₁₂.₅₉D₆ ►in GL₆(F₁₃)

9	0	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	0
0	0	0	0	0	1

3	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	3	0	0
0	0	0	0	5	0
0	0	0	0	0	5

9	0	0	0	0	0
0	3	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	0	12

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

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

C₃xC₁₂.₅₉D₆ in GAP, Magma, Sage, TeX

C_3\times C_{12}._{59}D_6

% in TeX

G:=Group("C3xC12.59D6");

// GroupNames label

G:=SmallGroup(432,713);

// by ID

G=gap.SmallGroup(432,713);

# by ID

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

// 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,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;

// generators/relations

G = C3xC12.59D6 order 432 = 24·33

Direct product of C3 and C12.59D6

G = C₃xC₁₂.₅₉D₆ order 432 = 2⁴·3³

Direct product of C₃ and C₁₂.₅₉D₆