C6^2.74C2^3 - GroupNames

G = C₆².₇₄C₂³ order 288 = 2⁵·3²

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

Aliases: C₆².₇₄C₂³, D₆:₇(C₄xS₃), Dic₃²:₂C₂, C₃²:₁₁(C₄xD₄), C₃:D₁₂:₃C₄, Dic₃:₂(C₄xS₃), C₆.₁₄₂(S₃xD₄), Dic₃:C₄:₂₁S₃, D₆:Dic₃:₂₂C₂, (C₃xDic₃):₁₄D₄, (C₂xC₁₂).₂₆₁D₆, C₃:₄(Dic₃:₅D₄), C₆.₁₄(C₄oD₁₂), Dic₃:₇(C₃:D₄), (C₂xDic₃).₇₅D₆, (C₂²xS₃).₆₆D₆, C₆.₁₁D₁₂:₁₆C₂, (C₆xC₁₂).₂₃₅C₂², C₆.₁₅(Q₈:₃S₃), C₂.₅(D₆.₆D₆), (C₆xDic₃).₆₈C₂², (S₃xC₂xC₄):₁₁S₃, C₂.₂₂(C₄xS₃²), (C₂xC₄).₄₉S₃², C₃:₂(C₄xC₃:D₄), C₆.₂₁(S₃xC₂xC₄), (S₃xC₂xC₁₂):₁₉C₂, C₂.₃(S₃xC₃:D₄), (S₃xC₆):₁₃(C₂xC₄), C₂².₄₂(C₂xS₃²), C₆.₃₆(C₂xC₃:D₄), (C₃xDic₃):₄(C₂xC₄), (C₃xC₆).₁₀₁(C₂xD₄), (S₃xC₂xC₆).₇₉C₂², (C₃xDic₃:C₄):₂₁C₂, (C₂xC₃:D₁₂).₈C₂, (C₃xC₆).₄₄(C₄oD₄), (C₂xC₆.D₆):₁₁C₂, (C₂xC₆).₉₃(C₂²xS₃), (C₃xC₆).₂₀(C₂²xC₄), (C₂²xC₃:S₃).₂₀C₂², (C₂xC₃:Dic₃).₅₁C₂², (C₂xC₃:S₃):₃(C₂xC₄), SmallGroup(288,552)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₆².₇₄C₂³

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₆² — S₃xC₂xC₆ — C₂xC₃:D₁₂ — C₆².₇₄C₂³

Lower central

C₃² — C₃xC₆ — C₆².₇₄C₂³

Upper central

C₁ — C₂² — C₂xC₄

Generators and relations for C₆².₇₄C₂³
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=b³c, ce=ec, ede^-1=b³d >

Subgroups: 794 in 205 conjugacy classes, 62 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₄xD₄, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₄xDic₃, Dic₃:C₄, D₆:C₄, C₆.D₄, C₃xC₄:C₄, S₃xC₂xC₄, S₃xC₂xC₄, C₂xD₁₂, C₂xC₃:D₄, C₂²xC₁₂, C₆.D₆, C₃:D₁₂, S₃xC₁₂, C₆xDic₃, C₂xC₃:Dic₃, C₆xC₁₂, S₃xC₂xC₆, C₂²xC₃:S₃, Dic₃:₅D₄, C₄xC₃:D₄, Dic₃², D₆:Dic₃, C₃xDic₃:C₄, C₆.₁₁D₁₂, C₂xC₆.D₆, C₂xC₃:D₁₂, S₃xC₂xC₁₂, C₆².₇₄C₂³
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, D₆, C₂²xC₄, C₂xD₄, C₄oD₄, C₄xS₃, C₃:D₄, C₂²xS₃, C₄xD₄, S₃², S₃xC₂xC₄, C₄oD₁₂, S₃xD₄, Q₈:₃S₃, C₂xC₃:D₄, C₂xS₃², Dic₃:₅D₄, C₄xC₃:D₄, D₆.₆D₆, C₄xS₃², S₃xC₃:D₄, C₆².₇₄C₂³

Smallest permutation representation of C₆².₇₄C₂³
►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 36)(2 31)(3 32)(4 33)(5 34)(6 35)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(13 38)(14 39)(15 40)(16 41)(17 42)(18 37)(25 46)(26 47)(27 48)(28 43)(29 44)(30 45)

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	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	2	3	3	3	4	4	4	4	4	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	18	18	2	2	4	2	2	3	3	3	3	6	6	6	6	18	18	2	···	2	4	4	4	6	6	6	6	2	2	2	2	4	···	4	6	6	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	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	D₆	C₄oD₄	C₄xS₃	C₃:D₄	C₄xS₃	C₄oD₁₂	S₃²	S₃xD₄	Q₈:₃S₃	C₂xS₃²	D₆.₆D₆	C₄xS₃²	S₃xC₃:D₄
kernel	C₆².₇₄C₂³	Dic₃²	D₆:Dic₃	C₃xDic₃:C₄	C₆.₁₁D₁₂	C₂xC₆.D₆	C₂xC₃:D₁₂	S₃xC₂xC₁₂	C₃:D₁₂	Dic₃:C₄	S₃xC₂xC₄	C₃xDic₃	C₂xDic₃	C₂xC₁₂	C₂²xS₃	C₃xC₆	Dic₃	Dic₃	D₆	C₆	C₂xC₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	1	1	1	1	1	1	1	8	1	1	2	3	2	1	2	4	4	4	4	1	1	1	1	2	2	2

Matrix representation of C₆².₇₄C₂³ ►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	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	1	0
0	0	0	0	0	1

0	12	0	0	0	0
12	0	0	0	0	0
0	0	12	2	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	11	0	0
0	0	1	12	0	0
0	0	0	0	0	12
0	0	0	0	12	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	2	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],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

C₆².₇₄C₂³ in GAP, Magma, Sage, TeX

C_6^2._{74}C_2^3

% in TeX

G:=Group("C6^2.74C2^3");

// GroupNames label

G:=SmallGroup(288,552);

// by ID

G=gap.SmallGroup(288,552);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,219,58,1356,9414]);

// 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=b^3*c,c*e=e*c,e*d*e^-1=b^3*d>;

// generators/relations

G = C62.74C23 order 288 = 25·32

69th non-split extension by C62 of C23 acting via C23/C2=C22

G = C₆².₇₄C₂³ order 288 = 2⁵·3²

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