C3xC3^2:2D8 - GroupNames

G = C₃×C₃²⋊₂D₈ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₂D₈

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₃²⋊₂D₈, C₃³⋊₄D₈, C₁₂.₉₆S₃², (C₃×D₁₂)⋊₁C₆, D₁₂⋊₂(C₃×S₃), (C₃×D₁₂)⋊₁S₃, C₃²⋊₄(C₃×D₈), C₁₂.₂₅(S₃×C₆), C₃²⋊₄C₈⋊₈C₆, (C₃×C₁₂).₁₀₉D₆, (C₃²×D₁₂)⋊₁C₂, (C₃²×C₆).₁₈D₄, C₃²⋊₁₀(D₄⋊S₃), C₆.₂₇(D₆⋊S₃), (C₃²×C₁₂).₁C₂², C₄.₈(C₃×S₃²), C₃⋊₂(C₃×D₄⋊S₃), C₆.₇(C₃×C₃⋊D₄), (C₃×C₆).₁₇(C₃×D₄), (C₃×C₁₂).₃₅(C₂×C₆), (C₃×C₃²⋊₄C₈)⋊₇C₂, C₂.₃(C₃×D₆⋊S₃), (C₃×C₆).₈₂(C₃⋊D₄), SmallGroup(432,418)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃×C₃²⋊₂D₈

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — C₃²×D₁₂ — C₃×C₃²⋊₂D₈

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×C₃²⋊₂D₈

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×C₃²⋊₂D₈
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=b^-1, be=eb, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 496 in 134 conjugacy classes, 36 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₆, C₈, D₄, C₃², C₃², C₃², C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, D₈, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, D₁₂, C₃×D₄, C₃³, C₃×C₁₂, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₆², D₄⋊S₃, C₃×D₈, S₃×C₃², C₃²×C₆, C₃×C₃⋊C₈, C₃²⋊₄C₈, C₃×D₁₂, C₃×D₁₂, D₄×C₃², C₃²×C₁₂, S₃×C₃×C₆, C₃²⋊₂D₈, C₃×D₄⋊S₃, C₃×C₃²⋊₄C₈, C₃²×D₁₂, C₃×C₃²⋊₂D₈
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, D₈, C₃×S₃, C₃⋊D₄, C₃×D₄, S₃², S₃×C₆, D₄⋊S₃, C₃×D₈, D₆⋊S₃, C₃×C₃⋊D₄, C₃×S₃², C₃²⋊₂D₈, C₃×D₄⋊S₃, C₃×D₆⋊S₃, C₃×C₃²⋊₂D₈

Smallest permutation representation of C₃×C₃²⋊₂D₈
►On 48 points

Generators in S₄₈

(1 11 46)(2 12 47)(3 13 48)(4 14 41)(5 15 42)(6 16 43)(7 9 44)(8 10 45)(17 33 31)(18 34 32)(19 35 25)(20 36 26)(21 37 27)(22 38 28)(23 39 29)(24 40 30)

(1 11 46)(2 47 12)(3 13 48)(4 41 14)(5 15 42)(6 43 16)(7 9 44)(8 45 10)(17 33 31)(18 32 34)(19 35 25)(20 26 36)(21 37 27)(22 28 38)(23 39 29)(24 30 40)

(1 46 11)(2 12 47)(3 48 13)(4 14 41)(5 42 15)(6 16 43)(7 44 9)(8 10 45)(17 33 31)(18 32 34)(19 35 25)(20 26 36)(21 37 27)(22 28 38)(23 39 29)(24 30 40)

(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)

(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 33)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)

G:=sub<Sym(48)| (1,11,46)(2,12,47)(3,13,48)(4,14,41)(5,15,42)(6,16,43)(7,9,44)(8,10,45)(17,33,31)(18,34,32)(19,35,25)(20,36,26)(21,37,27)(22,38,28)(23,39,29)(24,40,30), (1,11,46)(2,47,12)(3,13,48)(4,41,14)(5,15,42)(6,43,16)(7,9,44)(8,45,10)(17,33,31)(18,32,34)(19,35,25)(20,26,36)(21,37,27)(22,28,38)(23,39,29)(24,30,40), (1,46,11)(2,12,47)(3,48,13)(4,14,41)(5,42,15)(6,16,43)(7,44,9)(8,10,45)(17,33,31)(18,32,34)(19,35,25)(20,26,36)(21,37,27)(22,28,38)(23,39,29)(24,30,40), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,33)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)>;

