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G = D36:5C2  order 144 = 24·32

The semidirect product of D36 and C2 acting through Inn(D36)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D36:5C2, C4.16D18, C12.45D6, Dic18:5C2, C18.4C23, C22.2D18, C36.16C22, D18.1C22, Dic9.2C22, (C2xC4):3D9, (C2xC36):4C2, (C4xD9):4C2, C9:1(C4oD4), C9:D4:3C2, C3.(C4oD12), (C2xC6).27D6, (C2xC12).11S3, C2.5(C22xD9), C6.22(C22xS3), (C2xC18).11C22, SmallGroup(144,40)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — D36:5C2
Chief series
C1 — C3 — C9 — C18 — D18 — C4xD9 — D36:5C2
Lower central
C9 — C18 — D36:5C2
Upper central
C1 — C4 — C2xC4

Generators and relations for D36:5C2
 G = < a,b,c | a36=b2=c2=1, bab=a-1, ac=ca, cbc=a18b >

Subgroups: 219 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C9, Dic3, C12, D6, C2xC6, C4oD4, D9, C18, C18, Dic6, C4xS3, D12, C3:D4, C2xC12, Dic9, C36, D18, C2xC18, C4oD12, Dic18, C4xD9, D36, C9:D4, C2xC36, D36:5C2
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, D9, C22xS3, D18, C4oD12, C22xD9, D36:5C2

Smallest permutation representation of D36:5C2
►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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)

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

D36:5C2 is a maximal subgroup of
 C42:4D9  Dic18:C4  D36.2C4  D72:7C2  D36.C4  C8:D18  C8.D18  D36:6C22  C36.C23  D4.9D18  D4:6D18  Q8.15D18  C4oD4xD9  D4:8D18  D4.10D18  D108:5C2  D6.D18  D36:5S3  Dic9.D6  D18.3D6  D36:6C6  C36.70D6
D36:5C2 is a maximal quotient of
 C4xDic18  C36.6Q8  C42:2D9  C4xD36  C42:7D9  C42:3D9  C23.8D18  C23.9D18  D18:D4  Dic9.D4  Dic9.Q8  D18.D4  D18:Q8  C4:C4:D9  C36.49D4  C23.26D18  C4xC9:D4  C23.28D18  C36:7D4  D108:5C2  D6.D18  D36:5S3  Dic9.D6  D18.3D6  C36.70D6

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order122223444446669991212121218···1836···36
size11218182112181822222222222···22···2

42 irreducible representations

dim111111222222222
type++++++++++++
imageC1C2C2C2C2C2S3D6D6C4oD4D9D18D18C4oD12D36:5C2
kernelD36:5C2Dic18C4xD9D36C9:D4C2xC36C2xC12C12C2xC6C9C2xC4C4C22C3C1
# reps1121211212363412

Matrix representation of D36:5C2 ►in GL2(F37) generated by

128
294
,
176
2620
,
714
2330
G:=sub<GL(2,GF(37))| [12,29,8,4],[17,26,6,20],[7,23,14,30] >;
 

D36:5C2 in GAP, Magma, Sage, TeX 

D_{36}\rtimes_5C_2
 
% in TeX
 
G:=Group("D36:5C2");
 
// GroupNames label
 
G:=SmallGroup(144,40);
 
// by ID
 
G=gap.SmallGroup(144,40);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,2404,208,3461]);
 
// Polycyclic
 
G:=Group<a,b,c|a^36=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^18*b>;
 
// generators/relations
 

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