C3xD12.S3 - GroupNames

G = C₃×D₁₂.S₃ order 432 = 2⁴·3³

Direct product of C₃ and D₁₂.S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃×D₁₂.S₃, C₃³⋊₁₀SD₁₆, C₁₂.₇₅S₃², C₁₂.₂₈(S₃×C₆), (C₃×D₁₂).₆C₆, D₁₂.₁(C₃×S₃), (C₃×C₆).₇₂D₁₂, C₆.₂₅(C₃×D₁₂), C₃²⋊₄Q₈⋊₈C₆, (C₃×D₁₂).₁₂S₃, (C₃×C₁₂).₁₇₃D₆, C₃²⋊₇(C₃×SD₁₆), (C₃²×C₆).₂₁D₄, (C₃²×D₁₂).₂C₂, C₃²⋊₁₄(C₂₄⋊C₂), C₆.₄₃(C₃⋊D₁₂), C₃²⋊₁₀(D₄.S₃), (C₃²×C₁₂).₄C₂², (C₃×C₃⋊C₈)⋊₂C₆, (C₃×C₃⋊C₈)⋊₉S₃, C₃⋊C₈⋊₂(C₃×S₃), C₄.₂(C₃×S₃²), C₃⋊₃(C₃×C₂₄⋊C₂), (C₃²×C₃⋊C₈)⋊₅C₂, C₃⋊₁(C₃×D₄.S₃), C₆.₂(C₃×C₃⋊D₄), (C₃×C₆).₂₀(C₃×D₄), (C₃×C₁₂).₃₈(C₂×C₆), C₂.₅(C₃×C₃⋊D₁₂), (C₃×C₃²⋊₄Q₈)⋊₁C₂, (C₃×C₆).₇₁(C₃⋊D₄), SmallGroup(432,421)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃×D₁₂.S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — C₃²×D₁₂ — C₃×D₁₂.S₃

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×D₁₂.S₃

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×D₁₂.S₃
G = < a,b,c,d,e | a³=b¹²=c²=d³=1, e²=b⁹, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece^-1=b³c, ede^-1=d^-1 >

Subgroups: 440 in 122 conjugacy classes, 36 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, D₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, SD₁₆, C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, D₁₂, C₃×D₄, C₃×Q₈, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₆², C₂₄⋊C₂, D₄.S₃, C₃×SD₁₆, S₃×C₃², C₃²×C₆, C₃×C₃⋊C₈, C₃×C₃⋊C₈, C₃×C₂₄, C₃×Dic₆, C₃×D₁₂, C₃×D₁₂, C₃²⋊₄Q₈, D₄×C₃², C₃×C₃⋊Dic₃, C₃²×C₁₂, S₃×C₃×C₆, D₁₂.S₃, C₃×C₂₄⋊C₂, C₃×D₄.S₃, C₃²×C₃⋊C₈, C₃²×D₁₂, C₃×C₃²⋊₄Q₈, C₃×D₁₂.S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, SD₁₆, C₃×S₃, D₁₂, C₃⋊D₄, C₃×D₄, S₃², S₃×C₆, C₂₄⋊C₂, D₄.S₃, C₃×SD₁₆, C₃⋊D₁₂, C₃×D₁₂, C₃×C₃⋊D₄, C₃×S₃², D₁₂.S₃, C₃×C₂₄⋊C₂, C₃×D₄.S₃, C₃×C₃⋊D₁₂, C₃×D₁₂.S₃

Smallest permutation representation of C₃×D₁₂.S₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6S	8A	8B	12A	···	12H	12I	···	12Q	12R	12S	24A	···	24P
order	1	2	2	3	3	3	···	3	3	3	3	4	4	6	6	6	···	6	6	6	6	6	···	6	8	8	12	···	12	12	···	12	12	12	24	···	24
size	1	1	12	1	1	2	···	2	4	4	4	2	36	1	1	2	···	2	4	4	4	12	···	12	6	6	2	···	2	4	···	4	36	36	6	···	6

72 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+					+	+	+	+				+									+	-	+		-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	S₃	D₄	D₆	SD₁₆	C₃×S₃	C₃×S₃	D₁₂	C₃⋊D₄	C₃×D₄	S₃×C₆	C₂₄⋊C₂	C₃×SD₁₆	C₃×D₁₂	C₃×C₃⋊D₄	C₃×C₂₄⋊C₂	S₃²	D₄.S₃	C₃⋊D₁₂	C₃×S₃²	D₁₂.S₃	C₃×D₄.S₃	C₃×C₃⋊D₁₂	C₃×D₁₂.S₃
kernel	C₃×D₁₂.S₃	C₃²×C₃⋊C₈	C₃²×D₁₂	C₃×C₃²⋊₄Q₈	D₁₂.S₃	C₃×C₃⋊C₈	C₃×D₁₂	C₃²⋊₄Q₈	C₃×C₃⋊C₈	C₃×D₁₂	C₃²×C₆	C₃×C₁₂	C₃³	C₃⋊C₈	D₁₂	C₃×C₆	C₃×C₆	C₃×C₆	C₁₂	C₃²	C₃²	C₆	C₆	C₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	8	1	1	1	2	2	2	2	4

Matrix representation of C₃×D₁₂.S₃ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	71	0	0	0	0
1	72	0	0	0	0
0	0	0	1	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

5	9	0	0	0	0
46	68	0	0	0	0
0	0	0	72	0	0
0	0	72	0	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

0	61	0	0	0	0
6	61	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,46,0,0,0,0,9,68,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,6,0,0,0,0,61,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₃×D₁₂.S₃ in GAP, Magma, Sage, TeX

C_3\times D_{12}.S_3

% in TeX

G:=Group("C3xD12.S3");

// GroupNames label

G:=SmallGroup(432,421);

// by ID

G=gap.SmallGroup(432,421);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^3=1,e^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;

// generators/relations

G = C3×D12.S3 order 432 = 24·33

Direct product of C3 and D12.S3

G = C₃×D₁₂.S₃ order 432 = 2⁴·3³

Direct product of C₃ and D₁₂.S₃