C6^2.99C2^3 - GroupNames

G = C₆².₉₉C₂³ order 288 = 2⁵·3²

94^th non-split extension by C₆² of C₂³ acting via C₂³/C₂=C₂²

Aliases: C₆².₉₉C₂³, Dic₃²⋊₂₃C₂, C₂³.₃₀S₃², C₆².₅₃(C₂×C₄), (C₂²×C₆).₆₄D₆, (C₂×Dic₃).₈₁D₆, C₆.D₄.₂S₃, C₆.₆₆(D₄⋊₂S₃), C₆².C₂²⋊₁₇C₂, C₂.₄(D₆.₄D₆), (C₂×C₆²).₁₈C₂², C₃²⋊₁₀(C₄²⋊C₂), C₃⋊₃(C₂³.₁₆D₆), (C₆×Dic₃).₇₂C₂², C₂².₆(C₆.D₆), C₆.₃₆(S₃×C₂×C₄), (C₂×C₆).₂₀(C₄×S₃), (C₂×C₃⋊Dic₃)⋊₈C₄, C₂².₄₉(C₂×S₃²), (C₃×C₆).₇₅(C₄○D₄), C₃⋊Dic₃.₄₅(C₂×C₄), (C₃×C₆).₆₃(C₂²×C₄), C₂.₁₃(C₂×C₆.D₆), (C₂×C₆).₁₁₈(C₂²×S₃), (C₃×C₆.D₄).₆C₂, (C₂²×C₃⋊Dic₃).₅C₂, (C₂×C₃⋊Dic₃).₁₄₃C₂², SmallGroup(288,605)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₆².₉₉C₂³

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₆×Dic₃ — Dic₃² — C₆².₉₉C₂³

Lower central

C₃² — C₃×C₆ — C₆².₉₉C₂³

Upper central

C₁ — C₂² — C₂³

Generators and relations for C₆².₉₉C₂³
G = < a,b,c,d,e | a⁶=b⁶=e²=1, c²=a³, d²=a³b³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc^-1=b^-1, bd=db, be=eb, cd=dc, ece=b³c, ede=b³d >

Subgroups: 530 in 179 conjugacy classes, 60 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, C₂³, C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₃×C₆, C₃×C₆, C₃×C₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₂²×C₆, C₄²⋊C₂, C₃×Dic₃, C₃⋊Dic₃, C₆², C₆², C₆², C₄×Dic₃, Dic₃⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₂²×Dic₃, C₆×Dic₃, C₂×C₃⋊Dic₃, C₂×C₃⋊Dic₃, C₂×C₆², C₂³.₁₆D₆, Dic₃², C₆².C₂², C₃×C₆.D₄, C₂²×C₃⋊Dic₃, C₆².₉₉C₂³
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₆, C₂²×C₄, C₄○D₄, C₄×S₃, C₂²×S₃, C₄²⋊C₂, S₃², S₃×C₂×C₄, D₄⋊₂S₃, C₆.D₆, C₂×S₃², C₂³.₁₆D₆, D₆.₄D₆, C₂×C₆.D₆, C₆².₉₉C₂³

Smallest permutation representation of C₆².₉₉C₂³
►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 17 5 15 3 13)(2 18 6 16 4 14)(7 47 11 45 9 43)(8 48 12 46 10 44)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 38 33 40 35 42)(32 39 34 41 36 37)

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	···	4H	4I	4J	4K	4L	4M	4N	6A	···	6F	6G	···	6Q	12A	···	12H
order	1	2	2	2	2	2	3	3	3	4	···	4	4	4	4	4	4	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	2	2	2	2	4	6	···	6	9	9	9	9	18	18	2	···	2	4	···	4	12	···	12

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+		+	+	+			+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₄	S₃	D₆	D₆	C₄○D₄	C₄×S₃	S₃²	D₄⋊₂S₃	C₆.D₆	C₂×S₃²	D₆.₄D₆
kernel	C₆².₉₉C₂³	Dic₃²	C₆².C₂²	C₃×C₆.D₄	C₂²×C₃⋊Dic₃	C₂×C₃⋊Dic₃	C₆.D₄	C₂×Dic₃	C₂²×C₆	C₃×C₆	C₂×C₆	C₂³	C₆	C₂²	C₂²	C₂
# reps	1	2	2	2	1	8	2	4	2	4	8	1	4	2	1	4

Matrix representation of C₆².₉₉C₂³ ►in GL₆(𝔽₁₃)

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	1	12
0	0	0	0	1	0

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

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

0	1	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	0	5
0	0	0	0	5	0

12	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	0
0	0	0	0	0	1

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

C₆².₉₉C₂³ in GAP, Magma, Sage, TeX

C_6^2._{99}C_2^3

% in TeX

G:=Group("C6^2.99C2^3");

// GroupNames label

G:=SmallGroup(288,605);

// by ID

G=gap.SmallGroup(288,605);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,176,422,219,1356,9414]);

// Polycyclic

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

// generators/relations

G = C62.99C23 order 288 = 25·32

94th non-split extension by C62 of C23 acting via C23/C2=C22

G = C₆².₉₉C₂³ order 288 = 2⁵·3²

94^th non-split extension by C₆² of C₂³ acting via C₂³/C₂=C₂²