C3xC3:S3:3C8 - GroupNames

G = C₃×C₃⋊S₃⋊₃C₈ order 432 = 2⁴·3³

Direct product of C₃ and C₃⋊S₃⋊₃C₈

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

Aliases: C₃×C₃⋊S₃⋊₃C₈, C₃⋊S₃⋊₄C₂₄, C₃³⋊₆(C₂×C₈), (C₃×C₁₂).₈C₁₂, C₃²⋊₄(C₂×C₂₄), C₃²⋊₂C₈⋊₅C₆, (C₃²×C₁₂).₄C₄, C₁₂.₁₁(C₃²⋊C₄), (C₃×C₃⋊S₃)⋊₄C₈, (C₄×C₃⋊S₃).₉C₆, C₂.₁(C₆×C₃²⋊C₄), C₄.₃(C₃×C₃²⋊C₄), (C₆×C₃⋊S₃).₁₀C₄, (C₂×C₃⋊S₃).₇C₁₂, C₆.₁₉(C₂×C₃²⋊C₄), (C₁₂×C₃⋊S₃).₁₂C₂, (C₃×C₆).₁₀(C₂×C₁₂), (C₃²×C₆).₈(C₂×C₄), (C₃×C₃²⋊₂C₈)⋊₁₁C₂, C₃⋊Dic₃.₁₂(C₂×C₆), (C₃×C₃⋊Dic₃).₃₇C₂², SmallGroup(432,628)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃×C₃⋊S₃⋊₃C₈

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₃×C₃⋊Dic₃ — C₃×C₃²⋊₂C₈ — C₃×C₃⋊S₃⋊₃C₈

Lower central

C₃² — C₃×C₃⋊S₃⋊₃C₈

Upper central

C₁ — C₁₂

Generators and relations for C₃×C₃⋊S₃⋊₃C₈
G = < a,b,c,d,e | a³=b³=c³=d²=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b^-1, ebe^-1=bc^-1, dcd=c^-1, ece^-1=b^-1c^-1, de=ed >

Subgroups: 364 in 88 conjugacy classes, 28 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₂₄, C₄×S₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, C₂×C₂₄, C₃×C₃⋊S₃, C₃²×C₆, C₃²⋊₂C₈, S₃×C₁₂, C₄×C₃⋊S₃, C₃×C₃⋊Dic₃, C₃²×C₁₂, C₆×C₃⋊S₃, C₃⋊S₃⋊₃C₈, C₃×C₃²⋊₂C₈, C₁₂×C₃⋊S₃, C₃×C₃⋊S₃⋊₃C₈
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₈, C₂×C₄, C₁₂, C₂×C₆, C₂×C₈, C₂₄, C₂×C₁₂, C₃²⋊C₄, C₂×C₂₄, C₂×C₃²⋊C₄, C₃×C₃²⋊C₄, C₃⋊S₃⋊₃C₈, C₆×C₃²⋊C₄, C₃×C₃⋊S₃⋊₃C₈

Smallest permutation representation of C₃×C₃⋊S₃⋊₃C₈
►On 48 points

Generators in S₄₈

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

(2 14 48)(4 42 16)(6 10 44)(8 46 12)(17 27 40)(19 34 29)(21 31 36)(23 38 25)

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

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

(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)

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	4A	4B	4C	4D	6A	6B	6C	···	6H	6I	6J	6K	6L	8A	···	8H	12A	12B	12C	12D	12E	···	12P	12Q	12R	12S	12T	24A	···	24P
order	1	2	2	2	3	3	3	···	3	4	4	4	4	6	6	6	···	6	6	6	6	6	8	···	8	12	12	12	12	12	···	12	12	12	12	12	24	···	24
size	1	1	9	9	1	1	4	···	4	1	1	9	9	1	1	4	···	4	9	9	9	9	9	···	9	1	1	1	1	4	···	4	9	9	9	9	9	···	9

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	4	4	4	4	4	4
type	+	+	+										+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₈	C₁₂	C₁₂	C₂₄	C₃²⋊C₄	C₂×C₃²⋊C₄	C₃×C₃²⋊C₄	C₃⋊S₃⋊₃C₈	C₆×C₃²⋊C₄	C₃×C₃⋊S₃⋊₃C₈
kernel	C₃×C₃⋊S₃⋊₃C₈	C₃×C₃²⋊₂C₈	C₁₂×C₃⋊S₃	C₃⋊S₃⋊₃C₈	C₃²×C₁₂	C₆×C₃⋊S₃	C₃²⋊₂C₈	C₄×C₃⋊S₃	C₃×C₃⋊S₃	C₃×C₁₂	C₂×C₃⋊S₃	C₃⋊S₃	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	2	1	2	2	2	4	2	8	4	4	16	2	2	4	4	4	8

Matrix representation of C₃×C₃⋊S₃⋊₃C₈ ►in GL₅(𝔽₇₃)

8	0	0	0	0
0	64	0	0	0
0	0	64	0	0
0	0	0	64	0
0	0	0	0	64

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	8	0
0	0	0	0	64

1	0	0	0	0
0	8	0	0	0
0	0	64	0	0
0	0	0	8	0
0	0	0	0	64

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

22	0	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0
0	1	0	0	0

G:=sub<GL(5,GF(73))| [8,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,72,0],[22,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0] >;

C₃×C₃⋊S₃⋊₃C₈ in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes S_3\rtimes_3C_8

% in TeX

G:=Group("C3xC3:S3:3C8");

// GroupNames label

G:=SmallGroup(432,628);

// by ID

G=gap.SmallGroup(432,628);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,176,80,14117,362,18822,1203]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1,e*b*e^-1=b*c^-1,d*c*d=c^-1,e*c*e^-1=b^-1*c^-1,d*e=e*d>;

// generators/relations

G = C3×C3⋊S3⋊3C8 order 432 = 24·33

Direct product of C3 and C3⋊S3⋊3C8

G = C₃×C₃⋊S₃⋊₃C₈ order 432 = 2⁴·3³

Direct product of C₃ and C₃⋊S₃⋊₃C₈