C2.Dic3^2 - GroupNames

G = C₂.Dic₃² order 288 = 2⁵·3²

2^nd central stem extension by C₂ of Dic₃²

Aliases: C₃:C₈:₄Dic₃, C₂.₄Dic₃², C₁₂.₉₈(C₄xS₃), C₃:₁(C₂₄:C₄), (C₃xC₆).₉C₄², C₆.₄(C₄xDic₃), C₆.₁(C₈:S₃), C₃²:₄(C₈:C₄), (C₂xC₁₂).₂₉₄D₆, C₆².₂₄(C₂xC₄), C₄.₂₁(S₃xDic₃), (C₃xC₆).₄M₄(2), C₁₂.₃₂(C₂xDic₃), (C₆xC₁₂).₁₉₉C₂², C₂.₁(C₁₂.₃₁D₆), C₂².₈(C₆.D₆), (C₃xC₃:C₈):₇C₄, (C₂xC₃:C₈).₈S₃, (C₂xC₄).₁₂₇S₃², (C₆xC₃:C₈).₁₉C₂, (C₂xC₆).₂₆(C₄xS₃), (C₃xC₁₂).₈₄(C₂xC₄), (C₂xC₃:Dic₃).₇C₄, (C₄xC₃:Dic₃).₁₀C₂, SmallGroup(288,203)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₂.Dic₃²

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₆xC₁₂ — C₆xC₃:C₈ — C₂.Dic₃²

Lower central

C₃² — C₃xC₆ — C₂.Dic₃²

Upper central

C₁ — C₂xC₄

Generators and relations for C₂.Dic₃²
G = < a,b,c,d | a¹²=c³=1, b⁴=a⁶, d²=a⁹, bab^-1=a⁵, ac=ca, ad=da, bc=cb, dbd^-1=a⁶b, dcd^-1=c^-1 >

Subgroups: 266 in 95 conjugacy classes, 48 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₃², Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₄², C₂xC₈, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₈:C₄, C₃:Dic₃, C₃xC₁₂, C₆², C₂xC₃:C₈, C₄xDic₃, C₂xC₂₄, C₃xC₃:C₈, C₂xC₃:Dic₃, C₆xC₁₂, C₂₄:C₄, C₆xC₃:C₈, C₄xC₃:Dic₃, C₂.Dic₃²
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, Dic₃, D₆, C₄², M₄(2), C₄xS₃, C₂xDic₃, C₈:C₄, S₃², C₈:S₃, C₄xDic₃, S₃xDic₃, C₆.D₆, C₂₄:C₄, C₁₂.₃₁D₆, Dic₃², C₂.Dic₃²

Smallest permutation representation of C₂.Dic₃²
►On 96 points

Generators in S₉₆

(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)

(1 60 93 22 7 54 87 16)(2 53 94 15 8 59 88 21)(3 58 95 20 9 52 89 14)(4 51 96 13 10 57 90 19)(5 56 85 18 11 50 91 24)(6 49 86 23 12 55 92 17)(25 38 67 80 31 44 61 74)(26 43 68 73 32 37 62 79)(27 48 69 78 33 42 63 84)(28 41 70 83 34 47 64 77)(29 46 71 76 35 40 65 82)(30 39 72 81 36 45 66 75)

(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)

