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G = C2×D36order 144 = 24·32

Direct product of C2 and D36

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D36, C42D18, C181D4, C6.6D12, C362C22, C12.44D6, D181C22, C18.3C23, C22.10D18, C91(C2×D4), (C2×C4)⋊2D9, C3.(C2×D12), (C2×C36)⋊3C2, (C2×C12).7S3, (C2×C6).26D6, (C22×D9)⋊1C2, C2.4(C22×D9), C6.21(C22×S3), (C2×C18).10C22, SmallGroup(144,39)

Series: Derived Chief Lower central Upper central

C1C18 — C2×D36
C1C3C9C18D18C22×D9 — C2×D36
C9C18 — C2×D36
C1C22C2×C4

Generators and relations for C2×D36
 G = < a,b,c | a2=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 375 in 81 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], C9, C12 [×2], D6 [×8], C2×C6, C2×D4, D9 [×4], C18, C18 [×2], D12 [×4], C2×C12, C22×S3 [×2], C36 [×2], D18 [×4], D18 [×4], C2×C18, C2×D12, D36 [×4], C2×C36, C22×D9 [×2], C2×D36
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, D12 [×2], C22×S3, D18 [×3], C2×D12, D36 [×2], C22×D9, C2×D36

Smallest permutation representation of C2×D36
On 72 points
Generators in S72
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)

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

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

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

C2×D36 is a maximal subgroup of
C18.D8  C2.D72  C36.48D4  C426D9  C427D9  C223D36  D18⋊D4  D36⋊C4  D18.D4  C4⋊D36  C8⋊D18  C367D4  C36⋊D4  C36.23D4  D4⋊D18  C2×D4×D9  D48D18
C2×D36 is a maximal quotient of
C362Q8  C426D9  C427D9  C223D36  C22.4D36  C4⋊D36  D182Q8  D727C2  C8⋊D18  C8.D18  C367D4

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order122222223446669991212121218···1836···36
size11111818181822222222222222···22···2

42 irreducible representations

dim1111222222222
type+++++++++++++
imageC1C2C2C2S3D4D6D6D9D12D18D18D36
kernelC2×D36D36C2×C36C22×D9C2×C12C18C12C2×C6C2×C4C6C4C22C2
# reps14121221346312

Matrix representation of C2×D36 in GL3(𝔽37) generated by

3600
010
001
,
100
03329
0825
,
100
0825
03329
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[1,0,0,0,33,8,0,29,25],[1,0,0,0,8,33,0,25,29] >;

C2×D36 in GAP, Magma, Sage, TeX

C_2\times D_{36}
% in TeX

G:=Group("C2xD36");
// GroupNames label

G:=SmallGroup(144,39);
// by ID

G=gap.SmallGroup(144,39);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^2=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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