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G = D365C2order 144 = 24·32

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D365C2, C4.16D18, C12.45D6, Dic185C2, C18.4C23, C22.2D18, C36.16C22, D18.1C22, Dic9.2C22, (C2×C4)⋊3D9, (C2×C36)⋊4C2, (C4×D9)⋊4C2, C91(C4○D4), C9⋊D43C2, C3.(C4○D12), (C2×C6).27D6, (C2×C12).11S3, C2.5(C22×D9), C6.22(C22×S3), (C2×C18).11C22, SmallGroup(144,40)

Series: Derived Chief Lower central Upper central

C1C18 — D365C2
C1C3C9C18D18C4×D9 — D365C2
C9C18 — D365C2
C1C4C2×C4

Generators and relations for D365C2
 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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C4○D4, D9 [×2], C18, C18, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, Dic9 [×2], C36 [×2], D18 [×2], C2×C18, C4○D12, Dic18, C4×D9 [×2], D36, C9⋊D4 [×2], C2×C36, D365C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, D9, C22×S3, D18 [×3], C4○D12, C22×D9, D365C2

Smallest permutation representation of D365C2
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 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)

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

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

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

D365C2 is a maximal subgroup of
C424D9  Dic18⋊C4  D36.2C4  D727C2  D36.C4  C8⋊D18  C8.D18  D366C22  C36.C23  D4.9D18  D46D18  Q8.15D18  C4○D4×D9  D48D18  D4.10D18  D1085C2  D6.D18  D365S3  Dic9.D6  D18.3D6  D366C6  C36.70D6
D365C2 is a maximal quotient of
C4×Dic18  C36.6Q8  C422D9  C4×D36  C427D9  C423D9  C23.8D18  C23.9D18  D18⋊D4  Dic9.D4  Dic9.Q8  D18.D4  D18⋊Q8  C4⋊C4⋊D9  C36.49D4  C23.26D18  C4×C9⋊D4  C23.28D18  C367D4  D1085C2  D6.D18  D365S3  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++++++++++++
imageC1C2C2C2C2C2S3D6D6C4○D4D9D18D18C4○D12D365C2
kernelD365C2Dic18C4×D9D36C9⋊D4C2×C36C2×C12C12C2×C6C9C2×C4C4C22C3C1
# reps1121211212363412

Matrix representation of D365C2 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] >;

D365C2 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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