(1 77 10 74 7 83 4 80)(2 78 11 75 8 84 5 81)(3 79 12 76 9 73 6 82)(13 67 22 64 19 61 16 70)(14 68 23 65 20 62 17 71)(15 69 24 66 21 63 18 72)(25 60 34 57 31 54 28 51)(26 49 35 58 32 55 29 52)(27 50 36 59 33 56 30 53)(37 86 46 95 43 92 40 89)(38 87 47 96 44 93 41 90)(39 88 48 85 45 94 42 91)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,60,93,22,7,54,87,16)(2,53,94,15,8,59,88,21)(3,58,95,20,9,52,89,14)(4,51,96,13,10,57,90,19)(5,56,85,18,11,50,91,24)(6,49,86,23,12,55,92,17)(25,38,67,80,31,44,61,74)(26,43,68,73,32,37,62,79)(27,48,69,78,33,42,63,84)(28,41,70,83,34,47,64,77)(29,46,71,76,35,40,65,82)(30,39,72,81,36,45,66,75), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,77,10,74,7,83,4,80)(2,78,11,75,8,84,5,81)(3,79,12,76,9,73,6,82)(13,67,22,64,19,61,16,70)(14,68,23,65,20,62,17,71)(15,69,24,66,21,63,18,72)(25,60,34,57,31,54,28,51)(26,49,35,58,32,55,29,52)(27,50,36,59,33,56,30,53)(37,86,46,95,43,92,40,89)(38,87,47,96,44,93,41,90)(39,88,48,85,45,94,42,91)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,60,93,22,7,54,87,16)(2,53,94,15,8,59,88,21)(3,58,95,20,9,52,89,14)(4,51,96,13,10,57,90,19)(5,56,85,18,11,50,91,24)(6,49,86,23,12,55,92,17)(25,38,67,80,31,44,61,74)(26,43,68,73,32,37,62,79)(27,48,69,78,33,42,63,84)(28,41,70,83,34,47,64,77)(29,46,71,76,35,40,65,82)(30,39,72,81,36,45,66,75), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,77,10,74,7,83,4,80)(2,78,11,75,8,84,5,81)(3,79,12,76,9,73,6,82)(13,67,22,64,19,61,16,70)(14,68,23,65,20,62,17,71)(15,69,24,66,21,63,18,72)(25,60,34,57,31,54,28,51)(26,49,35,58,32,55,29,52)(27,50,36,59,33,56,30,53)(37,86,46,95,43,92,40,89)(38,87,47,96,44,93,41,90)(39,88,48,85,45,94,42,91) );

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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,60,93,22,7,54,87,16),(2,53,94,15,8,59,88,21),(3,58,95,20,9,52,89,14),(4,51,96,13,10,57,90,19),(5,56,85,18,11,50,91,24),(6,49,86,23,12,55,92,17),(25,38,67,80,31,44,61,74),(26,43,68,73,32,37,62,79),(27,48,69,78,33,42,63,84),(28,41,70,83,34,47,64,77),(29,46,71,76,35,40,65,82),(30,39,72,81,36,45,66,75)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,77,10,74,7,83,4,80),(2,78,11,75,8,84,5,81),(3,79,12,76,9,73,6,82),(13,67,22,64,19,61,16,70),(14,68,23,65,20,62,17,71),(15,69,24,66,21,63,18,72),(25,60,34,57,31,54,28,51),(26,49,35,58,32,55,29,52),(27,50,36,59,33,56,30,53),(37,86,46,95,43,92,40,89),(38,87,47,96,44,93,41,90),(39,88,48,85,45,94,42,91)]])

60 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	6G	6H	6I	8A	···	8H	12A	···	12H	12I	12J	12K	12L	24A	···	24P
order	1	2	2	2	3	3	3	4	4	4	4	4	4	4	4	6	···	6	6	6	6	8	···	8	12	···	12	12	12	12	12	24	···	24
size	1	1	1	1	2	2	4	1	1	1	1	18	18	18	18	2	···	2	4	4	4	6	···	6	2	···	2	4	4	4	4	6	···	6

60 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+			+	-	+					+	-	+
image	C₁	C₂	C₂	C₄	C₄	S₃	Dic₃	D₆	M₄(2)	C₄xS₃	C₄xS₃	C₈:S₃	S₃²	S₃xDic₃	C₆.D₆	C₁₂.₃₁D₆
kernel	C₂.Dic₃²	C₆xC₃:C₈	C₄xC₃:Dic₃	C₃xC₃:C₈	C₂xC₃:Dic₃	C₂xC₃:C₈	C₃:C₈	C₂xC₁₂	C₃xC₆	C₁₂	C₂xC₆	C₆	C₂xC₄	C₄	C₂²	C₂
# reps	1	2	1	8	4	2	4	2	4	4	4	16	1	2	1	4

Matrix representation of C₂.Dic₃² ►in GL₆(F₇₃)

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

0	1	0	0	0	0
46	0	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	72	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

49	71	0	0	0	0
19	24	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,46,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[49,19,0,0,0,0,71,24,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂.Dic₃² in GAP, Magma, Sage, TeX

C_2.{\rm Dic}_3^2

% in TeX

G:=Group("C2.Dic3^2");

// GroupNames label

G:=SmallGroup(288,203);

// by ID

G=gap.SmallGroup(288,203);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,92,100,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^12=c^3=1,b^4=a^6,d^2=a^9,b*a*b^-1=a^5,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;

// generators/relations

G = C2.Dic32 order 288 = 25·32

2nd central stem extension by C2 of Dic32

G = C₂.Dic₃² order 288 = 2⁵·3²

2^nd central stem extension by C₂ of Dic₃²