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## G = C22×3- 1+2order 108 = 22·33

### Direct product of C22 and 3- 1+2

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

Aliases: C22×3- 1+2, C182C6, C62.3C3, C3.2C62, C92(C2×C6), (C2×C18)⋊3C3, C6.5(C3×C6), (C3×C6).4C6, C32.(C2×C6), (C2×C6).7C32, SmallGroup(108,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×3- 1+2
 Chief series C1 — C3 — C32 — 3- 1+2 — C2×3- 1+2 — C22×3- 1+2
 Lower central C1 — C3 — C22×3- 1+2
 Upper central C1 — C2×C6 — C22×3- 1+2

Generators and relations for C22×3- 1+2
G = < a,b,c,d | a2=b2=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of C22×3- 1+2
On 36 points
Generators in S36
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 31 34)(30 36 33)

G:=sub<Sym(36)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,31,34)(30,36,33)>;

G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,31,34)(30,36,33) );

G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24),(28,31,34),(30,36,33)]])

C22×3- 1+2 is a maximal subgroup of   Dic9⋊C6  C62.C32  3- 1+2⋊A4  C62.6C32  C62.9C32

44 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A ··· 6F 6G ··· 6L 9A ··· 9F 18A ··· 18R order 1 2 2 2 3 3 3 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 1 1 3 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

44 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 3- 1+2 C2×3- 1+2 kernel C22×3- 1+2 C2×3- 1+2 C2×C18 C62 C18 C3×C6 C22 C2 # reps 1 3 6 2 18 6 2 6

Matrix representation of C22×3- 1+2 in GL4(𝔽19) generated by

 1 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 7 0 0 0 0 0 1 0 0 0 0 11 0 1 0 0
,
 11 0 0 0 0 1 0 0 0 0 11 0 0 0 0 7
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,0,0,1,0,1,0,0,0,0,11,0],[11,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7] >;

C22×3- 1+2 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xES-(3,1)");
// GroupNames label

G:=SmallGroup(108,31);
// by ID

G=gap.SmallGroup(108,31);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,147,253]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=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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