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## G = C2×C18.D6order 432 = 24·33

### Direct product of C2 and C18.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C18.D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C18.D6 — C2×C18.D6
 Lower central C3×C9 — C2×C18.D6
 Upper central C1 — C22

Generators and relations for C2×C18.D6
G = < a,b,c,d | a2=b18=d2=1, c6=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 1260 in 194 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, D9, C18, C18, C18, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C3×C9, Dic9, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C2×C3⋊S3, C62, S3×C2×C4, C9⋊S3, C3×C18, C3×C18, C4×D9, C2×Dic9, C2×C36, C22×D9, C6.D6, C6×Dic3, C6×Dic3, C22×C3⋊S3, C3×Dic9, C9×Dic3, C2×C9⋊S3, C6×C18, C2×C4×D9, C2×C6.D6, C18.D6, C6×Dic9, Dic3×C18, C22×C9⋊S3, C2×C18.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C4×S3, C22×S3, D18, S32, S3×C2×C4, C4×D9, C22×D9, C6.D6, C2×S32, S3×D9, C2×C4×D9, C2×C6.D6, C18.D6, C2×S3×D9, C2×C18.D6

Smallest permutation representation of C2×C18.D6
On 72 points
Generators in S72
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(37 72)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)
(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 59 16 62 13 65 10 68 7 71 4 56)(2 58 17 61 14 64 11 67 8 70 5 55)(3 57 18 60 15 63 12 66 9 69 6 72)(19 50 34 53 31 38 28 41 25 44 22 47)(20 49 35 52 32 37 29 40 26 43 23 46)(21 48 36 51 33 54 30 39 27 42 24 45)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 18)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 36)(34 35)(37 38)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 27 27 27 27 2 2 4 3 3 3 3 9 9 9 9 2 ··· 2 4 4 4 2 2 2 4 4 4 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 D6 D9 C4×S3 C4×S3 D18 D18 C4×D9 S32 C6.D6 C2×S32 S3×D9 C18.D6 C2×S3×D9 kernel C2×C18.D6 C18.D6 C6×Dic9 Dic3×C18 C22×C9⋊S3 C2×C9⋊S3 C2×Dic9 C6×Dic3 Dic9 C2×C18 C3×Dic3 C62 C2×Dic3 C18 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 1 2 1 2 1 3 4 4 6 3 12 1 2 1 3 6 3

Matrix representation of C2×C18.D6 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 36 0 0 0 0 36 0 0 0 0 26 31 0 0 6 20
,
 6 31 0 0 6 0 0 0 0 0 1 36 0 0 0 36
,
 1 0 0 0 1 36 0 0 0 0 1 36 0 0 0 36
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,26,6,0,0,31,20],[6,6,0,0,31,0,0,0,0,0,1,0,0,0,36,36],[1,1,0,0,0,36,0,0,0,0,1,0,0,0,36,36] >;

C2×C18.D6 in GAP, Magma, Sage, TeX

C_2\times C_{18}.D_6
% in TeX

G:=Group("C2xC18.D6");
// GroupNames label

G:=SmallGroup(432,306);
// by ID

G=gap.SmallGroup(432,306);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=d^2=1,c^6=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^5>;
// generators/relations

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