Dic3xC3:S3 - GroupNames

G = Dic₃×C₃⋊S₃ order 216 = 2³·3³

Direct product of Dic₃ and C₃⋊S₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: Dic₃×C₃⋊S₃, C₆.₁₀S₃², C₃³⋊₇(C₂×C₄), C₃²⋊₆(C₄×S₃), C₃⋊₁(S₃×Dic₃), (C₃×C₆).₄₄D₆, C₃³⋊₅C₄⋊₂C₂, (C₃×Dic₃)⋊₂S₃, C₃²⋊₈(C₂×Dic₃), (C₃²×Dic₃)⋊₅C₂, (C₃²×C₆).₇C₂², C₃⋊₃(C₄×C₃⋊S₃), (C₃×C₃⋊S₃)⋊₃C₄, C₂.₂(S₃×C₃⋊S₃), C₆.₂(C₂×C₃⋊S₃), (C₆×C₃⋊S₃).₂C₂, (C₂×C₃⋊S₃).₄S₃, SmallGroup(216,125)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — Dic₃×C₃⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃²×Dic₃ — Dic₃×C₃⋊S₃

Lower central

C₃³ — Dic₃×C₃⋊S₃

Upper central

C₁ — C₂

Generators and relations for Dic₃×C₃⋊S₃
G = < a,b,c,d,e | a⁶=c³=d³=e²=1, b²=a³, bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 484 in 120 conjugacy classes, 38 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, C₃×C₃⋊S₃, C₃²×C₆, S₃×Dic₃, C₄×C₃⋊S₃, C₃²×Dic₃, C₃³⋊₅C₄, C₆×C₃⋊S₃, Dic₃×C₃⋊S₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Dic₃, D₆, C₃⋊S₃, C₄×S₃, C₂×Dic₃, S₃², C₂×C₃⋊S₃, S₃×Dic₃, C₄×C₃⋊S₃, S₃×C₃⋊S₃, Dic₃×C₃⋊S₃

Smallest permutation representation of Dic₃×C₃⋊S₃
►On 72 points

Generators in S₇₂

(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)

(1 52 4 49)(2 51 5 54)(3 50 6 53)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 46 16 43)(14 45 17 48)(15 44 18 47)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 65 28 62)(26 64 29 61)(27 63 30 66)(31 68 34 71)(32 67 35 70)(33 72 36 69)

(1 19 15)(2 20 16)(3 21 17)(4 22 18)(5 23 13)(6 24 14)(7 67 65)(8 68 66)(9 69 61)(10 70 62)(11 71 63)(12 72 64)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(29 37 36)(30 38 31)(43 51 58)(44 52 59)(45 53 60)(46 54 55)(47 49 56)(48 50 57)

(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 65 63)(62 66 64)(67 71 69)(68 72 70)

(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 56)(8 57)(9 58)(10 59)(11 60)(12 55)(13 26)(14 27)(15 28)(16 29)(17 30)(18 25)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

G:=sub<Sym(72)| (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), (1,52,4,49)(2,51,5,54)(3,50,6,53)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,46,16,43)(14,45,17,48)(15,44,18,47)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,65,28,62)(26,64,29,61)(27,63,30,66)(31,68,34,71)(32,67,35,70)(33,72,36,69), (1,19,15)(2,20,16)(3,21,17)(4,22,18)(5,23,13)(6,24,14)(7,67,65)(8,68,66)(9,69,61)(10,70,62)(11,71,63)(12,72,64)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(29,37,36)(30,38,31)(43,51,58)(44,52,59)(45,53,60)(46,54,55)(47,49,56)(48,50,57), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,65,63)(62,66,64)(67,71,69)(68,72,70), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,56)(8,57)(9,58)(10,59)(11,60)(12,55)(13,26)(14,27)(15,28)(16,29)(17,30)(18,25)(19,42)(20,37)(21,38)(22,39)(23,40)(24,41)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)>;

