C6^2.20C2^3 - GroupNames

G = C₆².₂₀C₂³ order 288 = 2⁵·3²

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

Aliases: C₆².₂₀C₂³, Dic₃:C₄:₁S₃, (S₃xC₆).₂₈D₄, C₆.₁₃₄(S₃xD₄), D₆:Dic₃:₂₉C₂, (C₂xC₁₂).₂₅₇D₆, (C₂xDic₃).₇D₆, C₆.₁₈(C₄oD₁₂), C₃:₆(D₆.D₄), D₆.₁₁(C₃:D₄), C₆.₅(Q₈:₃S₃), C₆².C₂²:₂C₂, (C₂²xS₃).₅₉D₆, C₆.D₁₂:₂₀C₂, C₆.₁₁D₁₂:₁₃C₂, (C₆xC₁₂).₂₁₁C₂², C₂.₆(D₆.D₆), C₂.₈(D₆.₆D₆), C₃:₁(C₂³.₂₈D₆), C₃²:₂(C₂².D₄), (C₆xDic₃).₁₀₆C₂², (S₃xC₂xC₄):₇S₃, (C₂xC₄).₃₉S₃², (S₃xC₂xC₁₂):₁₅C₂, C₂.₉(S₃xC₃:D₄), C₂².₇₈(C₂xS₃²), (C₃xC₆).₇₈(C₂xD₄), C₆.₂₇(C₂xC₃:D₄), (C₃xDic₃:C₄):₃C₂, (C₃xC₆).₉(C₄oD₄), (S₃xC₂xC₆).₆₉C₂², (C₂xC₃:D₁₂).₁C₂, (C₂xC₆).₃₉(C₂²xS₃), (C₂²xC₃:S₃).₈C₂², (C₂xC₃:Dic₃).₂₀C₂², SmallGroup(288,498)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — C₆².₂₀C₂³

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — S₃xC₂xC₆ — D₆:Dic₃ — C₆².₂₀C₂³

Lower central

C₃² — C₆² — C₆².₂₀C₂³

Upper central

C₁ — C₂² — C₂xC₄

Generators and relations for C₆².₂₀C₂³
G = < a,b,c,d,e | a⁶=b⁶=c²=1, d²=b³, e²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=b^-1, bd=db, be=eb, dcd^-1=a³c, ce=ec, ede^-1=b³d >

Subgroups: 730 in 173 conjugacy classes, 48 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₃², Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂².D₄, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₆², Dic₃:C₄, Dic₃:C₄, D₆:C₄, C₆.D₄, C₃xC₄:C₄, S₃xC₂xC₄, C₂xD₁₂, C₂xC₃:D₄, C₂²xC₁₂, C₃:D₁₂, S₃xC₁₂, C₆xDic₃, C₂xC₃:Dic₃, C₆xC₁₂, S₃xC₂xC₆, C₂²xC₃:S₃, D₆.D₄, C₂³.₂₈D₆, D₆:Dic₃, C₆.D₁₂, C₆².C₂², C₃xDic₃:C₄, C₆.₁₁D₁₂, C₂xC₃:D₁₂, S₃xC₂xC₁₂, C₆².₂₀C₂³
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₄oD₄, C₃:D₄, C₂²xS₃, C₂².D₄, S₃², C₄oD₁₂, S₃xD₄, Q₈:₃S₃, C₂xC₃:D₄, C₂xS₃², D₆.D₄, C₂³.₂₈D₆, D₆.D₆, D₆.₆D₆, S₃xC₃:D₄, C₆².₂₀C₂³

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

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	6A	···	6F	6G	6H	6I	6J	6K	6L	6M	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N	12O	12P	12Q	12R
order	1	2	2	2	2	2	2	3	3	3	4	4	4	4	4	4	4	6	···	6	6	6	6	6	6	6	6	12	12	12	12	12	···	12	12	12	12	12	12	12	12	12
size	1	1	1	1	6	6	36	2	2	4	2	2	6	6	12	12	36	2	···	2	4	4	4	6	6	6	6	2	2	2	2	4	···	4	6	6	6	6	12	12	12	12

48 irreducible representations

dim	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	S₃	S₃	D₄	D₆	D₆	D₆	C₄oD₄	C₃:D₄	C₄oD₁₂	S₃²	S₃xD₄	Q₈:₃S₃	C₂xS₃²	D₆.D₆	D₆.₆D₆	S₃xC₃:D₄
kernel	C₆².₂₀C₂³	D₆:Dic₃	C₆.D₁₂	C₆².C₂²	C₃xDic₃:C₄	C₆.₁₁D₁₂	C₂xC₃:D₁₂	S₃xC₂xC₁₂	Dic₃:C₄	S₃xC₂xC₄	S₃xC₆	C₂xDic₃	C₂xC₁₂	C₂²xS₃	C₃xC₆	D₆	C₆	C₂xC₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	1	2	3	2	1	4	4	12	1	1	1	1	2	2	2

Matrix representation of C₆².₂₀C₂³ ►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	1	12
0	0	0	0	1	0

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

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	12	0	0
0	0	12	0	0	0
0	0	0	0	2	9
0	0	0	0	4	11

6	3	0	0	0	0
5	7	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	3	3
0	0	0	0	6	10

7	10	0	0	0	0
3	6	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	5	0
0	0	0	0	0	5

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[6,5,0,0,0,0,3,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,3,10],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

C₆².₂₀C₂³ in GAP, Magma, Sage, TeX

C_6^2._{20}C_2^3

% in TeX

G:=Group("C6^2.20C2^3");

// GroupNames label

G:=SmallGroup(288,498);

// by ID

G=gap.SmallGroup(288,498);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,58,1356,9414]);

// Polycyclic

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

// generators/relations

G = C62.20C23 order 288 = 25·32

15th non-split extension by C62 of C23 acting via C23/C2=C22

G = C₆².₂₀C₂³ order 288 = 2⁵·3²

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