C2xD6.4D6 - GroupNames

G = C₂×D₆.₄D₆ order 288 = 2⁵·3²

Direct product of C₂ and D₆.₄D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₂×D₆.₄D₆, C₆².₁₄₃C₂³, C₃⋊D₄⋊₈D₆, C₂³.₃₇S₃², C₆⋊₄(D₄⋊₂S₃), (C₃×C₆).₃₀C₂⁴, C₆.₃₀(S₃×C₂³), (S₃×C₆).₁₇C₂³, (C₂×Dic₃).₈₉D₆, (S₃×Dic₃)⋊₉C₂², D₆.₁₇(C₂²×S₃), (C₂²×S₃).₅₇D₆, (C₂²×C₆).₁₀₀D₆, D₆⋊S₃⋊₁₇C₂², C₃²⋊₂Q₈⋊₁₆C₂², C₃⋊Dic₃.₄₃C₂³, (C₂×C₆²).₇₈C₂², (C₃×Dic₃).₁₈C₂³, Dic₃.₁₆(C₂²×S₃), (C₆×Dic₃).₄₉C₂², (C₆×C₃⋊D₄)⋊₅C₂, (C₂×S₃×Dic₃)⋊₅C₂, (C₃×C₆)⋊₆(C₄○D₄), C₃⋊₅(C₂×D₄⋊₂S₃), (C₂×C₃⋊D₄)⋊₁₀S₃, C₂².₁₄(C₂×S₃²), C₂.₃₁(C₂²×S₃²), C₃²⋊₁₂(C₂×C₄○D₄), (S₃×C₂×C₆).₆₇C₂², (C₂×D₆⋊S₃)⋊₁₆C₂, (C₂×C₃²⋊₂Q₈)⋊₁₈C₂, (C₃×C₃⋊D₄)⋊₁₁C₂², (C₂×C₆).₁₅₈(C₂²×S₃), (C₂²×C₃⋊Dic₃)⋊₁₃C₂, (C₂×C₃⋊Dic₃)⋊₂₄C₂², SmallGroup(288,971)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₂×D₆.₄D₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — S₃×C₆ — S₃×Dic₃ — C₂×S₃×Dic₃ — C₂×D₆.₄D₆

Lower central

C₃² — C₃×C₆ — C₂×D₆.₄D₆

Upper central

C₁ — C₂² — C₂³

Generators and relations for C₂×D₆.₄D₆
G = < a,b,c,d,e | a²=b⁶=c²=1, d⁶=e²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=ebe^-1=b^-1, dcd^-1=bc, ece^-1=b⁴c, ede^-1=d⁵ >

Subgroups: 1106 in 355 conjugacy classes, 116 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₆, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂×C₄○D₄, C₃×Dic₃, C₃⋊Dic₃, S₃×C₆, S₃×C₆, C₆², C₆², C₆², C₂×Dic₆, S₃×C₂×C₄, D₄⋊₂S₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂×C₃⋊D₄, C₆×D₄, S₃×Dic₃, D₆⋊S₃, C₃²⋊₂Q₈, C₆×Dic₃, C₃×C₃⋊D₄, C₂×C₃⋊Dic₃, C₂×C₃⋊Dic₃, S₃×C₂×C₆, C₂×C₆², C₂×D₄⋊₂S₃, C₂×S₃×Dic₃, D₆.₄D₆, C₂×D₆⋊S₃, C₂×C₃²⋊₂Q₈, C₆×C₃⋊D₄, C₂²×C₃⋊Dic₃, C₂×D₆.₄D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, S₃², D₄⋊₂S₃, S₃×C₂³, C₂×S₃², C₂×D₄⋊₂S₃, D₆.₄D₆, C₂²×S₃², C₂×D₆.₄D₆

Smallest permutation representation of C₂×D₆.₄D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	···	6F	6G	···	6Q	6R	6S	6T	6U	12A	12B	12C	12D
order	1	2	2	2	2	2	2	2	2	2	3	3	3	4	4	4	4	4	4	4	4	4	4	6	···	6	6	···	6	6	6	6	6	12	12	12	12
size	1	1	1	1	2	2	6	6	6	6	2	2	4	6	6	6	6	9	9	9	9	18	18	2	···	2	4	···	4	12	12	12	12	12	12	12	12

48 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₆	D₆	C₄○D₄	S₃²	D₄⋊₂S₃	C₂×S₃²	D₆.₄D₆
kernel	C₂×D₆.₄D₆	C₂×S₃×Dic₃	D₆.₄D₆	C₂×D₆⋊S₃	C₂×C₃²⋊₂Q₈	C₆×C₃⋊D₄	C₂²×C₃⋊Dic₃	C₂×C₃⋊D₄	C₂×Dic₃	C₃⋊D₄	C₂²×S₃	C₂²×C₆	C₃×C₆	C₂³	C₆	C₂²	C₂
# reps	1	2	8	1	1	2	1	2	2	8	2	2	4	1	4	3	4

Matrix representation of C₂×D₆.₄D₆ ►in GL₆(𝔽₁₃)

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	12	0
0	0	0	0	0	12

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	12
0	0	0	0	1	12

0	12	0	0	0	0
12	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	1
0	0	0	0	0	1

0	1	0	0	0	0
12	0	0	0	0	0
0	0	0	12	0	0
0	0	1	12	0	0
0	0	0	0	0	1
0	0	0	0	1	0

8	0	0	0	0	0
0	8	0	0	0	0
0	0	1	12	0	0
0	0	0	12	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂×D₆.₄D₆ in GAP, Magma, Sage, TeX

C_2\times D_6._4D_6

% in TeX

G:=Group("C2xD6.4D6");

// GroupNames label

G:=SmallGroup(288,971);

// by ID

G=gap.SmallGroup(288,971);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,346,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^6=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^4*c,e*d*e^-1=d^5>;

// generators/relations

G = C2×D6.4D6 order 288 = 25·32

Direct product of C2 and D6.4D6

G = C₂×D₆.₄D₆ order 288 = 2⁵·3²

Direct product of C₂ and D₆.₄D₆