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## G = C2×C4×3- 1+2order 216 = 23·33

### Direct product of C2×C4 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C4×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C4×3- 1+2 — C2×C4×3- 1+2
 Lower central C1 — C3 — C2×C4×3- 1+2
 Upper central C1 — C2×C12 — C2×C4×3- 1+2

Generators and relations for C2×C4×3- 1+2
G = < a,b,c,d | a2=b4=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 80 in 64 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C18, C3×C6, C3×C6, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C2×3- 1+2, C2×3- 1+2, C2×C36, C6×C12, C4×3- 1+2, C22×3- 1+2, C2×C4×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C3×C6, C2×C12, 3- 1+2, C3×C12, C62, C2×3- 1+2, C6×C12, C4×3- 1+2, C22×3- 1+2, C2×C4×3- 1+2

Smallest permutation representation of C2×C4×3- 1+2
On 72 points
Generators in S72
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 34)(20 35)(21 36)(22 28)(23 29)(24 30)(25 31)(26 32)(27 33)(55 70)(56 71)(57 72)(58 64)(59 65)(60 66)(61 67)(62 68)(63 69)
(1 18 19 70)(2 10 20 71)(3 11 21 72)(4 12 22 64)(5 13 23 65)(6 14 24 66)(7 15 25 67)(8 16 26 68)(9 17 27 69)(28 58 40 46)(29 59 41 47)(30 60 42 48)(31 61 43 49)(32 62 44 50)(33 63 45 51)(34 55 37 52)(35 56 38 53)(36 57 39 54)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(20 26 23)(21 24 27)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(56 62 59)(57 60 63)(65 71 68)(66 69 72)

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

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

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

C2×C4×3- 1+2 is a maximal subgroup of   C36.C12  Dic9⋊C12  C36⋊C12  D18⋊C12  D366C6

88 conjugacy classes

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

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + + image C1 C2 C2 C3 C3 C4 C6 C6 C6 C6 C12 C12 3- 1+2 C2×3- 1+2 C2×3- 1+2 C4×3- 1+2 kernel C2×C4×3- 1+2 C4×3- 1+2 C22×3- 1+2 C2×C36 C6×C12 C2×3- 1+2 C36 C2×C18 C3×C12 C62 C18 C3×C6 C2×C4 C4 C22 C2 # reps 1 2 1 6 2 4 12 6 4 2 24 8 2 4 2 8

Matrix representation of C2×C4×3- 1+2 in GL4(𝔽37) generated by

 36 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 6 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31
,
 1 0 0 0 0 1 9 0 0 1 36 10 0 0 36 0
,
 26 0 0 0 0 1 0 0 0 1 10 0 0 11 0 26
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[1,0,0,0,0,1,1,0,0,9,36,36,0,0,10,0],[26,0,0,0,0,1,1,11,0,0,10,0,0,0,0,26] >;

C2×C4×3- 1+2 in GAP, Magma, Sage, TeX

C_2\times C_4\times 3_-^{1+2}
% in TeX

G:=Group("C2xC4xES-(3,1)");
// GroupNames label

G:=SmallGroup(216,75);
// by ID

G=gap.SmallGroup(216,75);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,338,519]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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