C2xC3^3:4C8 - GroupNames

G = C₂×C₃³⋊₄C₈ order 432 = 2⁴·3³

Direct product of C₂ and C₃³⋊₄C₈

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

Aliases: C₂×C₃³⋊₄C₈, C₆².₁₂Dic₃, C₆⋊(C₃²⋊₂C₈), (C₃²×C₆)⋊₄C₈, C₃³⋊₁₅(C₂×C₈), (C₃×C₆²).₃C₄, C₃⋊Dic₃.₄₂D₆, C₃⋊Dic₃.₇Dic₃, C₂².₂(C₃³⋊C₄), (C₃×C₆)⋊₄(C₃⋊C₈), C₃²⋊₈(C₂×C₃⋊C₈), C₃⋊₂(C₂×C₃²⋊₂C₈), C₆.₁₂(C₂×C₃²⋊C₄), (C₂×C₆).₄(C₃²⋊C₄), C₂.₃(C₂×C₃³⋊C₄), (C₆×C₃⋊Dic₃).₁₄C₂, (C₃×C₃⋊Dic₃).₁₀C₄, (C₂×C₃⋊Dic₃).₁₀S₃, (C₃²×C₆).₁₉(C₂×C₄), (C₃×C₆).₂₆(C₂×Dic₃), (C₃×C₃⋊Dic₃).₅₀C₂², SmallGroup(432,639)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×C₃³⋊₄C₈

Chief series

C₁ — C₃ — C₃³ — C₃²×C₆ — C₃×C₃⋊Dic₃ — C₃³⋊₄C₈ — C₂×C₃³⋊₄C₈

Lower central

C₃³ — C₂×C₃³⋊₄C₈

Upper central

C₁ — C₂²

Generators and relations for C₂×C₃³⋊₄C₈
G = < a,b,c,d,e | a²=b³=c³=d³=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bc^-1, cd=dc, ece^-1=b^-1c^-1, ede^-1=d^-1 >

Subgroups: 392 in 96 conjugacy classes, 29 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₈, C₂×C₄, C₃², C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×C₈, C₃×C₆, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂×Dic₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₆², C₆², C₂×C₃⋊C₈, C₃²×C₆, C₃²×C₆, C₃²⋊₂C₈, C₆×Dic₃, C₂×C₃⋊Dic₃, C₃×C₃⋊Dic₃, C₃×C₆², C₂×C₃²⋊₂C₈, C₃³⋊₄C₈, C₆×C₃⋊Dic₃, C₂×C₃³⋊₄C₈
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, Dic₃, D₆, C₂×C₈, C₃⋊C₈, C₂×Dic₃, C₃²⋊C₄, C₂×C₃⋊C₈, C₃²⋊₂C₈, C₂×C₃²⋊C₄, C₃³⋊C₄, C₂×C₃²⋊₂C₈, C₃³⋊₄C₈, C₂×C₃³⋊C₄, C₂×C₃³⋊₄C₈

Smallest permutation representation of C₂×C₃³⋊₄C₈
►On 48 points

Generators in S₄₈

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

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

(1 17 13)(3 15 19)(5 21 9)(7 11 23)(25 33 42)(27 44 35)(29 37 46)(31 48 39)

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	···	3G	4A	4B	4C	4D	6A	6B	6C	6D	···	6U	8A	···	8H	12A	12B	12C	12D
order	1	2	2	2	3	3	···	3	4	4	4	4	6	6	6	6	···	6	8	···	8	12	12	12	12
size	1	1	1	1	2	4	···	4	9	9	9	9	2	2	2	4	···	4	27	···	27	18	18	18	18

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+				+	-	+	-		+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₈	S₃	Dic₃	D₆	Dic₃	C₃⋊C₈	C₃²⋊C₄	C₃²⋊₂C₈	C₂×C₃²⋊C₄	C₃³⋊C₄	C₃³⋊₄C₈	C₂×C₃³⋊C₄
kernel	C₂×C₃³⋊₄C₈	C₃³⋊₄C₈	C₆×C₃⋊Dic₃	C₃×C₃⋊Dic₃	C₃×C₆²	C₃²×C₆	C₂×C₃⋊Dic₃	C₃⋊Dic₃	C₃⋊Dic₃	C₆²	C₃×C₆	C₂×C₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	2	1	2	2	8	1	1	1	1	4	2	4	2	4	8	4

Matrix representation of C₂×C₃³⋊₄C₈ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	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	64	0	0	19
0	0	0	8	19	0
0	0	0	0	64	0
0	0	0	0	0	8

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	48	48
0	0	0	64	6	67
0	0	0	0	1	0
0	0	0	0	0	1

0	72	0	0	0	0
1	72	0	0	0	0
0	0	8	0	54	54
0	0	0	8	19	54
0	0	0	0	64	0
0	0	0	0	0	64

0	22	0	0	0	0
22	0	0	0	0	0
0	0	67	6	0	0
0	0	67	67	0	0
0	0	0	63	6	67
0	0	10	0	6	6

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,19,64,0,0,0,19,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,48,6,1,0,0,0,48,67,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,54,19,64,0,0,0,54,54,0,64],[0,22,0,0,0,0,22,0,0,0,0,0,0,0,67,67,0,10,0,0,6,67,63,0,0,0,0,0,6,6,0,0,0,0,67,6] >;

C₂×C₃³⋊₄C₈ in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_4C_8

% in TeX

G:=Group("C2xC3^3:4C8");

// GroupNames label

G:=SmallGroup(432,639);

// by ID

G=gap.SmallGroup(432,639);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,2804,298,2693,1027,14118]);

// Polycyclic

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

// generators/relations

G = C2×C33⋊4C8 order 432 = 24·33

Direct product of C2 and C33⋊4C8

G = C₂×C₃³⋊₄C₈ order 432 = 2⁴·3³

Direct product of C₂ and C₃³⋊₄C₈