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G = C2×Dic9order 72 = 23·32

Direct product of C2 and Dic9

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×Dic9, C18⋊C4, C6.9D6, C22.D9, C2.2D18, C6.2Dic3, C18.4C22, C92(C2×C4), (C2×C18).C2, (C2×C6).2S3, C3.(C2×Dic3), SmallGroup(72,7)

Series: Derived Chief Lower central Upper central

C1C9 — C2×Dic9
C1C3C9C18Dic9 — C2×Dic9
C9 — C2×Dic9
C1C22

Generators and relations for C2×Dic9
 G = < a,b,c | a2=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

9C4
9C4
9C2×C4
3Dic3
3Dic3
3C2×Dic3

Character table of C2×Dic9

 class 12A2B2C34A4B4C4D6A6B6C9A9B9C18A18B18C18D18E18F18G18H18I
 size 111129999222222222222222
ρ1111111111111111111111111    trivial
ρ211-1-111-11-11-1-11111-1-1-1-1-1-111    linear of order 2
ρ311-1-11-11-111-1-11111-1-1-1-1-1-111    linear of order 2
ρ411111-1-1-1-1111111111111111    linear of order 2
ρ51-1-111-iii-i-1-11111-1-1111-1-1-1-1    linear of order 4
ρ61-11-11-i-iii-11-1111-11-1-1-111-1-1    linear of order 4
ρ71-11-11ii-i-i-11-1111-11-1-1-111-1-1    linear of order 4
ρ81-1-111i-i-ii-1-11111-1-1111-1-1-1-1    linear of order 4
ρ922-2-2-10000-111ζ9594ζ989ζ9792ζ97929899594989979297929594ζ9594ζ989    orthogonal lifted from D18
ρ1022-2-2200002-2-2-1-1-1-1111111-1-1    orthogonal lifted from D6
ρ11222220000222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122222-10000-1-1-1ζ9594ζ989ζ9792ζ9792ζ989ζ9594ζ989ζ9792ζ9792ζ9594ζ9594ζ989    orthogonal lifted from D9
ρ132222-10000-1-1-1ζ9792ζ9594ζ989ζ989ζ9594ζ9792ζ9594ζ989ζ989ζ9792ζ9792ζ9594    orthogonal lifted from D9
ρ1422-2-2-10000-111ζ9792ζ9594ζ989ζ9899594979295949899899792ζ9792ζ9594    orthogonal lifted from D18
ρ1522-2-2-10000-111ζ989ζ9792ζ9594ζ95949792989979295949594989ζ989ζ9792    orthogonal lifted from D18
ρ162222-10000-1-1-1ζ989ζ9792ζ9594ζ9594ζ9792ζ989ζ9792ζ9594ζ9594ζ989ζ989ζ9792    orthogonal lifted from D9
ρ172-22-220000-22-2-1-1-11-1111-1-111    symplectic lifted from Dic3, Schur index 2
ρ182-2-2220000-2-22-1-1-111-1-1-11111    symplectic lifted from Dic3, Schur index 2
ρ192-22-2-100001-11ζ9792ζ9594ζ989989ζ959497929594989ζ989ζ979297929594    symplectic lifted from Dic9, Schur index 2
ρ202-22-2-100001-11ζ989ζ9792ζ95949594ζ979298997929594ζ9594ζ9899899792    symplectic lifted from Dic9, Schur index 2
ρ212-22-2-100001-11ζ9594ζ989ζ97929792ζ98995949899792ζ9792ζ95949594989    symplectic lifted from Dic9, Schur index 2
ρ222-2-22-1000011-1ζ9792ζ9594ζ9899899594ζ9792ζ9594ζ989989979297929594    symplectic lifted from Dic9, Schur index 2
ρ232-2-22-1000011-1ζ9594ζ989ζ97929792989ζ9594ζ989ζ9792979295949594989    symplectic lifted from Dic9, Schur index 2
ρ242-2-22-1000011-1ζ989ζ9792ζ959495949792ζ989ζ9792ζ959495949899899792    symplectic lifted from Dic9, Schur index 2

Smallest permutation representation of C2×Dic9
Regular action on 72 points
Generators in S72
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 19)(16 20)(17 21)(18 22)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 55)(52 56)(53 57)(54 58)
(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 41 10 50)(2 40 11 49)(3 39 12 48)(4 38 13 47)(5 37 14 46)(6 54 15 45)(7 53 16 44)(8 52 17 43)(9 51 18 42)(19 67 28 58)(20 66 29 57)(21 65 30 56)(22 64 31 55)(23 63 32 72)(24 62 33 71)(25 61 34 70)(26 60 35 69)(27 59 36 68)

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

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

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

C2×Dic9 is a maximal subgroup of   Dic9⋊C4  C4⋊Dic9  D18⋊C4  C18.D4  C2×C4×D9  D42D9  Q8⋊Dic9
C2×Dic9 is a maximal quotient of   C4.Dic9  C4⋊Dic9  C18.D4

Matrix representation of C2×Dic9 in GL3(𝔽37) generated by

100
0360
0036
,
3600
01120
01731
,
600
02617
0611
G:=sub<GL(3,GF(37))| [1,0,0,0,36,0,0,0,36],[36,0,0,0,11,17,0,20,31],[6,0,0,0,26,6,0,17,11] >;

C2×Dic9 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_9
% in TeX

G:=Group("C2xDic9");
// GroupNames label

G:=SmallGroup(72,7);
// by ID

G=gap.SmallGroup(72,7);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,20,803,138,1204]);
// Polycyclic

G:=Group<a,b,c|a^2=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C2×Dic9 in TeX
Character table of C2×Dic9 in TeX

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