C2xC3:S3:3C8 - GroupNames

G = C₂×C₃⋊S₃⋊₃C₈ order 288 = 2⁵·3²

Direct product of C₂ and C₃⋊S₃⋊₃C₈

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

Aliases: C₂×C₃⋊S₃⋊₃C₈, (C₆×C₁₂).₉C₄, C₃²⋊₃(C₂²×C₈), C₆².₁₃(C₂×C₄), C₃²⋊₂C₈⋊₁₀C₂², C₃⋊Dic₃.₂₉C₂³, (C₂×C₃⋊S₃)⋊₇C₈, C₃⋊S₃⋊₄(C₂×C₈), (C₃×C₆)⋊₂(C₂×C₈), (C₄×C₃⋊S₃).₁₅C₄, C₄.₂₁(C₂×C₃²⋊C₄), (C₃×C₁₂).₁₈(C₂×C₄), C₂.₁(C₂²×C₃²⋊C₄), (C₂×C₃²⋊₂C₈)⋊₁₀C₂, (C₄×C₃⋊S₃).₉₅C₂², (C₂²×C₃⋊S₃).₁₄C₄, (C₂×C₄).₁₁(C₃²⋊C₄), C₃⋊Dic₃.₅₀(C₂×C₄), (C₃×C₆).₂₄(C₂²×C₄), C₂².₁₅(C₂×C₃²⋊C₄), (C₂×C₃⋊Dic₃).₁₇₂C₂², (C₂×C₄×C₃⋊S₃).₂₅C₂, (C₂×C₃⋊S₃).₄₄(C₂×C₄), SmallGroup(288,929)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₃⋊S₃⋊₃C₈

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₃²⋊₂C₈ — C₂×C₃²⋊₂C₈ — C₂×C₃⋊S₃⋊₃C₈

Lower central

C₃² — C₂×C₃⋊S₃⋊₃C₈

Upper central

C₁ — C₂×C₄

Generators and relations for C₂×C₃⋊S₃⋊₃C₈
G = < a,b,c,d,e | a²=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: 576 in 130 conjugacy classes, 46 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, C₂×C₄, C₂³, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₈, C₂²×C₄, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂²×C₈, C₃⋊Dic₃, C₃×C₁₂, C₂×C₃⋊S₃, C₆², S₃×C₂×C₄, C₃²⋊₂C₈, C₄×C₃⋊S₃, C₂×C₃⋊Dic₃, C₆×C₁₂, C₂²×C₃⋊S₃, C₃⋊S₃⋊₃C₈, C₂×C₃²⋊₂C₈, C₂×C₄×C₃⋊S₃, C₂×C₃⋊S₃⋊₃C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, C₂²×C₈, C₃²⋊C₄, C₂×C₃²⋊C₄, C₃⋊S₃⋊₃C₈, C₂²×C₃²⋊C₄, C₂×C₃⋊S₃⋊₃C₈

Smallest permutation representation of C₂×C₃⋊S₃⋊₃C₈
►On 48 points

Generators in S₄₈

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

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

(2 22 32)(4 26 24)(6 18 28)(8 30 20)(10 42 36)(12 38 44)(14 46 40)(16 34 48)

(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	8A	···	8P	12A	···	12H
order	1	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	4	4	6	···	6	8	···	8	12	···	12
size	1	1	1	1	9	9	9	9	4	4	1	1	1	1	9	9	9	9	4	···	4	9	···	9	4	···	4

48 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4	4	4
type	+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₈	C₃²⋊C₄	C₂×C₃²⋊C₄	C₂×C₃²⋊C₄	C₃⋊S₃⋊₃C₈
kernel	C₂×C₃⋊S₃⋊₃C₈	C₃⋊S₃⋊₃C₈	C₂×C₃²⋊₂C₈	C₂×C₄×C₃⋊S₃	C₄×C₃⋊S₃	C₆×C₁₂	C₂²×C₃⋊S₃	C₂×C₃⋊S₃	C₂×C₄	C₄	C₂²	C₂
# reps	1	4	2	1	4	2	2	16	2	4	2	8

Matrix representation of C₂×C₃⋊S₃⋊₃C₈ ►in GL₅(𝔽₇₃)

72	0	0	0	0
0	72	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	72

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

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

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

72	0	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	27	0	0
0	27	0	0	0

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,0,0,0,0,27,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0] >;

C₂×C₃⋊S₃⋊₃C₈ in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes S_3\rtimes_3C_8

% in TeX

G:=Group("C2xC3:S3:3C8");

// GroupNames label

G:=SmallGroup(288,929);

// by ID

G=gap.SmallGroup(288,929);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,100,80,9413,362,12550,1203]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=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 = C2×C3⋊S3⋊3C8 order 288 = 25·32

Direct product of C2 and C3⋊S3⋊3C8

G = C₂×C₃⋊S₃⋊₃C₈ order 288 = 2⁵·3²

Direct product of C₂ and C₃⋊S₃⋊₃C₈