Dic3:6Dic6 - GroupNames

G = Dic₃:₆Dic₆ order 288 = 2⁵·3²

2^nd semidirect product of Dic₃ and Dic₆ acting through Inn(Dic₃)

Aliases: Dic₃:₆Dic₆, C₆².₁₄C₂³, C₃²:₆(C₄xQ₈), C₃:₃(C₄xDic₆), Dic₃².₉C₂, C₆.₁₇(S₃xQ₈), C₁₂.₃₅(C₄xS₃), (C₃xDic₃):₇Q₈, C₆.₅(C₂xDic₆), C₂.₃(S₃xDic₆), (C₂xC₁₂).₁₂₈D₆, C₄:Dic₃.₁₄S₃, (C₄xDic₃).₂S₃, C₃²:₄Q₈:₁₀C₄, C₆.₃₇(C₄oD₁₂), (C₆xC₁₂).₈₈C₂², (Dic₃xC₁₂).₆C₂, C₄.₅(C₆.D₆), C₃:₂(Dic₆:C₄), C₂.₂(D₁₂:₅S₃), C₆.₁₄(D₄:₂S₃), (C₂xDic₃).₁₀₉D₆, C₆².C₂².₁₁C₂, (C₆xDic₃).₁₅₀C₂², (C₂xC₄).₇₁S₃², C₆.₂₉(S₃xC₂xC₄), C₂².₂₁(C₂xS₃²), (C₃xC₆).₁₂(C₂xQ₈), (C₃xC₁₂).₆₃(C₂xC₄), (C₃xC₆).₅(C₄oD₄), C₂.₇(C₂xC₆.D₆), (C₃xC₄:Dic₃).₁₁C₂, C₃:Dic₃.₂₁(C₂xC₄), (C₂xC₆).₃₃(C₂²xS₃), (C₃xC₆).₄₉(C₂²xC₄), (C₂xC₃²:₄Q₈).₁₂C₂, (C₂xC₃:Dic₃).₁₅C₂², SmallGroup(288,492)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — Dic₃:₆Dic₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — C₆xDic₃ — Dic₃² — Dic₃:₆Dic₆

Lower central

C₃² — C₃xC₆ — Dic₃:₆Dic₆

Upper central

C₁ — C₂² — C₂xC₄

Generators and relations for Dic₃:₆Dic₆
G = < a,b,c,d | a⁶=c¹²=1, b²=a³, d²=c⁶, bab^-1=dad^-1=a^-1, ac=ca, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 482 in 159 conjugacy classes, 64 normal (34 characteristic)
C₁, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₂xC₄, C₂xC₄, Q₈, C₃², Dic₃, Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₄², C₄:C₄, C₂xQ₈, C₃xC₆, Dic₆, C₂xDic₃, C₂xDic₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₄xQ₈, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₆², C₄xDic₃, C₄xDic₃, Dic₃:C₄, C₄:Dic₃, C₄xC₁₂, C₃xC₄:C₄, C₂xDic₆, C₆xDic₃, C₆xDic₃, C₃²:₄Q₈, C₂xC₃:Dic₃, C₆xC₁₂, C₄xDic₆, Dic₆:C₄, Dic₃², C₆².C₂², Dic₃xC₁₂, C₃xC₄:Dic₃, C₂xC₃²:₄Q₈, Dic₃:₆Dic₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, Q₈, C₂³, D₆, C₂²xC₄, C₂xQ₈, C₄oD₄, Dic₆, C₄xS₃, C₂²xS₃, C₄xQ₈, S₃², C₂xDic₆, S₃xC₂xC₄, C₄oD₁₂, D₄:₂S₃, S₃xQ₈, C₆.D₆, C₂xS₃², C₄xDic₆, Dic₆:C₄, S₃xDic₆, D₁₂:₅S₃, C₂xC₆.D₆, Dic₃:₆Dic₆

Smallest permutation representation of Dic₃:₆Dic₆
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	···	4L	4M	4N	4O	4P	6A	···	6F	6G	6H	6I	12A	12B	12C	12D	12E	···	12J	12K	···	12R	12S	12T	12U	12V
order	1	2	2	2	3	3	3	4	4	4	4	4	4	4	···	4	4	4	4	4	6	···	6	6	6	6	12	12	12	12	12	···	12	12	···	12	12	12	12	12
size	1	1	1	1	2	2	4	2	2	3	3	3	3	6	···	6	18	18	18	18	2	···	2	4	4	4	2	2	2	2	4	···	4	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	4	4
type	+	+	+	+	+	+		+	+	-	+	+		-			+	-	-	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	S₃	S₃	Q₈	D₆	D₆	C₄oD₄	Dic₆	C₄xS₃	C₄oD₁₂	S₃²	D₄:₂S₃	S₃xQ₈	C₆.D₆	C₂xS₃²	S₃xDic₆	D₁₂:₅S₃
kernel	Dic₃:₆Dic₆	Dic₃²	C₆².C₂²	Dic₃xC₁₂	C₃xC₄:Dic₃	C₂xC₃²:₄Q₈	C₃²:₄Q₈	C₄xDic₃	C₄:Dic₃	C₃xDic₃	C₂xDic₃	C₂xC₁₂	C₃xC₆	Dic₃	C₁₂	C₆	C₂xC₄	C₆	C₆	C₄	C₂²	C₂	C₂
# reps	1	2	2	1	1	1	8	1	1	2	4	2	2	4	8	4	1	1	1	2	1	2	2

Matrix representation of Dic₃:₆Dic₆ ►in GL₆(F₁₃)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	0	1
0	0	0	0	1	0

12	3	0	0	0	0
8	1	0	0	0	0
0	0	12	12	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic₃:₆Dic₆ in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_6{\rm Dic}_6

% in TeX

G:=Group("Dic3:6Dic6");

// GroupNames label

G:=SmallGroup(288,492);

// by ID

G=gap.SmallGroup(288,492);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,142,1356,9414]);

// Polycyclic

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

// generators/relations

G = Dic3:6Dic6 order 288 = 25·32

2nd semidirect product of Dic3 and Dic6 acting through Inn(Dic3)

G = Dic₃:₆Dic₆ order 288 = 2⁵·3²

2^nd semidirect product of Dic₃ and Dic₆ acting through Inn(Dic₃)