C3:C8.22D6 - GroupNames

G = C₃:C₈.₂₂D₆ order 288 = 2⁵·3²

11^st non-split extension by C₃:C₈ of D₆ acting via D₆/S₃=C₂

Aliases: C₃:C₈.₂₂D₆, C₃:₃(C₈oD₁₂), C₁₂.₄₂(C₄xS₃), C₃²:₆(C₈oD₄), C₁₂:S₃.₂C₄, C₃:₃(D₁₂.C₄), (C₂xC₁₂).₂₉₀D₆, C₄.Dic₃:₁₁S₃, C₆².₄₇(C₂xC₄), C₃²:₇D₄.₂C₄, C₃²:₄Q₈.₂C₄, (C₆xC₁₂).₆₈C₂², C₄.₄(C₆.D₆), C₁₂.₃₁D₆:₁₁C₂, C₁₂.₂₉D₆:₁₀C₂, (C₃xC₁₂).₁₄₆C₂³, C₁₂.₁₄₅(C₂²xS₃), C₁₂.₅₉D₆.₄C₂, C₂².₁(C₆.D₆), (C₂xC₃:C₈):₄S₃, (C₆xC₃:C₈):₁₈C₂, C₄.₉₂(C₂xS₃²), (C₂xC₄).₆₃S₃², C₆.₂₆(S₃xC₂xC₄), (C₂xC₆).₁₈(C₄xS₃), (C₃xC₁₂).₅₈(C₂xC₄), (C₃xC₃:C₈).₂₇C₂², C₂.₄(C₂xC₆.D₆), (C₄xC₃:S₃).₅₉C₂², (C₃xC₄.Dic₃):₁₁C₂, C₃:Dic₃.₁₉(C₂xC₄), (C₃xC₆).₄₂(C₂²xC₄), (C₂xC₃:S₃).₁₅(C₂xC₄), SmallGroup(288,465)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃:C₈.₂₂D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₃xC₃:C₈ — C₁₂.₂₉D₆ — C₃:C₈.₂₂D₆

Lower central

C₃² — C₃xC₆ — C₃:C₈.₂₂D₆

Upper central

C₁ — C₄ — C₂xC₄

Generators and relations for C₃:C₈.₂₂D₆
G = < a,b,c,d | a³=b⁸=1, c⁶=b², d²=b⁴, bab^-1=cac^-1=dad^-1=a^-1, cbc^-1=dbd^-1=b⁵, dcd^-1=b⁴c⁵ >

Subgroups: 506 in 144 conjugacy classes, 52 normal (36 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), C₄oD₄, C₃:S₃, C₃xC₆, C₃xC₆, C₃:C₈, C₃:C₈, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₈oD₄, C₃:Dic₃, C₃xC₁₂, C₂xC₃:S₃, C₆², S₃xC₈, C₈:S₃, C₂xC₃:C₈, C₄.Dic₃, C₂xC₂₄, C₃xM₄(2), C₄oD₁₂, C₃xC₃:C₈, C₃xC₃:C₈, C₃²:₄Q₈, C₄xC₃:S₃, C₁₂:S₃, C₃²:₇D₄, C₆xC₁₂, C₈oD₁₂, D₁₂.C₄, C₁₂.₂₉D₆, C₁₂.₃₁D₆, C₆xC₃:C₈, C₃xC₄.Dic₃, C₁₂.₅₉D₆, C₃:C₈.₂₂D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₆, C₂²xC₄, C₄xS₃, C₂²xS₃, C₈oD₄, S₃², S₃xC₂xC₄, C₆.D₆, C₂xS₃², C₈oD₁₂, D₁₂.C₄, C₂xC₆.D₆, C₃:C₈.₂₂D₆

Smallest permutation representation of C₃:C₈.₂₂D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	8C	8D	8E	···	8J	12A	···	12F	12G	···	12K	24A	···	24H	24I	24J	24K	24L
order	1	2	2	2	2	3	3	3	4	4	4	4	4	6	6	6	6	6	6	6	6	8	8	8	8	8	···	8	12	···	12	12	···	12	24	···	24	24	24	24	24
size	1	1	2	18	18	2	2	4	1	1	2	18	18	2	2	2	2	4	4	4	4	3	3	3	3	6	···	6	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	4	4	4	4	4	4
type	+	+	+	+	+	+				+	+	+	+					+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	S₃	D₆	D₆	C₄xS₃	C₄xS₃	C₈oD₄	C₈oD₁₂	S₃²	C₆.D₆	C₂xS₃²	C₆.D₆	D₁₂.C₄	C₃:C₈.₂₂D₆
kernel	C₃:C₈.₂₂D₆	C₁₂.₂₉D₆	C₁₂.₃₁D₆	C₆xC₃:C₈	C₃xC₄.Dic₃	C₁₂.₅₉D₆	C₃²:₄Q₈	C₁₂:S₃	C₃²:₇D₄	C₂xC₃:C₈	C₄.Dic₃	C₃:C₈	C₂xC₁₂	C₁₂	C₂xC₆	C₃²	C₃	C₂xC₄	C₄	C₄	C₂²	C₃	C₁
# reps	1	2	2	1	1	1	2	2	4	1	1	4	2	4	4	4	8	1	1	1	1	2	4

Matrix representation of C₃:C₈.₂₂D₆ ►in GL₆(F₇₃)

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	72	1
0	0	0	0	72	0

46	0	0	0	0	0
0	46	0	0	0	0
0	0	0	51	0	0
0	0	22	0	0	0
0	0	0	0	46	27
0	0	0	0	0	27

0	46	0	0	0	0
27	27	0	0	0	0
0	0	22	0	0	0
0	0	0	51	0	0
0	0	0	0	27	46
0	0	0	0	0	46

0	72	0	0	0	0
72	0	0	0	0	0
0	0	27	0	0	0
0	0	0	46	0	0
0	0	0	0	72	1
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,22,0,0,0,0,51,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27],[0,27,0,0,0,0,46,27,0,0,0,0,0,0,22,0,0,0,0,0,0,51,0,0,0,0,0,0,27,0,0,0,0,0,46,46],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;

C₃:C₈.₂₂D₆ in GAP, Magma, Sage, TeX

C_3\rtimes C_8._{22}D_6

% in TeX

G:=Group("C3:C8.22D6");

// GroupNames label

G:=SmallGroup(288,465);

// by ID

G=gap.SmallGroup(288,465);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,219,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^2,d^2=b^4,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^4*c^5>;

// generators/relations

G = C3:C8.22D6 order 288 = 25·32

11st non-split extension by C3:C8 of D6 acting via D6/S3=C2

G = C₃:C₈.₂₂D₆ order 288 = 2⁵·3²

11^st non-split extension by C₃:C₈ of D₆ acting via D₆/S₃=C₂