C6^2.6C2^3 - GroupNames

G = C₆².₆C₂³ order 288 = 2⁵·3²

1^st non-split extension by C₆² of C₂³ acting via C₂³/C₂=C₂²

Aliases: C₆².₆C₂³, Dic₃²⋊₉C₂, (C₄×Dic₃)⋊₈S₃, C₆.D₆⋊₂C₄, Dic₃⋊C₄⋊₂₀S₃, C₆.₁(C₄○D₁₂), (C₂×C₁₂).₁₈₄D₆, Dic₃.₉(C₄×S₃), C₃⋊₂(C₄²⋊₂S₃), (Dic₃×C₁₂)⋊₁₈C₂, C₆.₁(Q₈⋊₃S₃), Dic₃⋊Dic₃⋊₂₁C₂, C₆.₂₉(D₄⋊₂S₃), (C₆×C₁₂).₂₀₈C₂², (C₂×Dic₃).₁₀₈D₆, C₃²⋊₂(C₄²⋊C₂), C₆.D₁₂.₅C₂, C₂.₁(D₆.₆D₆), C₆.₁₁D₁₂.₇C₂, C₂.₁(D₆.₃D₆), (C₆×Dic₃).₅₀C₂², C₂.₉(C₄×S₃²), C₆.₇(S₃×C₂×C₄), (C₂×C₄).₃₇S₃², C₂².₁₅(C₂×S₃²), C₃⋊₁(C₄⋊C₄⋊₇S₃), (C₃×Dic₃⋊C₄)⋊₁C₂, (C₃×C₆).₁(C₄○D₄), (C₃×C₆).₇(C₂²×C₄), (C₂×C₆).₂₅(C₂²×S₃), (C₃×Dic₃).₆(C₂×C₄), (C₂×C₆.D₆).₁C₂, (C₂²×C₃⋊S₃).₆C₂², (C₂×C₃⋊Dic₃).₉C₂², (C₂×C₃⋊S₃).₁₆(C₂×C₄), SmallGroup(288,484)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₆².₆C₂³

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₆×Dic₃ — Dic₃² — C₆².₆C₂³

Lower central

C₃² — C₃×C₆ — C₆².₆C₂³

Upper central

C₁ — C₂² — C₂×C₄

Generators and relations for C₆².₆C₂³
G = < a,b,c,d,e | a⁶=b⁶=1, c²=e²=b³, d²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc^-1=b^-1, bd=db, be=eb, cd=dc, ece^-1=a³c, de=ed >

Subgroups: 634 in 169 conjugacy classes, 58 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₃⋊S₃, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₄²⋊C₂, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², C₄×Dic₃, C₄×Dic₃, Dic₃⋊C₄, Dic₃⋊C₄, C₄⋊Dic₃, D₆⋊C₄, C₄×C₁₂, C₃×C₄⋊C₄, S₃×C₂×C₄, C₆.D₆, C₆×Dic₃, C₂×C₃⋊Dic₃, C₆×C₁₂, C₂²×C₃⋊S₃, C₄²⋊₂S₃, C₄⋊C₄⋊₇S₃, Dic₃², C₆.D₁₂, Dic₃⋊Dic₃, Dic₃×C₁₂, C₃×Dic₃⋊C₄, C₆.₁₁D₁₂, C₂×C₆.D₆, C₆².₆C₂³
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₆, C₂²×C₄, C₄○D₄, C₄×S₃, C₂²×S₃, C₄²⋊C₂, S₃², S₃×C₂×C₄, C₄○D₁₂, D₄⋊₂S₃, Q₈⋊₃S₃, C₂×S₃², C₄²⋊₂S₃, C₄⋊C₄⋊₇S₃, D₆.₆D₆, C₄×S₃², D₆.₃D₆, C₆².₆C₂³

Smallest permutation representation of C₆².₆C₂³
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	···	4L	4M	4N	6A	···	6F	6G	6H	6I	12A	12B	12C	12D	12E	···	12J	12K	···	12R	12S	12T	12U	12V
order	1	2	2	2	2	2	3	3	3	4	4	4	4	4	4	4	···	4	4	4	6	···	6	6	6	6	12	12	12	12	12	···	12	12	···	12	12	12	12	12
size	1	1	1	1	18	18	2	2	4	2	2	3	3	3	3	6	···	6	18	18	2	···	2	4	4	4	2	2	2	2	4	···	4	6	···	6	12	12	12	12

54 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+		+	+	+	+				+	-	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	S₃	S₃	D₆	D₆	C₄○D₄	C₄×S₃	C₄○D₁₂	S₃²	D₄⋊₂S₃	Q₈⋊₃S₃	C₂×S₃²	D₆.₆D₆	C₄×S₃²	D₆.₃D₆
kernel	C₆².₆C₂³	Dic₃²	C₆.D₁₂	Dic₃⋊Dic₃	Dic₃×C₁₂	C₃×Dic₃⋊C₄	C₆.₁₁D₁₂	C₂×C₆.D₆	C₆.D₆	C₄×Dic₃	Dic₃⋊C₄	C₂×Dic₃	C₂×C₁₂	C₃×C₆	Dic₃	C₆	C₂×C₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	1	1	1	1	1	1	1	8	1	1	4	2	4	8	8	1	1	1	1	2	2	2

Matrix representation of C₆².₆C₂³ ►in GL₆(𝔽₁₃)

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

1	12	0	0	0	0
1	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	0	12

0	8	0	0	0	0
8	0	0	0	0	0
0	0	5	3	0	0
0	0	5	8	0	0
0	0	0	0	8	0
0	0	0	0	0	8

1	0	0	0	0	0
0	1	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	12	1
0	0	0	0	0	1

5	0	0	0	0	0
0	5	0	0	0	0
0	0	1	11	0	0
0	0	0	12	0	0
0	0	0	0	8	0
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,5,5,0,0,0,0,3,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C₆².₆C₂³ in GAP, Magma, Sage, TeX

C_6^2._6C_2^3

% in TeX

G:=Group("C6^2.6C2^3");

// GroupNames label

G:=SmallGroup(288,484);

// by ID

G=gap.SmallGroup(288,484);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,254,100,1356,9414]);

// Polycyclic

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

// generators/relations

G = C62.6C23 order 288 = 25·32

1st non-split extension by C62 of C23 acting via C23/C2=C22

G = C₆².₆C₂³ order 288 = 2⁵·3²

1^st non-split extension by C₆² of C₂³ acting via C₂³/C₂=C₂²