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## G = C22×C32⋊C4order 144 = 24·32

### Direct product of C22 and C32⋊C4

Aliases: C22×C32⋊C4, C622C4, C3⋊S3.3C23, C322(C22×C4), (C2×C3⋊S3)⋊5C4, C3⋊S33(C2×C4), (C3×C6)⋊1(C2×C4), (C22×C3⋊S3).5C2, (C2×C3⋊S3).20C22, SmallGroup(144,191)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C22×C32⋊C4
 Lower central C32 — C22×C32⋊C4
 Upper central C1 — C22

Generators and relations for C22×C32⋊C4
G = < a,b,c,d,e | a2=b2=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 382 in 86 conjugacy classes, 32 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C4 [×4], C22, C22 [×6], S3 [×8], C6 [×6], C2×C4 [×6], C23, C32, D6 [×12], C2×C6 [×2], C22×C4, C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C22×S3 [×2], C32⋊C4 [×4], C2×C3⋊S3 [×6], C62, C2×C32⋊C4 [×6], C22×C3⋊S3, C22×C32⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4

Character table of C22×C32⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F size 1 1 1 1 9 9 9 9 4 4 9 9 9 9 9 9 9 9 4 4 4 4 4 4 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 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 1 1 -1 -1 -1 -1 1 1 -i -i i -i i -i i i 1 1 1 1 1 1 linear of order 4 ρ10 1 -1 1 -1 1 1 -1 -1 1 1 -i i i -i -i i i -i -1 -1 -1 1 1 -1 linear of order 4 ρ11 1 -1 1 -1 1 1 -1 -1 1 1 i -i -i i i -i -i i -1 -1 -1 1 1 -1 linear of order 4 ρ12 1 1 1 1 -1 -1 -1 -1 1 1 i i -i i -i i -i -i 1 1 1 1 1 1 linear of order 4 ρ13 1 -1 -1 1 1 -1 1 -1 1 1 -i i -i i i -i i -i 1 -1 -1 -1 -1 1 linear of order 4 ρ14 1 1 -1 -1 -1 1 1 -1 1 1 -i -i -i i -i i i i -1 1 1 -1 -1 -1 linear of order 4 ρ15 1 1 -1 -1 -1 1 1 -1 1 1 i i i -i i -i -i -i -1 1 1 -1 -1 -1 linear of order 4 ρ16 1 -1 -1 1 1 -1 1 -1 1 1 i -i i -i -i i -i i 1 -1 -1 -1 -1 1 linear of order 4 ρ17 4 -4 -4 4 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 1 2 -1 2 -1 -2 orthogonal lifted from C2×C32⋊C4 ρ18 4 4 4 4 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 1 -2 1 -2 1 -2 orthogonal lifted from C32⋊C4 ρ19 4 4 -4 -4 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 -1 -2 1 2 -1 2 orthogonal lifted from C2×C32⋊C4 ρ20 4 -4 4 -4 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 -1 2 -1 -2 1 2 orthogonal lifted from C2×C32⋊C4 ρ21 4 4 -4 -4 0 0 0 0 -2 1 0 0 0 0 0 0 0 0 2 1 -2 -1 2 -1 orthogonal lifted from C2×C32⋊C4 ρ22 4 -4 4 -4 0 0 0 0 -2 1 0 0 0 0 0 0 0 0 2 -1 2 1 -2 -1 orthogonal lifted from C2×C32⋊C4 ρ23 4 -4 -4 4 0 0 0 0 -2 1 0 0 0 0 0 0 0 0 -2 -1 2 -1 2 1 orthogonal lifted from C2×C32⋊C4 ρ24 4 4 4 4 0 0 0 0 -2 1 0 0 0 0 0 0 0 0 -2 1 -2 1 -2 1 orthogonal lifted from C32⋊C4

Permutation representations of C22×C32⋊C4
On 24 points - transitive group 24T241
Generators in S24
(1 6)(2 5)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 20)
(1 8)(2 7)(3 5)(4 6)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)
(1 15 13)(2 14 16)(3 24 22)(4 21 23)(5 18 20)(6 19 17)(7 11 9)(8 12 10)
(2 16 14)(3 22 24)(5 20 18)(7 9 11)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

