C2xS3xDic6 - GroupNames

G = C₂xS₃xDic₆ order 288 = 2⁵·3²

Direct product of C₂, S₃ and Dic₆

direct product, metabelian, supersoluble, monomial

Aliases: C₂xS₃xDic₆, C₆².₁₂₆C₂³, C₆:₂(S₃xQ₈), (S₃xC₆):₁₁Q₈, C₆:₁(C₂xDic₆), (C₄xS₃).₃₉D₆, C₆.₁(S₃xC₂³), (C₃xC₆).₁C₂⁴, (C₆xDic₆):₁₇C₂, (C₂xC₁₂).₂₈₄D₆, C₃²:₂(C₂²xQ₈), C₃:₁(C₂²xDic₆), (S₃xC₆).₁₉C₂³, (C₂xDic₃).₈₆D₆, D₆.₂₄(C₂²xS₃), (C₂²xS₃).₇₈D₆, C₃²:₂Q₈:₉C₂², (S₃xC₁₂).₅₃C₂², (C₃xC₁₂).₁₁₀C₂³, (C₆xC₁₂).₁₅₆C₂², C₁₂.₁₂₉(C₂²xS₃), (C₃xDic₆):₂₄C₂², C₃:Dic₃.₁₃C₂³, (C₃xDic₃).₁C₂³, (S₃xDic₃).₅C₂², C₃²:₄Q₈:₁₉C₂², (C₆xDic₃).₄₃C₂², Dic₃.₁₈(C₂²xS₃), C₃:₂(C₂xS₃xQ₈), C₄.₅₉(C₂xS₃²), (C₂xC₄).₈₅S₃², (S₃xC₂xC₄).₆S₃, (C₃xC₆):₂(C₂xQ₈), (C₃xS₃):₁(C₂xQ₈), C₂.₄(C₂²xS₃²), (S₃xC₂xC₁₂).₁₄C₂, C₂².₅₈(C₂xS₃²), (C₂xS₃xDic₃).₄C₂, (C₂xC₃²:₂Q₈):₁₄C₂, (S₃xC₂xC₆).₁₀₂C₂², (C₂xC₃²:₄Q₈):₁₈C₂, (C₂xC₆).₁₄₅(C₂²xS₃), (C₂xC₃:Dic₃).₁₀₁C₂², SmallGroup(288,942)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₂xS₃xDic₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — S₃xDic₃ — C₂xS₃xDic₃ — C₂xS₃xDic₆

Lower central

C₃² — C₃xC₆ — C₂xS₃xDic₆

Upper central

C₁ — C₂² — C₂xC₄

Generators and relations for C₂xS₃xDic₆
G = < a,b,c,d,e | a²=b³=c²=d¹²=1, e²=d⁶, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 994 in 331 conjugacy classes, 132 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xQ₈, C₃xS₃, C₃xC₆, C₃xC₆, Dic₆, Dic₆, C₄xS₃, C₄xS₃, C₂xDic₃, C₂xDic₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²xQ₈, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, C₆², C₂xDic₆, C₂xDic₆, S₃xC₂xC₄, S₃xC₂xC₄, S₃xQ₈, C₂²xDic₃, C₂²xC₁₂, C₆xQ₈, S₃xDic₃, C₃²:₂Q₈, C₃xDic₆, S₃xC₁₂, C₆xDic₃, C₆xDic₃, C₃²:₄Q₈, C₂xC₃:Dic₃, C₆xC₁₂, S₃xC₂xC₆, C₂²xDic₆, C₂xS₃xQ₈, S₃xDic₆, C₂xS₃xDic₃, C₂xC₃²:₂Q₈, C₆xDic₆, S₃xC₂xC₁₂, C₂xC₃²:₄Q₈, C₂xS₃xDic₆
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂xQ₈, C₂⁴, Dic₆, C₂²xS₃, C₂²xQ₈, S₃², C₂xDic₆, S₃xQ₈, S₃xC₂³, C₂xS₃², C₂²xDic₆, C₂xS₃xQ₈, S₃xDic₆, C₂²xS₃², C₂xS₃xDic₆

Smallest permutation representation of C₂xS₃xDic₆
►On 96 points

Generators in S₉₆

(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 85)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 49)(46 50)(47 51)(48 52)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)

(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)

(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 73)(33 74)(34 75)(35 76)(36 77)(49 93)(50 94)(51 95)(52 96)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)

(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)

(1 51 7 57)(2 50 8 56)(3 49 9 55)(4 60 10 54)(5 59 11 53)(6 58 12 52)(13 84 19 78)(14 83 20 77)(15 82 21 76)(16 81 22 75)(17 80 23 74)(18 79 24 73)(25 38 31 44)(26 37 32 43)(27 48 33 42)(28 47 34 41)(29 46 35 40)(30 45 36 39)(61 89 67 95)(62 88 68 94)(63 87 69 93)(64 86 70 92)(65 85 71 91)(66 96 72 90)