G:=Group( (1,11,46)(2,12,47)(3,13,48)(4,14,41)(5,15,42)(6,16,43)(7,9,44)(8,10,45)(17,33,31)(18,34,32)(19,35,25)(20,36,26)(21,37,27)(22,38,28)(23,39,29)(24,40,30), (1,11,46)(2,47,12)(3,13,48)(4,41,14)(5,15,42)(6,43,16)(7,9,44)(8,45,10)(17,33,31)(18,32,34)(19,35,25)(20,26,36)(21,37,27)(22,28,38)(23,39,29)(24,30,40), (1,46,11)(2,12,47)(3,48,13)(4,14,41)(5,42,15)(6,16,43)(7,44,9)(8,10,45)(17,33,31)(18,32,34)(19,35,25)(20,26,36)(21,37,27)(22,28,38)(23,39,29)(24,30,40), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,33)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43) );

G=PermutationGroup([[(1,11,46),(2,12,47),(3,13,48),(4,14,41),(5,15,42),(6,16,43),(7,9,44),(8,10,45),(17,33,31),(18,34,32),(19,35,25),(20,36,26),(21,37,27),(22,38,28),(23,39,29),(24,40,30)], [(1,11,46),(2,47,12),(3,13,48),(4,41,14),(5,15,42),(6,43,16),(7,9,44),(8,45,10),(17,33,31),(18,32,34),(19,35,25),(20,26,36),(21,37,27),(22,28,38),(23,39,29),(24,30,40)], [(1,46,11),(2,12,47),(3,48,13),(4,14,41),(5,42,15),(6,16,43),(7,44,9),(8,10,45),(17,33,31),(18,32,34),(19,35,25),(20,26,36),(21,37,27),(22,28,38),(23,39,29),(24,30,40)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,33),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)]])

63 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	3I	3J	3K	4	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6AA	8A	8B	12A	12B	12C	···	12N	24A	24B	24C	24D
order	1	2	2	2	3	3	3	···	3	3	3	3	4	6	6	6	···	6	6	6	6	6	···	6	8	8	12	12	12	···	12	24	24	24	24
size	1	1	12	12	1	1	2	···	2	4	4	4	2	1	1	2	···	2	4	4	4	12	···	12	18	18	2	2	4	···	4	18	18	18	18

63 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+				+	+	+	+							+	+	-
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₆	D₈	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	C₃×D₈	C₃×C₃⋊D₄	S₃²	D₄⋊S₃	D₆⋊S₃	C₃×S₃²	C₃²⋊₂D₈	C₃×D₄⋊S₃	C₃×D₆⋊S₃	C₃×C₃²⋊₂D₈
kernel	C₃×C₃²⋊₂D₈	C₃×C₃²⋊₄C₈	C₃²×D₁₂	C₃²⋊₂D₈	C₃²⋊₄C₈	C₃×D₁₂	C₃×D₁₂	C₃²×C₆	C₃×C₁₂	C₃³	D₁₂	C₃×C₆	C₃×C₆	C₁₂	C₃²	C₆	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	2	2	2	4	2	1	2	2	4	4	2	4	4	8	1	2	1	2	2	4	2	4

Matrix representation of C₃×C₃²⋊₂D₈ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	64	0	0	0
0	0	0	64	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	72
0	0	0	0	1	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	72	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

10	59	0	0	0	0
0	22	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	0	1
0	0	0	0	1	0

10	59	0	0	0	0
54	63	0	0	0	0
0	0	72	0	0	0
0	0	1	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,59,22,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,54,0,0,0,0,59,63,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃×C₃²⋊₂D₈ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2D_8

% in TeX

G:=Group("C3xC3^2:2D8");

// GroupNames label

G:=SmallGroup(432,418);

// by ID

G=gap.SmallGroup(432,418);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,1011,514,80,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,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 = C3×C32⋊2D8 order 432 = 24·33

Direct product of C3 and C32⋊2D8

G = C₃×C₃²⋊₂D₈ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₂D₈