Dic18:C4 - GroupNames

G = Dic₁₈:C₄ order 288 = 2⁵·3²

4^th semidirect product of Dic₁₈ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₃₆:₄C₄, C₃₆.₅₄D₄, Dic₁₈:₄C₄, M₄(2):₄D₉, C₂².₃D₃₆, C₉:₂C₄wrC₂, C₄.₃(C₄xD₉), C₁₂.₆(C₄xS₃), C₃₆.₆(C₂xC₄), (C₂xC₁₈).₁D₄, (C₂xC₆).₃D₁₂, (C₄xDic₉):₁C₂, (C₂xC₄).₄₀D₁₈, (C₂xC₁₂).₄₄D₆, C₃.(D₁₂:C₄), C₆.₁₇(D₆:C₄), C₄.₂₉(C₉:D₄), (C₉xM₄(2)):₈C₂, C₂.₁₁(D₁₈:C₄), D₃₆:₅C₂.₂C₂, (C₂xC₃₆).₂₂C₂², (C₃xM₄(2)).₈S₃, C₁₂.₁₂₄(C₃:D₄), C₁₈.₁₀(C₂²:C₄), SmallGroup(288,32)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — Dic₁₈:C₄

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₂xC₃₆ — D₃₆:₅C₂ — Dic₁₈:C₄

Lower central

C₉ — C₁₈ — C₃₆ — Dic₁₈:C₄

Upper central

C₁ — C₄ — C₂xC₄ — M₄(2)

Generators and relations for Dic₁₈:C₄
G = < a,b,c | a³⁶=c⁴=1, b²=a¹⁸, bab^-1=a^-1, cac^-1=a¹⁷, cbc^-1=a²⁷b >

Subgroups: 332 in 66 conjugacy classes, 26 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₉, Dic₃, C₁₂, D₆, C₂xC₆, C₄², M₄(2), C₄oD₄, D₉, C₁₈, C₁₈, C₂₄, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₄wrC₂, Dic₉, C₃₆, D₁₈, C₂xC₁₈, C₄xDic₃, C₃xM₄(2), C₄oD₁₂, C₇₂, Dic₁₈, C₄xD₉, D₃₆, C₂xDic₉, C₉:D₄, C₂xC₃₆, D₁₂:C₄, C₄xDic₉, C₉xM₄(2), D₃₆:₅C₂, Dic₁₈:C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, D₆, C₂²:C₄, D₉, C₄xS₃, D₁₂, C₃:D₄, C₄wrC₂, D₁₈, D₆:C₄, C₄xD₉, D₃₆, C₉:D₄, D₁₂:C₄, D₁₈:C₄, Dic₁₈:C₄

Smallest permutation representation of Dic₁₈:C₄
►On 72 points

Generators in S₇₂

(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)

(1 37 19 55)(2 72 20 54)(3 71 21 53)(4 70 22 52)(5 69 23 51)(6 68 24 50)(7 67 25 49)(8 66 26 48)(9 65 27 47)(10 64 28 46)(11 63 29 45)(12 62 30 44)(13 61 31 43)(14 60 32 42)(15 59 33 41)(16 58 34 40)(17 57 35 39)(18 56 36 38)

(1 28 19 10)(2 9 20 27)(3 26 21 8)(4 7 22 25)(5 24 23 6)(11 18 29 36)(12 35 30 17)(13 16 31 34)(14 33 32 15)(38 54)(39 71)(40 52)(41 69)(42 50)(43 67)(44 48)(45 65)(47 63)(49 61)(51 59)(53 57)(56 72)(58 70)(60 68)(62 66)

G:=sub<Sym(72)| (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), (1,37,19,55)(2,72,20,54)(3,71,21,53)(4,70,22,52)(5,69,23,51)(6,68,24,50)(7,67,25,49)(8,66,26,48)(9,65,27,47)(10,64,28,46)(11,63,29,45)(12,62,30,44)(13,61,31,43)(14,60,32,42)(15,59,33,41)(16,58,34,40)(17,57,35,39)(18,56,36,38), (1,28,19,10)(2,9,20,27)(3,26,21,8)(4,7,22,25)(5,24,23,6)(11,18,29,36)(12,35,30,17)(13,16,31,34)(14,33,32,15)(38,54)(39,71)(40,52)(41,69)(42,50)(43,67)(44,48)(45,65)(47,63)(49,61)(51,59)(53,57)(56,72)(58,70)(60,68)(62,66)>;

