Dic3.Dic6 - GroupNames

G = Dic₃.Dic₆ order 288 = 2⁵·3²

1^st non-split extension by Dic₃ of Dic₆ acting via Dic₆/Dic₃=C₂

Aliases: Dic₃.₁Dic₆, C₆².₁₅C₂³, Dic₃².₆C₂, C₆.₁₈(S₃×Q₈), (C₂×C₁₂).₁₃D₆, C₆.₆(C₂×Dic₆), C₂.₈(S₃×Dic₆), C₆.₄(C₄○D₁₂), Dic₃⋊C₄.₃S₃, (C₆×C₁₂).₂C₂², (C₃×Dic₃).₁Q₈, (C₂×Dic₃).₃D₆, C₃⋊₂(Dic₃.Q₈), C₆.₄(Q₈⋊₃S₃), C₃⋊₂(C₄.Dic₆), C₆.₅₉(D₄⋊₂S₃), C₃²⋊₂(C₄².C₂), C₁₂⋊Dic₃.₄C₂, C₂.₆(D₆.₆D₆), C₂.₆(D₆.₄D₆), C₆².C₂².₇C₂, (C₆×Dic₃).₁C₂², Dic₃⋊Dic₃.₁₁C₂, (C₂×C₄).₁₃S₃², C₂².₇₄(C₂×S₃²), (C₃×C₆).₁₃(C₂×Q₈), (C₃×C₆).₅₇(C₄○D₄), (C₃×Dic₃⋊C₄).₅C₂, (C₂×C₆).₃₄(C₂²×S₃), (C₂×C₃⋊Dic₃).₁₆C₂², SmallGroup(288,493)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — Dic₃.Dic₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₆×Dic₃ — Dic₃² — Dic₃.Dic₆

Lower central

C₃² — C₆² — Dic₃.Dic₆

Upper central

C₁ — C₂² — C₂×C₄

Generators and relations for Dic₃.Dic₆
G = < a,b,c,d | a⁶=c¹²=1, b²=a³, d²=c⁶, bab^-1=a^-1, ac=ca, ad=da, cbc^-1=a³b, bd=db, dcd^-1=a³c^-1 >

Subgroups: 394 in 125 conjugacy classes, 48 normal (44 characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₂×C₄, C₂×C₄, C₃², Dic₃, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₄², C₄⋊C₄, C₃×C₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₄².C₂, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₆², C₄×Dic₃, Dic₃⋊C₄, Dic₃⋊C₄, C₄⋊Dic₃, C₃×C₄⋊C₄, C₆×Dic₃, C₂×C₃⋊Dic₃, C₆×C₁₂, Dic₃.Q₈, C₄.Dic₆, Dic₃², Dic₃⋊Dic₃, C₆².C₂², C₃×Dic₃⋊C₄, C₁₂⋊Dic₃, Dic₃.Dic₆
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, C₄○D₄, Dic₆, C₂²×S₃, C₄².C₂, S₃², C₂×Dic₆, C₄○D₁₂, D₄⋊₂S₃, S₃×Q₈, Q₈⋊₃S₃, C₂×S₃², Dic₃.Q₈, C₄.Dic₆, S₃×Dic₆, D₆.₆D₆, D₆.₄D₆, Dic₃.Dic₆

Smallest permutation representation of Dic₃.Dic₆
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	···	6F	6G	6H	6I	12A	···	12H	12I	···	12P
order	1	2	2	2	3	3	3	4	4	4	4	4	4	4	4	4	4	6	···	6	6	6	6	12	···	12	12	···	12
size	1	1	1	1	2	2	4	4	6	6	6	6	12	12	18	18	36	2	···	2	4	4	4	4	···	4	12	···	12

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	-	+	+		-		+	-	-	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	Q₈	D₆	D₆	C₄○D₄	Dic₆	C₄○D₁₂	S₃²	D₄⋊₂S₃	S₃×Q₈	Q₈⋊₃S₃	C₂×S₃²	S₃×Dic₆	D₆.₆D₆	D₆.₄D₆
kernel	Dic₃.Dic₆	Dic₃²	Dic₃⋊Dic₃	C₆².C₂²	C₃×Dic₃⋊C₄	C₁₂⋊Dic₃	Dic₃⋊C₄	C₃×Dic₃	C₂×Dic₃	C₂×C₁₂	C₃×C₆	Dic₃	C₆	C₂×C₄	C₆	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	1	2	1	2	1	2	2	4	2	4	4	4	1	2	1	1	1	2	2	2

Matrix representation of Dic₃.Dic₆ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	10	0	0
0	0	0	0	1	0
0	0	0	0	0	1

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	10	0	0
0	0	9	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12

8	11	0	0	0	0
0	5	0	0	0	0
0	0	5	0	0	0
0	0	0	8	0	0
0	0	0	0	12	12
0	0	0	0	1	0

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

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,8,0,0,0,0,3,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;

Dic₃.Dic₆ in GAP, Magma, Sage, TeX

{\rm Dic}_3.{\rm Dic}_6

% in TeX

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

// GroupNames label

G:=SmallGroup(288,493);

// by ID

G=gap.SmallGroup(288,493);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,100,1356,9414]);

// Polycyclic

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

// generators/relations

G = Dic3.Dic6 order 288 = 25·32

1st non-split extension by Dic3 of Dic6 acting via Dic6/Dic3=C2

G = Dic₃.Dic₆ order 288 = 2⁵·3²

1^st non-split extension by Dic₃ of Dic₆ acting via Dic₆/Dic₃=C₂