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## G = C2×D36order 144 = 24·32

### Direct product of C2 and D36

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

 Derived series C1 — C18 — C2×D36
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×D36
 Lower central C9 — C18 — C2×D36
 Upper central C1 — C22 — C2×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 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 36A ··· 36L order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 9 9 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 18 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D9 D12 D18 D18 D36 kernel C2×D36 D36 C2×C36 C22×D9 C2×C12 C18 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 1 2 2 1 3 4 6 3 12

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

 36 0 0 0 1 0 0 0 1
,
 1 0 0 0 33 29 0 8 25
,
 1 0 0 0 8 25 0 33 29
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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