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), (1,37,19,55)(2,72,20,54)(3,71,21,53)(4,70,22,52)(5,69,23,51)(6,68,24,50)(7,67,25,49)(8,66,26,48)(9,65,27,47)(10,64,28,46)(11,63,29,45)(12,62,30,44)(13,61,31,43)(14,60,32,42)(15,59,33,41)(16,58,34,40)(17,57,35,39)(18,56,36,38), (1,28,19,10)(2,9,20,27)(3,26,21,8)(4,7,22,25)(5,24,23,6)(11,18,29,36)(12,35,30,17)(13,16,31,34)(14,33,32,15)(38,54)(39,71)(40,52)(41,69)(42,50)(43,67)(44,48)(45,65)(47,63)(49,61)(51,59)(53,57)(56,72)(58,70)(60,68)(62,66) );

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)], [(1,37,19,55),(2,72,20,54),(3,71,21,53),(4,70,22,52),(5,69,23,51),(6,68,24,50),(7,67,25,49),(8,66,26,48),(9,65,27,47),(10,64,28,46),(11,63,29,45),(12,62,30,44),(13,61,31,43),(14,60,32,42),(15,59,33,41),(16,58,34,40),(17,57,35,39),(18,56,36,38)], [(1,28,19,10),(2,9,20,27),(3,26,21,8),(4,7,22,25),(5,24,23,6),(11,18,29,36),(12,35,30,17),(13,16,31,34),(14,33,32,15),(38,54),(39,71),(40,52),(41,69),(42,50),(43,67),(44,48),(45,65),(47,63),(49,61),(51,59),(53,57),(56,72),(58,70),(60,68),(62,66)]])

54 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	8A	8B	9A	9B	9C	12A	12B	12C	18A	18B	18C	18D	18E	18F	24A	24B	24C	24D	36A	···	36F	36G	36H	36I	72A	···	72L
order	1	2	2	2	3	4	4	4	4	4	4	4	4	6	6	8	8	9	9	9	12	12	12	18	18	18	18	18	18	24	24	24	24	36	···	36	36	36	36	72	···	72
size	1	1	2	36	2	1	1	2	18	18	18	18	36	2	4	4	4	2	2	2	2	2	4	2	2	2	4	4	4	4	4	4	4	2	···	2	4	4	4	4	···	4

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+			+	+	+	+	+			+		+			+
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	D₆	D₉	C₄xS₃	C₃:D₄	D₁₂	C₄wrC₂	D₁₈	C₄xD₉	C₉:D₄	D₃₆	D₁₂:C₄	Dic₁₈:C₄
kernel	Dic₁₈:C₄	C₄xDic₉	C₉xM₄(2)	D₃₆:₅C₂	Dic₁₈	D₃₆	C₃xM₄(2)	C₃₆	C₂xC₁₈	C₂xC₁₂	M₄(2)	C₁₂	C₁₂	C₂xC₆	C₉	C₂xC₄	C₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	1	2	2	1	1	1	1	3	2	2	2	4	3	6	6	6	2	6

Matrix representation of Dic₁₈:C₄ ►in GL₄(F₇₃) generated by

28	70	0	0
3	31	0	0
0	0	27	0
0	0	0	46

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

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

G:=sub<GL(4,GF(73))| [28,3,0,0,70,31,0,0,0,0,27,0,0,0,0,46],[1,72,0,0,0,72,0,0,0,0,0,46,0,0,46,0],[1,72,0,0,0,72,0,0,0,0,46,0,0,0,0,1] >;

Dic₁₈:C₄ in GAP, Magma, Sage, TeX

{\rm Dic}_{18}\rtimes C_4

% in TeX

G:=Group("Dic18:C4");

// GroupNames label

G:=SmallGroup(288,32);

// by ID

G=gap.SmallGroup(288,32);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c|a^36=c^4=1,b^2=a^18,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^27*b>;

// generators/relations

G = Dic18:C4 order 288 = 25·32

4th semidirect product of Dic18 and C4 acting via C4/C2=C2

G = Dic₁₈:C₄ order 288 = 2⁵·3²

4^th semidirect product of Dic₁₈ and C₄ acting via C₄/C₂=C₂