C3^3:6D4 - GroupNames

G = C₃³:₆D₄ order 216 = 2³·3³

3^rd semidirect product of C₃³ and D₄ acting via D₄/C₂=C₂²

Aliases: C₃³:₆D₄, C₆.₁₂S₃², (S₃xC₆):₃S₃, D₆:₁(C₃:S₃), (C₃xC₆).₃₁D₆, C₃³:₅C₄:₃C₂, C₃:₂(D₆:S₃), C₃²:₆(C₃:D₄), C₃:₂(C₃²:₇D₄), (C₃²xC₆).₉C₂², (S₃xC₃xC₆):₃C₂, (C₆xC₃:S₃):₂C₂, (C₂xC₃:S₃):₄S₃, C₂.₄(S₃xC₃:S₃), C₆.₄(C₂xC₃:S₃), SmallGroup(216,127)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²xC₆ — C₃³:₆D₄

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₆ — S₃xC₃xC₆ — C₃³:₆D₄

Lower central

C₃³ — C₃²xC₆ — C₃³:₆D₄

Upper central

C₁ — C₂

Generators and relations for C₃³:₆D₄
G = < a,b,c,d,e | a³=b³=c³=d⁴=e²=1, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, bc=cb, dbd^-1=b^-1, be=eb, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 532 in 120 conjugacy classes, 34 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₆, D₄, C₃², C₃², C₃², Dic₃, D₆, D₆, C₂xC₆, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, C₃:D₄, C₃³, C₃:Dic₃, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₆², S₃xC₃², C₃xC₃:S₃, C₃²xC₆, D₆:S₃, C₃²:₇D₄, C₃³:₅C₄, S₃xC₃xC₆, C₆xC₃:S₃, C₃³:₆D₄
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, C₃:S₃, C₃:D₄, S₃², C₂xC₃:S₃, D₆:S₃, C₃²:₇D₄, S₃xC₃:S₃, C₃³:₆D₄

Smallest permutation representation of C₃³:₆D₄
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

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

C₃³:₆D₄ is a maximal subgroup of
S₃xD₆:S₃ D₆:S₃² D₆.S₃² D₆.₄S₃² (C₃xD₁₂):S₃ C₁₂.₇₃S₃² C₁₂.₅₇S₃² C₁₂:S₃² C₆².₉₁D₆ S₃xC₃²:₇D₄ C₃:S₃xC₃:D₄
C₃³:₆D₄ is a maximal quotient of
C₃³:₆D₈ C₃³:₁₂SD₁₆ C₃³:₁₃SD₁₆ C₃³:₆Q₁₆ C₆².₇₇D₆ C₆².₇₈D₆ C₆².₈₁D₆

33 conjugacy classes

class	1	2A	2B	2C	3A	···	3E	3F	3G	3H	3I	4	6A	···	6E	6F	6G	6H	6I	6J	···	6Q	6R	6S
order	1	2	2	2	3	···	3	3	3	3	3	4	6	···	6	6	6	6	6	6	···	6	6	6
size	1	1	6	18	2	···	2	4	4	4	4	54	2	···	2	4	4	4	4	6	···	6	18	18

33 irreducible representations

dim	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	S₃	S₃	D₄	D₆	C₃:D₄	S₃²	D₆:S₃
kernel	C₃³:₆D₄	C₃³:₅C₄	S₃xC₃xC₆	C₆xC₃:S₃	S₃xC₆	C₂xC₃:S₃	C₃³	C₃xC₆	C₃²	C₆	C₃
# reps	1	1	1	1	4	1	1	5	10	4	4

Matrix representation of C₃³:₆D₄ ►in GL₆(F₁₃)

12	12	0	0	0	0
1	0	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	12

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

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

11	9	0	0	0	0
4	2	0	0	0	0
0	0	0	12	0	0
0	0	12	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

C₃³:₆D₄ in GAP, Magma, Sage, TeX

C_3^3\rtimes_6D_4

% in TeX

G:=Group("C3^3:6D4");

// GroupNames label

G:=SmallGroup(216,127);

// by ID

G=gap.SmallGroup(216,127);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,730,5189]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = C33:6D4 order 216 = 23·33

3rd semidirect product of C33 and D4 acting via D4/C2=C22

G = C₃³:₆D₄ order 216 = 2³·3³

3^rd semidirect product of C₃³ and D₄ acting via D₄/C₂=C₂²