G:=sub<Sym(24)| (1,6)(2,5)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20), (1,8)(2,7)(3,5)(4,6)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (1,15,13)(2,14,16)(3,24,22)(4,21,23)(5,18,20)(6,19,17)(7,11,9)(8,12,10), (2,16,14)(3,22,24)(5,20,18)(7,9,11), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

G:=Group( (1,6)(2,5)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20), (1,8)(2,7)(3,5)(4,6)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (1,15,13)(2,14,16)(3,24,22)(4,21,23)(5,18,20)(6,19,17)(7,11,9)(8,12,10), (2,16,14)(3,22,24)(5,20,18)(7,9,11), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );

G=PermutationGroup([(1,6),(2,5),(3,7),(4,8),(9,22),(10,23),(11,24),(12,21),(13,17),(14,18),(15,19),(16,20)], [(1,8),(2,7),(3,5),(4,6),(9,16),(10,13),(11,14),(12,15),(17,23),(18,24),(19,21),(20,22)], [(1,15,13),(2,14,16),(3,24,22),(4,21,23),(5,18,20),(6,19,17),(7,11,9),(8,12,10)], [(2,16,14),(3,22,24),(5,20,18),(7,9,11)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)])

G:=TransitiveGroup(24,241);

On 24 points - transitive group 24T242
Generators in S24
(1 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(13 24)(14 21)(15 22)(16 23)
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)(17 21)(18 22)(19 23)(20 24)
(2 6 20)(4 18 8)(10 15 24)(12 22 13)
(1 5 19)(2 6 20)(3 17 7)(4 18 8)(9 14 23)(10 15 24)(11 21 16)(12 22 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

G:=sub<Sym(24)| (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,11)(10,12)(13,24)(14,21)(15,22)(16,23), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,21)(18,22)(19,23)(20,24), (2,6,20)(4,18,8)(10,15,24)(12,22,13), (1,5,19)(2,6,20)(3,17,7)(4,18,8)(9,14,23)(10,15,24)(11,21,16)(12,22,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

G:=Group( (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,11)(10,12)(13,24)(14,21)(15,22)(16,23), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,21)(18,22)(19,23)(20,24), (2,6,20)(4,18,8)(10,15,24)(12,22,13), (1,5,19)(2,6,20)(3,17,7)(4,18,8)(9,14,23)(10,15,24)(11,21,16)(12,22,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );

G=PermutationGroup([(1,3),(2,4),(5,17),(6,18),(7,19),(8,20),(9,11),(10,12),(13,24),(14,21),(15,22),(16,23)], [(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13),(17,21),(18,22),(19,23),(20,24)], [(2,6,20),(4,18,8),(10,15,24),(12,22,13)], [(1,5,19),(2,6,20),(3,17,7),(4,18,8),(9,14,23),(10,15,24),(11,21,16),(12,22,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)])

G:=TransitiveGroup(24,242);

C22×C32⋊C4 is a maximal subgroup of   C62.D4  C62.Q8  (C6×C12)⋊2C4  C22⋊F9  C62⋊D4  C62⋊Q8
C22×C32⋊C4 is a maximal quotient of   C3⋊S3⋊M4(2)  (C6×C12)⋊5C4  C62.(C2×C4)  C12⋊S3.C4

Matrix representation of C22×C32⋊C4 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 12 12
,
 1 0 0 0 0 0 12 12 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 12 12
,
 5 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 12 0 0 0 0 1 1 0 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,12],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,1,12],[5,0,0,0,0,0,0,0,12,1,0,0,0,0,1,0,12,0,0,0,0,0,12,0,0] >;

C22×C32⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes C_4
% in TeX

G:=Group("C2^2xC3^2:C4");
// GroupNames label

G:=SmallGroup(144,191);
// by ID

G=gap.SmallGroup(144,191);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,48,3364,142,4613,455]);
// Polycyclic

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

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