D6:Dic6 - GroupNames

G = D₆:Dic₆ order 288 = 2⁵·3²

1^st semidirect product of D₆ and Dic₆ acting via Dic₆/C₁₂=C₂

Aliases: D₆:₅Dic₆, C₆².₂₁C₂³, (S₃xC₆):₅Q₈, C₆.₂₀(S₃xQ₈), Dic₃:C₄:₂S₃, C₆.₁₃₅(S₃xD₄), C₆.₈(C₂xDic₆), C₃:₇(D₆:Q₈), (C₂xC₁₂).₂₅₈D₆, (C₂xDic₃).₈D₆, C₂.₁₀(S₃xDic₆), C₆.₁₉(C₄oD₁₂), C₃²:₃(C₂²:Q₈), D₆:Dic₃.₁₁C₂, (C₃xDic₃).₃₃D₄, (C₂²xS₃).₆₀D₆, Dic₃:Dic₃:₃₁C₂, C₃:₁(C₁₂.₄₈D₄), (C₆xC₁₂).₂₁₂C₂², C₆.Dic₆:₁₂C₂, C₂.₇(D₆.D₆), Dic₃.₁₄(C₃:D₄), (C₆xDic₃).₇₄C₂², (C₂xC₄).₄₀S₃², (S₃xC₂xC₄).₉S₃, (S₃xC₂xC₁₂).₂C₂, C₂².₇₉(C₂xS₃²), (C₃xC₆).₇₉(C₂xD₄), C₂.₁₀(S₃xC₃:D₄), C₆.₂₈(C₂xC₃:D₄), (C₃xC₆).₁₆(C₂xQ₈), (C₃xDic₃:C₄):₄C₂, (C₂xC₃²:₂Q₈):₁C₂, (S₃xC₂xC₆).₇₀C₂², (C₃xC₆).₁₀(C₄oD₄), (C₂xC₆).₄₀(C₂²xS₃), (C₂xC₃:Dic₃).₂₁C₂², SmallGroup(288,499)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — D₆:Dic₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — S₃xC₂xC₆ — D₆:Dic₃ — D₆:Dic₆

Lower central

C₃² — C₆² — D₆:Dic₆

Upper central

C₁ — C₂² — C₂xC₄

Generators and relations for D₆:Dic₆
G = < a,b,c,d | a⁶=b²=c¹²=1, d²=c⁶, bab=cac^-1=dad^-1=a^-1, cbc^-1=a⁴b, dbd^-1=ab, dcd^-1=c^-1 >

Subgroups: 554 in 161 conjugacy classes, 52 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xQ₈, C₃xS₃, C₃xC₆, Dic₆, C₄xS₃, C₂xDic₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₂²:Q₈, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₆², Dic₃:C₄, Dic₃:C₄, C₄:Dic₃, D₆:C₄, C₆.D₄, C₃xC₄:C₄, C₂xDic₆, S₃xC₂xC₄, C₂²xC₁₂, C₃²:₂Q₈, S₃xC₁₂, C₆xDic₃, C₂xC₃:Dic₃, C₆xC₁₂, S₃xC₂xC₆, D₆:Q₈, C₁₂.₄₈D₄, D₆:Dic₃, Dic₃:Dic₃, C₃xDic₃:C₄, C₆.Dic₆, C₂xC₃²:₂Q₈, S₃xC₂xC₁₂, D₆:Dic₆
Quotients: C₁, C₂, C₂², S₃, D₄, Q₈, C₂³, D₆, C₂xD₄, C₂xQ₈, C₄oD₄, Dic₆, C₃:D₄, C₂²xS₃, C₂²:Q₈, S₃², C₂xDic₆, C₄oD₁₂, S₃xD₄, S₃xQ₈, C₂xC₃:D₄, C₂xS₃², D₆:Q₈, C₁₂.₄₈D₄, S₃xDic₆, D₆.D₆, S₃xC₃:D₄, D₆:Dic₆

Smallest permutation representation of D₆:Dic₆
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	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	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	6	6	2	2	4	2	2	6	6	12	12	36	36	2	···	2	4	4	4	6	6	6	6	2	2	2	2	4	···	4	6	6	6	6	12	12	12	12

48 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	-	+	+	+			-		+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₄	Q₈	D₆	D₆	D₆	C₄oD₄	C₃:D₄	Dic₆	C₄oD₁₂	S₃²	S₃xD₄	S₃xQ₈	C₂xS₃²	S₃xDic₆	D₆.D₆	S₃xC₃:D₄
kernel	D₆:Dic₆	D₆:Dic₃	Dic₃:Dic₃	C₃xDic₃:C₄	C₆.Dic₆	C₂xC₃²:₂Q₈	S₃xC₂xC₁₂	Dic₃:C₄	S₃xC₂xC₄	C₃xDic₃	S₃xC₆	C₂xDic₃	C₂xC₁₂	C₂²xS₃	C₃xC₆	Dic₃	D₆	C₆	C₂xC₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	2	1	1	1	1	1	1	1	2	2	3	2	1	2	4	4	8	1	1	1	1	2	2	2

Matrix representation of D₆:Dic₆ ►in GL₆(F₁₃)

12	0	0	0	0	0
0	12	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

2	9	0	0	0	0
4	11	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	12	1

7	3	0	0	0	0
10	10	0	0	0	0
0	0	10	3	0	0
0	0	10	7	0	0
0	0	0	0	0	1
0	0	0	0	1	0

11	11	0	0	0	0
9	2	0	0	0	0
0	0	2	4	0	0
0	0	2	11	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[7,10,0,0,0,0,3,10,0,0,0,0,0,0,10,10,0,0,0,0,3,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,9,0,0,0,0,11,2,0,0,0,0,0,0,2,2,0,0,0,0,4,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D₆:Dic₆ in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_6

% in TeX

G:=Group("D6:Dic6");

// GroupNames label

G:=SmallGroup(288,499);

// by ID

G=gap.SmallGroup(288,499);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,142,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;

// generators/relations

G = D6:Dic6 order 288 = 25·32

1st semidirect product of D6 and Dic6 acting via Dic6/C12=C2

G = D₆:Dic₆ order 288 = 2⁵·3²

1^st semidirect product of D₆ and Dic₆ acting via Dic₆/C₁₂=C₂