D36:6C2^2 - GroupNames

G = D₃₆⋊₆C₂² order 288 = 2⁵·3²

4^th semidirect product of D₃₆ and C₂² acting via C₂²/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄.₇D₁₈, C₃₆.₁₆D₄, D₃₆⋊₆C₂², C₃₆.₁₃C₂³, Dic₁₈⋊₆C₂², D₄⋊D₉⋊₅C₂, (C₂×D₄)⋊₂D₉, C₉⋊C₈⋊₃C₂², (D₄×C₁₈)⋊₂C₂, C₉⋊₄(C₈⋊C₂²), D₄.D₉⋊₅C₂, (C₆×D₄).₅S₃, (C₃×D₄).₃₁D₆, (C₂×C₄).₁₄D₁₈, C₁₈.₄₇(C₂×D₄), (C₂×C₁₂).₅₇D₆, (C₂×C₁₈).₄₀D₄, C₄.Dic₉⋊₆C₂, D₃₆⋊₅C₂⋊₃C₂, C₄.₁₆(C₉⋊D₄), C₃.(D₁₂⋊₆C₂²), (D₄×C₉).₇C₂², C₄.₁₃(C₂²×D₉), C₁₂.₁₁(C₃⋊D₄), C₁₂.₅₂(C₂²×S₃), (C₂×C₃₆).₃₅C₂², C₂².₁₀(C₉⋊D₄), C₂.₁₀(C₂×C₉⋊D₄), C₆.₉₄(C₂×C₃⋊D₄), (C₂×C₆).₇₉(C₃⋊D₄), SmallGroup(288,143)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — D₃₆⋊₆C₂²

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — D₃₆ — D₃₆⋊₅C₂ — D₃₆⋊₆C₂²

Lower central

C₉ — C₁₈ — C₃₆ — D₃₆⋊₆C₂²

Upper central

C₁ — C₂ — C₂×C₄ — C₂×D₄

Generators and relations for D₃₆⋊₆C₂²
G = < a,b,c,d | a³⁶=b²=c²=d²=1, bab=a^-1, ac=ca, dad=a¹⁹, cbc=a¹⁸b, dbd=a²⁷b, cd=dc >

Subgroups: 404 in 102 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₉, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, D₉, C₁₈, C₁₈, C₃⋊C₈, Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂²×C₆, C₈⋊C₂², Dic₉, C₃₆, D₁₈, C₂×C₁₈, C₂×C₁₈, C₄.Dic₃, D₄⋊S₃, D₄.S₃, C₄○D₁₂, C₆×D₄, C₉⋊C₈, Dic₁₈, C₄×D₉, D₃₆, C₉⋊D₄, C₂×C₃₆, D₄×C₉, D₄×C₉, C₂²×C₁₈, D₁₂⋊₆C₂², C₄.Dic₉, D₄.D₉, D₄⋊D₉, D₃₆⋊₅C₂, D₄×C₁₈, D₃₆⋊₆C₂²
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, D₉, C₃⋊D₄, C₂²×S₃, C₈⋊C₂², D₁₈, C₂×C₃⋊D₄, C₉⋊D₄, C₂²×D₉, D₁₂⋊₆C₂², C₂×C₉⋊D₄, D₃₆⋊₆C₂²

Smallest permutation representation of D₃₆⋊₆C₂²
►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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)

(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

(2 20)(4 22)(6 24)(8 26)(10 28)(12 30)(14 32)(16 34)(18 36)(37 64)(38 47)(39 66)(40 49)(41 68)(42 51)(43 70)(44 53)(45 72)(46 55)(48 57)(50 59)(52 61)(54 63)(56 65)(58 67)(60 69)(62 71)

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

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

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

51 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	6A	6B	6C	6D	6E	6F	6G	8A	8B	9A	9B	9C	12A	12B	18A	···	18I	18J	···	18U	36A	···	36F
order	1	2	2	2	2	2	3	4	4	4	6	6	6	6	6	6	6	8	8	9	9	9	12	12	18	···	18	18	···	18	36	···	36
size	1	1	2	4	4	36	2	2	2	36	2	2	2	4	4	4	4	36	36	2	2	2	4	4	2	···	2	4	···	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+			+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₆	D₆	D₉	C₃⋊D₄	C₃⋊D₄	D₁₈	D₁₈	C₉⋊D₄	C₉⋊D₄	C₈⋊C₂²	D₁₂⋊₆C₂²	D₃₆⋊₆C₂²
kernel	D₃₆⋊₆C₂²	C₄.Dic₉	D₄.D₉	D₄⋊D₉	D₃₆⋊₅C₂	D₄×C₁₈	C₆×D₄	C₃₆	C₂×C₁₈	C₂×C₁₂	C₃×D₄	C₂×D₄	C₁₂	C₂×C₆	C₂×C₄	D₄	C₄	C₂²	C₉	C₃	C₁
# reps	1	1	2	2	1	1	1	1	1	1	2	3	2	2	3	6	6	6	1	2	6

Matrix representation of D₃₆⋊₆C₂² ►in GL₆(𝔽₇₃)

2	0	0	0	0	0
56	37	0	0	0	0
0	0	1	3	0	0
0	0	48	72	0	0
0	0	57	49	0	72
0	0	0	49	1	0

68	6	0	0	0	0
69	5	0	0	0	0
0	0	49	0	0	70
0	0	16	0	1	1
0	0	22	1	0	24
0	0	70	0	0	24

72	0	0	0	0	0
0	72	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	57	0	72	0
0	0	57	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	48	72	0	0
0	0	57	0	0	72
0	0	57	0	72	0

G:=sub<GL(6,GF(73))| [2,56,0,0,0,0,0,37,0,0,0,0,0,0,1,48,57,0,0,0,3,72,49,49,0,0,0,0,0,1,0,0,0,0,72,0],[68,69,0,0,0,0,6,5,0,0,0,0,0,0,49,16,22,70,0,0,0,0,1,0,0,0,0,1,0,0,0,0,70,1,24,24],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,57,57,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,48,57,57,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

D₃₆⋊₆C₂² in GAP, Magma, Sage, TeX

D_{36}\rtimes_6C_2^2

% in TeX

G:=Group("D36:6C2^2");

// GroupNames label

G:=SmallGroup(288,143);

// by ID

G=gap.SmallGroup(288,143);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,675,185,80,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^36=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^19,c*b*c=a^18*b,d*b*d=a^27*b,c*d=d*c>;

// generators/relations

G = D36⋊6C22 order 288 = 25·32

4th semidirect product of D36 and C22 acting via C22/C2=C2

G = D₃₆⋊₆C₂² order 288 = 2⁵·3²

4^th semidirect product of D₃₆ and C₂² acting via C₂²/C₂=C₂