G:=sub<Sym(96)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,49)(46,50)(47,51)(48,52)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,37)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,73)(33,74)(34,75)(35,76)(36,77)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,51,7,57)(2,50,8,56)(3,49,9,55)(4,60,10,54)(5,59,11,53)(6,58,12,52)(13,84,19,78)(14,83,20,77)(15,82,21,76)(16,81,22,75)(17,80,23,74)(18,79,24,73)(25,38,31,44)(26,37,32,43)(27,48,33,42)(28,47,34,41)(29,46,35,40)(30,45,36,39)(61,89,67,95)(62,88,68,94)(63,87,69,93)(64,86,70,92)(65,85,71,91)(66,96,72,90)>;

G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,49)(46,50)(47,51)(48,52)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,37)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,73)(33,74)(34,75)(35,76)(36,77)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,51,7,57)(2,50,8,56)(3,49,9,55)(4,60,10,54)(5,59,11,53)(6,58,12,52)(13,84,19,78)(14,83,20,77)(15,82,21,76)(16,81,22,75)(17,80,23,74)(18,79,24,73)(25,38,31,44)(26,37,32,43)(27,48,33,42)(28,47,34,41)(29,46,35,40)(30,45,36,39)(61,89,67,95)(62,88,68,94)(63,87,69,93)(64,86,70,92)(65,85,71,91)(66,96,72,90) );

G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,25),(11,26),(12,27),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,85),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,49),(46,50),(47,51),(48,52),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(71,73),(72,74)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,37),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,73),(33,74),(34,75),(35,76),(36,77),(49,93),(50,94),(51,95),(52,96),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,51,7,57),(2,50,8,56),(3,49,9,55),(4,60,10,54),(5,59,11,53),(6,58,12,52),(13,84,19,78),(14,83,20,77),(15,82,21,76),(16,81,22,75),(17,80,23,74),(18,79,24,73),(25,38,31,44),(26,37,32,43),(27,48,33,42),(28,47,34,41),(29,46,35,40),(30,45,36,39),(61,89,67,95),(62,88,68,94),(63,87,69,93),(64,86,70,92),(65,85,71,91),(66,96,72,90)]])

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	4A	4B	4C	···	4H	4I	4J	4K	4L	6A	···	6F	6G	6H	6I	6J	6K	6L	6M	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N	12O	12P	12Q	12R
order	1	2	2	2	2	2	2	2	3	3	3	4	4	4	···	4	4	4	4	4	6	···	6	6	6	6	6	6	6	6	12	12	12	12	12	···	12	12	12	12	12	12	12	12	12
size	1	1	1	1	3	3	3	3	2	2	4	2	2	6	···	6	18	18	18	18	2	···	2	4	4	4	6	6	6	6	2	2	2	2	4	···	4	6	6	6	6	12	12	12	12

54 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	-	+	+	+	+	+	-	+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	Q₈	D₆	D₆	D₆	D₆	D₆	Dic₆	S₃²	S₃xQ₈	C₂xS₃²	C₂xS₃²	S₃xDic₆
kernel	C₂xS₃xDic₆	S₃xDic₆	C₂xS₃xDic₃	C₂xC₃²:₂Q₈	C₆xDic₆	S₃xC₂xC₁₂	C₂xC₃²:₄Q₈	C₂xDic₆	S₃xC₂xC₄	S₃xC₆	Dic₆	C₄xS₃	C₂xDic₃	C₂xC₁₂	C₂²xS₃	D₆	C₂xC₄	C₆	C₄	C₂²	C₂
# reps	1	8	2	2	1	1	1	1	1	4	4	4	3	2	1	8	1	2	2	1	4

Matrix representation of C₂xS₃xDic₆ ►in GL₆(F₁₃)

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

1	0	0	0	0	0
0	1	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

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

12	11	0	0	0	0
1	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	12
0	0	0	0	1	0

6	5	0	0	0	0
3	7	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	1	0
0	0	0	0	12	12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[6,3,0,0,0,0,5,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C₂xS₃xDic₆ in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_6

% in TeX

G:=Group("C2xS3xDic6");

// GroupNames label

G:=SmallGroup(288,942);

// by ID

G=gap.SmallGroup(288,942);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,346,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = C2xS3xDic6 order 288 = 25·32

Direct product of C2, S3 and Dic6

G = C₂xS₃xDic₆ order 288 = 2⁵·3²

Direct product of C₂, S₃ and Dic₆