C3^3:8(C2xC4) - GroupNames

G = C₃³⋊₈(C₂×C₄) order 216 = 2³·3³

5^th semidirect product of C₃³ and C₂×C₄ acting via C₂×C₄/C₂=C₂²

metabelian, supersoluble, monomial, A-group

Aliases: C₆.₁₁S₃², C₃³⋊₈(C₂×C₄), C₃⋊Dic₃⋊₇S₃, C₃²⋊₇(C₄×S₃), (C₃×C₆).₃₀D₆, C₃³⋊C₂⋊₂C₄, (C₃×Dic₃)⋊₃S₃, Dic₃⋊₂(C₃⋊S₃), C₃⋊₁(C₆.D₆), (C₃²×Dic₃)⋊₆C₂, (C₃²×C₆).₈C₂², C₃⋊₁(C₄×C₃⋊S₃), C₆.₃(C₂×C₃⋊S₃), C₂.₃(S₃×C₃⋊S₃), (C₃×C₃⋊Dic₃)⋊₅C₂, (C₂×C₃³⋊C₂).C₂, SmallGroup(216,126)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃³⋊₈(C₂×C₄)

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃²×Dic₃ — C₃³⋊₈(C₂×C₄)

Lower central

C₃³ — C₃³⋊₈(C₂×C₄)

Upper central

C₁ — C₂

Generators and relations for C₃³⋊₈(C₂×C₄)
G = < a,b,c,d,e | a³=b³=c³=d²=e⁴=1, ab=ba, ac=ca, dad=eae^-1=a^-1, bc=cb, dbd=ebe^-1=b^-1, dcd=c^-1, ce=ec, de=ed >

Subgroups: 764 in 136 conjugacy classes, 36 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₄×S₃, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₂×C₃⋊S₃, C₃³⋊C₂, C₃²×C₆, C₆.D₆, C₄×C₃⋊S₃, C₃²×Dic₃, C₃×C₃⋊Dic₃, C₂×C₃³⋊C₂, C₃³⋊₈(C₂×C₄)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₆, C₃⋊S₃, C₄×S₃, S₃², C₂×C₃⋊S₃, C₆.D₆, C₄×C₃⋊S₃, S₃×C₃⋊S₃, C₃³⋊₈(C₂×C₄)

Smallest permutation representation of C₃³⋊₈(C₂×C₄)
►On 36 points

Generators in S₃₆

(1 9 27)(2 28 10)(3 11 25)(4 26 12)(5 20 23)(6 24 17)(7 18 21)(8 22 19)(13 34 30)(14 31 35)(15 36 32)(16 29 33)

(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)

(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)

(5 14)(6 15)(7 16)(8 13)(9 27)(10 28)(11 25)(12 26)(17 36)(18 33)(19 34)(20 35)(21 29)(22 30)(23 31)(24 32)

(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)

G:=sub<Sym(36)| (1,9,27)(2,28,10)(3,11,25)(4,26,12)(5,20,23)(6,24,17)(7,18,21)(8,22,19)(13,34,30)(14,31,35)(15,36,32)(16,29,33), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32), (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)>;

G:=Group( (1,9,27)(2,28,10)(3,11,25)(4,26,12)(5,20,23)(6,24,17)(7,18,21)(8,22,19)(13,34,30)(14,31,35)(15,36,32)(16,29,33), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32), (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) );

G=PermutationGroup([[(1,9,27),(2,28,10),(3,11,25),(4,26,12),(5,20,23),(6,24,17),(7,18,21),(8,22,19),(13,34,30),(14,31,35),(15,36,32),(16,29,33)], [(1,16,7),(2,8,13),(3,14,5),(4,6,15),(9,29,18),(10,19,30),(11,31,20),(12,17,32),(21,27,33),(22,34,28),(23,25,35),(24,36,26)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(5,14),(6,15),(7,16),(8,13),(9,27),(10,28),(11,25),(12,26),(17,36),(18,33),(19,34),(20,35),(21,29),(22,30),(23,31),(24,32)], [(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)]])

C₃³⋊₈(C₂×C₄) is a maximal subgroup of
C₃³⋊₅(C₂×C₈) C₃³⋊₂M₄(2) S₃×C₆.D₆ (S₃×C₆)⋊D₆ C₃³⋊₆(C₂×Q₈) (S₃×C₆).D₆ D₆.₃S₃² Dic₃.S₃² C₁₂.₄₀S₃² C₃²⋊₉(S₃×Q₈) C₁₂.₅₈S₃² C₄×S₃×C₃⋊S₃ C₆².₉₀D₆ C₆².₉₃D₆ C₆²⋊₂₃D₆
C₃³⋊₈(C₂×C₄) is a maximal quotient of
C₁₂.₆₉S₃² C₃³⋊₉M₄(2) Dic₃×C₃⋊Dic₃ C₆².₇₉D₆ C₆².₈₁D₆

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+		+	+	+		+	+
image	C₁	C₂	C₂	C₂	C₄	S₃	S₃	D₆	C₄×S₃	S₃²	C₆.D₆
kernel	C₃³⋊₈(C₂×C₄)	C₃²×Dic₃	C₃×C₃⋊Dic₃	C₂×C₃³⋊C₂	C₃³⋊C₂	C₃×Dic₃	C₃⋊Dic₃	C₃×C₆	C₃²	C₆	C₃
# reps	1	1	1	1	4	4	1	5	10	4	4

Matrix representation of C₃³⋊₈(C₂×C₄) ►in GL₆(𝔽₁₃)

11	3	0	0	0	0
12	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

1	10	0	0	0	0
1	11	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	12
0	0	0	0	1	12

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

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

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

C₃³⋊₈(C₂×C₄) in GAP, Magma, Sage, TeX

C_3^3\rtimes_8(C_2\times C_4)

% in TeX

G:=Group("C3^3:8(C2xC4)");

// GroupNames label

G:=SmallGroup(216,126);

// by ID

G=gap.SmallGroup(216,126);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,31,201,730,5189]);

// Polycyclic

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

// generators/relations

G = C33⋊8(C2×C4) order 216 = 23·33

5th semidirect product of C33 and C2×C4 acting via C2×C4/C2=C22

G = C₃³⋊₈(C₂×C₄) order 216 = 2³·3³

5^th semidirect product of C₃³ and C₂×C₄ acting via C₂×C₄/C₂=C₂²