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## G = C22×C3.A4order 144 = 24·32

### Direct product of C22 and C3.A4

Aliases: C22×C3.A4, C23⋊C18, C241C9, C22⋊(C2×C18), C6.7(C2×A4), (C2×C6).5A4, C3.(C22×A4), (C22×C6).3C6, (C23×C6).1C3, (C2×C6).2(C2×C6), SmallGroup(144,110)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C22×C3.A4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C22×C3.A4
 Lower central C22 — C22×C3.A4
 Upper central C1 — C2×C6

Generators and relations for C22×C3.A4
G = < a,b,c,d,e,f | a2=b2=c3=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 159 in 68 conjugacy classes, 25 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C23, C9, C2×C6, C2×C6, C24, C18, C22×C6, C22×C6, C3.A4, C2×C18, C23×C6, C2×C3.A4, C22×C3.A4
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C22×A4, C2×C3.A4, C22×C3.A4

Smallest permutation representation of C22×C3.A4
On 36 points
Generators in S36
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 19)(18 20)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 23)(2 33)(3 14)(4 26)(5 36)(6 17)(7 20)(8 30)(9 11)(10 21)(12 32)(13 24)(15 35)(16 27)(18 29)(19 28)(22 31)(25 34)
(1 12)(2 24)(3 34)(4 15)(5 27)(6 28)(7 18)(8 21)(9 31)(10 30)(11 22)(13 33)(14 25)(16 36)(17 19)(20 29)(23 32)(26 35)
(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)

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

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

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

C22×C3.A4 is a maximal subgroup of   C23.D18  C3.A42  C24⋊3- 1+2  C2423- 1+2  A4×C2×C18
C22×C3.A4 is a maximal quotient of   2+ 1+4⋊C9  2- 1+4⋊C9

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A ··· 6F 6G ··· 6N 9A ··· 9F 18A ··· 18R order 1 2 2 2 2 2 2 2 3 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 3 3 3 3 1 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 C9 C18 A4 C2×A4 C3.A4 C2×C3.A4 kernel C22×C3.A4 C2×C3.A4 C23×C6 C22×C6 C24 C23 C2×C6 C6 C22 C2 # reps 1 3 2 6 6 18 1 3 2 6

Matrix representation of C22×C3.A4 in GL4(𝔽19) generated by

 18 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 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 18 0 0 0 0 18 0 0 0 0 1
,
 9 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,1],[9,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C22×C3.A4 in GAP, Magma, Sage, TeX

C_2^2\times C_3.A_4
% in TeX

G:=Group("C2^2xC3.A4");
// GroupNames label

G:=SmallGroup(144,110);
// by ID

G=gap.SmallGroup(144,110);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,68,556,989]);
// Polycyclic

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

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