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## G = C2×Dic9order 72 = 23·32

### Direct product of C2 and Dic9

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

 Derived series C1 — C9 — C2×Dic9
 Chief series C1 — C3 — C9 — C18 — Dic9 — C2×Dic9
 Lower central C9 — C2×Dic9
 Upper central C1 — C22

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

Character table of C2×Dic9

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 9A 9B 9C 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 1 1 2 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 -1 -1 1 1 -i i i -i -1 -1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 -1 1 -1 1 -i -i i i -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 linear of order 4 ρ7 1 -1 1 -1 1 i i -i -i -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 i -i -i i -1 -1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 -2 -1 0 0 0 0 -1 1 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ10 2 2 -2 -2 2 0 0 0 0 2 -2 -2 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 0 0 0 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -1 0 0 0 0 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ13 2 2 2 2 -1 0 0 0 0 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ14 2 2 -2 -2 -1 0 0 0 0 -1 1 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ15 2 2 -2 -2 -1 0 0 0 0 -1 1 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ16 2 2 2 2 -1 0 0 0 0 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ17 2 -2 2 -2 2 0 0 0 0 -2 2 -2 -1 -1 -1 1 -1 1 1 1 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ18 2 -2 -2 2 2 0 0 0 0 -2 -2 2 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ19 2 -2 2 -2 -1 0 0 0 0 1 -1 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ20 2 -2 2 -2 -1 0 0 0 0 1 -1 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2 ρ21 2 -2 2 -2 -1 0 0 0 0 1 -1 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ22 2 -2 -2 2 -1 0 0 0 0 1 1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ95-ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ23 2 -2 -2 2 -1 0 0 0 0 1 1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ98-ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ24 2 -2 -2 2 -1 0 0 0 0 1 1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ97-ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2

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

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

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

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

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

 1 0 0 0 36 0 0 0 36
,
 36 0 0 0 11 20 0 17 31
,
 6 0 0 0 26 17 0 6 11
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

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