C2xC6xDic6 - GroupNames

G = C₂xC₆xDic₆ order 288 = 2⁵·3²

Direct product of C₂xC₆ and Dic₆

direct product, metabelian, supersoluble, monomial

Aliases: C₂xC₆xDic₆, C₆²:₁₁Q₈, C₆².₂₆₅C₂³, C₆:₁(C₆xQ₈), C₆.₁(C₂³xC₆), (C₂xC₁₂).₄₄₉D₆, C₃²:₆(C₂²xQ₈), C₂³.₄₃(S₃xC₆), C₆.₆₉(S₃xC₂³), (C₃xC₆).₃₈C₂⁴, (C₂²xC₁₂).₄₃S₃, C₁₂.₃₄(C₂²xC₆), (C₂²xC₁₂).₂₁C₆, (C₂²xC₆).₁₇₄D₆, (C₆xC₁₂).₃₂₇C₂², (C₃xC₁₂).₁₆₆C₂³, C₁₂.₂₂₁(C₂²xS₃), (C₂²xDic₃).₇C₆, Dic₃.₁(C₂²xC₆), (C₂xC₆²).₁₁₅C₂², (C₃xDic₃).₂₈C₂³, (C₆xDic₃).₁₆₂C₂², C₃:₁(Q₈xC₂xC₆), C₄.₃₂(S₃xC₂xC₆), (C₃xC₆):₅(C₂xQ₈), (C₂xC₆):₆(C₃xQ₈), (C₂xC₆xC₁₂).₁₇C₂, C₂.₃(S₃xC₂²xC₆), (C₂xC₄).₈₆(S₃xC₆), C₂².₂₈(S₃xC₂xC₆), (Dic₃xC₂xC₆).₁₃C₂, (C₂xC₁₂).₁₁₁(C₂xC₆), (C₂xC₆).₆₇(C₂²xC₆), (C₂²xC₆).₆₈(C₂xC₆), (C₂²xC₄).₁₂(C₃xS₃), (C₂xC₆).₃₄₅(C₂²xS₃), (C₂xDic₃).₄₃(C₂xC₆), SmallGroup(288,988)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂xC₆xDic₆

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — C₃xDic₃ — C₆xDic₃ — Dic₃xC₂xC₆ — C₂xC₆xDic₆

Lower central

C₃ — C₆ — C₂xC₆xDic₆

Upper central

C₁ — C₂²xC₆ — C₂²xC₁₂

Generators and relations for C₂xC₆xDic₆
G = < a,b,c,d | a²=b⁶=c¹²=1, d²=c⁶, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 570 in 339 conjugacy classes, 210 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₆, C₂xC₄, C₂xC₄, Q₈, C₂³, C₃², Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₃xC₆, C₃xC₆, Dic₆, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xC₆, C₂²xC₆, C₂²xQ₈, C₃xDic₃, C₃xC₁₂, C₆², C₂xDic₆, C₂²xDic₃, C₂²xC₁₂, C₂²xC₁₂, C₆xQ₈, C₃xDic₆, C₆xDic₃, C₆xC₁₂, C₂xC₆², C₂²xDic₆, Q₈xC₂xC₆, C₆xDic₆, Dic₃xC₂xC₆, C₂xC₆xC₁₂, C₂xC₆xDic₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, C₂³, D₆, C₂xC₆, C₂xQ₈, C₂⁴, C₃xS₃, Dic₆, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²xQ₈, S₃xC₆, C₂xDic₆, C₆xQ₈, S₃xC₂³, C₂³xC₆, C₃xDic₆, S₃xC₂xC₆, C₂²xDic₆, Q₈xC₂xC₆, C₆xDic₆, S₃xC₂²xC₆, C₂xC₆xDic₆

Smallest permutation representation of C₂xC₆xDic₆
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

108 conjugacy classes

class	1	2A	···	2G	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	···	4L	6A	···	6N	6O	···	6AI	12A	···	12AF	12AG	···	12AV
order	1	2	···	2	3	3	3	3	3	4	4	4	4	4	···	4	6	···	6	6	···	6	12	···	12	12	···	12
size	1	1	···	1	1	1	2	2	2	2	2	2	2	6	···	6	1	···	1	2	···	2	2	···	2	6	···	6

108 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+					+	-	+	+		-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	Q₈	D₆	D₆	C₃xS₃	Dic₆	C₃xQ₈	S₃xC₆	S₃xC₆	C₃xDic₆
kernel	C₂xC₆xDic₆	C₆xDic₆	Dic₃xC₂xC₆	C₂xC₆xC₁₂	C₂²xDic₆	C₂xDic₆	C₂²xDic₃	C₂²xC₁₂	C₂²xC₁₂	C₆²	C₂xC₁₂	C₂²xC₆	C₂²xC₄	C₂xC₆	C₂xC₆	C₂xC₄	C₂³	C₂²
# reps	1	12	2	1	2	24	4	2	1	4	6	1	2	8	8	12	2	16

Matrix representation of C₂xC₆xDic₆ ►in GL₅(F₁₃)

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

10	0	0	0	0
0	3	0	0	0
0	0	3	0	0
0	0	0	12	0
0	0	0	0	12

1	0	0	0	0
0	10	0	0	0
0	0	4	0	0
0	0	0	12	11
0	0	0	1	1

12	0	0	0	0
0	0	12	0	0
0	12	0	0	0
0	0	0	5	0
0	0	0	8	8

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

C₂xC₆xDic₆ in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_6

% in TeX

G:=Group("C2xC6xDic6");

// GroupNames label

G:=SmallGroup(288,988);

// by ID

G=gap.SmallGroup(288,988);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,1571,192,9414]);

// Polycyclic

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

// generators/relations

G = C2xC6xDic6 order 288 = 25·32

Direct product of C2xC6 and Dic6

G = C₂xC₆xDic₆ order 288 = 2⁵·3²

Direct product of C₂xC₆ and Dic₆