C6^2.94C2^3 - GroupNames

G = C₆².₉₄C₂³ order 288 = 2⁵·3²

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

Aliases: C₆².₉₄C₂³, Dic₃²:₁₉C₂, C₆²:₄(C₂xC₄), C₂³.₁₉S₃², C₃²:₁₃(C₄xD₄), C₆.₁₆₂(S₃xD₄), C₃²:₇D₄:₂C₄, (C₃xDic₃):₁₅D₄, (C₂²xC₆).₆₁D₆, C₆.₅₇(C₄oD₁₂), C₆.D₄:₁₅S₃, Dic₃:₈(C₃:D₄), (C₂xDic₃).₇₈D₆, (C₂²xDic₃):₄S₃, C₃:₆(Dic₃:₄D₄), C₆.D₁₂:₁₅C₂, C₆.₄₄(D₄:₂S₃), C₆².C₂²:₁₆C₂, C₂.₅(D₆.₃D₆), C₂²:₂(C₆.D₆), (C₂xC₆²).₁₃C₂², (C₆xDic₃).₇₁C₂², (C₂xC₆):₅(C₄xS₃), C₃:₄(C₄xC₃:D₄), C₆.₃₅(S₃xC₂xC₄), C₂.₅(S₃xC₃:D₄), C₃:Dic₃:₃(C₂xC₄), C₂².₄₆(C₂xS₃²), (Dic₃xC₂xC₆):₁₄C₂, C₆.₅₈(C₂xC₃:D₄), (C₃xC₆).₁₄₄(C₂xD₄), (C₃xC₆).₇₁(C₄oD₄), (C₂xC₆.D₆):₁₃C₂, (C₃xC₆).₆₀(C₂²xC₄), C₂.₁₂(C₂xC₆.D₆), (C₂xC₃²:₇D₄).₅C₂, (C₃xC₆.D₄):₁₇C₂, (C₂xC₆).₁₁₃(C₂²xS₃), (C₂²xC₃:S₃).₂₈C₂², (C₂xC₃:Dic₃).₅₈C₂², (C₂xC₃:S₃):₅(C₂xC₄), SmallGroup(288,600)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₆².₉₄C₂³

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — C₆xDic₃ — Dic₃² — C₆².₉₄C₂³

Lower central

C₃² — C₃xC₆ — C₆².₉₄C₂³

Upper central

C₁ — C₂² — C₂³

Generators and relations for C₆².₉₄C₂³
G = < a,b,c,d,e | a⁶=b⁶=e²=1, c²=b³, d²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc^-1=b^-1, bd=db, be=eb, cd=dc, ece=a³b³c, de=ed >

Subgroups: 802 in 215 conjugacy classes, 64 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₃:S₃, C₃xC₆, C₃xC₆, C₄xS₃, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₄xD₄, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₆², C₆², C₄xDic₃, Dic₃:C₄, D₆:C₄, C₆.D₄, C₃xC₂²:C₄, S₃xC₂xC₄, C₂²xDic₃, C₂xC₃:D₄, C₂²xC₁₂, C₆.D₆, C₆xDic₃, C₆xDic₃, C₂xC₃:Dic₃, C₃²:₇D₄, C₂²xC₃:S₃, C₂xC₆², Dic₃:₄D₄, C₄xC₃:D₄, Dic₃², C₆.D₁₂, C₆².C₂², C₃xC₆.D₄, C₂xC₆.D₆, Dic₃xC₂xC₆, C₂xC₃²:₇D₄, C₆².₉₄C₂³
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, D₆, C₂²xC₄, C₂xD₄, C₄oD₄, C₄xS₃, C₃:D₄, C₂²xS₃, C₄xD₄, S₃², S₃xC₂xC₄, C₄oD₁₂, S₃xD₄, D₄:₂S₃, C₂xC₃:D₄, C₆.D₆, C₂xS₃², Dic₃:₄D₄, C₄xC₃:D₄, D₆.₃D₆, C₂xC₆.D₆, S₃xC₃:D₄, C₆².₉₄C₂³

Smallest permutation representation of C₆².₉₄C₂³
►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 47 9 43 11 45)(8 48 10 44 12 46)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	4A	4B	4C	4D	4E	···	4J	4K	4L	6A	···	6J	6K	···	6S	12A	···	12H	12I	12J	12K	12L
order	1	2	2	2	2	2	2	2	3	3	3	4	4	4	4	4	···	4	4	4	6	···	6	6	···	6	12	···	12	12	12	12	12
size	1	1	1	1	2	2	18	18	2	2	4	3	3	3	3	6	···	6	18	18	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	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+		+	+	+	+	+					+	+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	S₃	S₃	D₄	D₆	D₆	C₄oD₄	C₃:D₄	C₄xS₃	C₄oD₁₂	S₃²	S₃xD₄	D₄:₂S₃	C₆.D₆	C₂xS₃²	D₆.₃D₆	S₃xC₃:D₄
kernel	C₆².₉₄C₂³	Dic₃²	C₆.D₁₂	C₆².C₂²	C₃xC₆.D₄	C₂xC₆.D₆	Dic₃xC₂xC₆	C₂xC₃²:₇D₄	C₃²:₇D₄	C₆.D₄	C₂²xDic₃	C₃xDic₃	C₂xDic₃	C₂²xC₆	C₃xC₆	Dic₃	C₂xC₆	C₆	C₂³	C₆	C₆	C₂²	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	8	1	1	2	4	2	2	4	8	4	1	1	1	2	1	2	2

Matrix representation of C₆².₉₄C₂³ ►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	0	12
0	0	0	0	1	1

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

0	5	0	0	0	0
5	0	0	0	0	0
0	0	7	10	0	0
0	0	8	6	0	0
0	0	0	0	8	0
0	0	0	0	0	8

5	0	0	0	0	0
0	5	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	0	5
0	0	0	0	5	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	6	3	0	0
0	0	10	7	0	0
0	0	0	0	12	0
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C₆².₉₄C₂³ in GAP, Magma, Sage, TeX

C_6^2._{94}C_2^3

% in TeX

G:=Group("C6^2.94C2^3");

// GroupNames label

G:=SmallGroup(288,600);

// by ID

G=gap.SmallGroup(288,600);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,1356,9414]);

// Polycyclic

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

// generators/relations

G = C62.94C23 order 288 = 25·32

89th non-split extension by C62 of C23 acting via C23/C2=C22

G = C₆².₉₄C₂³ order 288 = 2⁵·3²

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