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G = D36⋊5C2order 144 = 24·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D36⋊5C2
 Chief series C1 — C3 — C9 — C18 — D18 — C4×D9 — D36⋊5C2
 Lower central C9 — C18 — D36⋊5C2
 Upper central C1 — C4 — C2×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, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C4○D4, D9, C18, C18, Dic6, C4×S3, D12, C3⋊D4, C2×C12, Dic9, C36, D18, C2×C18, C4○D12, Dic18, C4×D9, D36, C9⋊D4, C2×C36, D365C2
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, 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 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)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 36A ··· 36L order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 9 9 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 2 18 18 2 1 1 2 18 18 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 D9 D18 D18 C4○D12 D36⋊5C2 kernel D36⋊5C2 Dic18 C4×D9 D36 C9⋊D4 C2×C36 C2×C12 C12 C2×C6 C9 C2×C4 C4 C22 C3 C1 # reps 1 1 2 1 2 1 1 2 1 2 3 6 3 4 12

Matrix representation of D365C2 in GL2(𝔽37) generated by

 12 8 29 4
,
 17 6 26 20
,
 7 14 23 30
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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