G:=Group( (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), (1,52,4,49)(2,51,5,54)(3,50,6,53)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,46,16,43)(14,45,17,48)(15,44,18,47)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,65,28,62)(26,64,29,61)(27,63,30,66)(31,68,34,71)(32,67,35,70)(33,72,36,69), (1,19,15)(2,20,16)(3,21,17)(4,22,18)(5,23,13)(6,24,14)(7,67,65)(8,68,66)(9,69,61)(10,70,62)(11,71,63)(12,72,64)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(29,37,36)(30,38,31)(43,51,58)(44,52,59)(45,53,60)(46,54,55)(47,49,56)(48,50,57), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,65,63)(62,66,64)(67,71,69)(68,72,70), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,56)(8,57)(9,58)(10,59)(11,60)(12,55)(13,26)(14,27)(15,28)(16,29)(17,30)(18,25)(19,42)(20,37)(21,38)(22,39)(23,40)(24,41)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72) );

G=PermutationGroup([[(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)], [(1,52,4,49),(2,51,5,54),(3,50,6,53),(7,42,10,39),(8,41,11,38),(9,40,12,37),(13,46,16,43),(14,45,17,48),(15,44,18,47),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,65,28,62),(26,64,29,61),(27,63,30,66),(31,68,34,71),(32,67,35,70),(33,72,36,69)], [(1,19,15),(2,20,16),(3,21,17),(4,22,18),(5,23,13),(6,24,14),(7,67,65),(8,68,66),(9,69,61),(10,70,62),(11,71,63),(12,72,64),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(29,37,36),(30,38,31),(43,51,58),(44,52,59),(45,53,60),(46,54,55),(47,49,56),(48,50,57)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,57,59),(56,58,60),(61,65,63),(62,66,64),(67,71,69),(68,72,70)], [(1,35),(2,36),(3,31),(4,32),(5,33),(6,34),(7,56),(8,57),(9,58),(10,59),(11,60),(12,55),(13,26),(14,27),(15,28),(16,29),(17,30),(18,25),(19,42),(20,37),(21,38),(22,39),(23,40),(24,41),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)]])

Dic₃×C₃⋊S₃ is a maximal subgroup of
C₃³⋊(C₄⋊C₄) S₃²×Dic₃ Dic₃⋊₆S₃² D₆⋊S₃² C₃³⋊₅(C₂×Q₈) D₆.S₃² D₆.₆S₃² C₁₂.₃₉S₃² C₁₂.₅₇S₃² C₄×S₃×C₃⋊S₃ C₆².₉₁D₆ C₆².₉₃D₆
Dic₃×C₃⋊S₃ is a maximal quotient of
C₃³⋊₈M₄(2) C₆².₇₈D₆ C₆².₈₂D₆

36 conjugacy classes

class	1	2A	2B	2C	3A	···	3E	3F	3G	3H	3I	4A	4B	4C	4D	6A	···	6E	6F	6G	6H	6I	6J	6K	12A	···	12H
order	1	2	2	2	3	···	3	3	3	3	3	4	4	4	4	6	···	6	6	6	6	6	6	6	12	···	12
size	1	1	9	9	2	···	2	4	4	4	4	3	3	27	27	2	···	2	4	4	4	4	18	18	6	···	6

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+		+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₄	S₃	S₃	Dic₃	D₆	C₄×S₃	S₃²	S₃×Dic₃
kernel	Dic₃×C₃⋊S₃	C₃²×Dic₃	C₃³⋊₅C₄	C₆×C₃⋊S₃	C₃×C₃⋊S₃	C₃×Dic₃	C₂×C₃⋊S₃	C₃⋊S₃	C₃×C₆	C₃²	C₆	C₃
# reps	1	1	1	1	4	4	1	2	5	8	4	4

Matrix representation of Dic₃×C₃⋊S₃ ►in GL₆(𝔽₁₃)

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

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	8
0	0	0	0	8	0

3	0	0	0	0	0
4	9	0	0	0	0
0	0	1	0	0	0
0	0	0	1	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	12	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

12	5	0	0	0	0
0	1	0	0	0	0
0	0	0	12	0	0
0	0	12	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12

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

Dic₃×C₃⋊S₃ in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\rtimes S_3

% in TeX

G:=Group("Dic3xC3:S3");

// GroupNames label

G:=SmallGroup(216,125);

// by ID

G=gap.SmallGroup(216,125);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,201,730,5189]);

// Polycyclic

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

// generators/relations

G = Dic3×C3⋊S3 order 216 = 23·33

Direct product of Dic3 and C3⋊S3

G = Dic₃×C₃⋊S₃ order 216 = 2³·3³

Direct product of Dic₃ and C₃⋊S₃