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G = C22×C32⋊C4order 144 = 24·32

Direct product of C22 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

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

C1C32 — C22×C32⋊C4
C1C32C3⋊S3C32⋊C4C2×C32⋊C4 — C22×C32⋊C4
C32 — C22×C32⋊C4
C1C22

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 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H6A6B6C6D6E6F
 size 111199994499999999444444
ρ1111111111111111111111111    trivial
ρ211-1-11-1-111111-1-1-1-111-111-1-1-1    linear of order 2
ρ31-11-1-1-11111-11-1-111-11-1-1-111-1    linear of order 2
ρ41-1-11-11-1111-1111-1-1-111-1-1-1-11    linear of order 2
ρ51-1-11-11-11111-1-1-1111-11-1-1-1-11    linear of order 2
ρ61-11-1-1-111111-111-1-11-1-1-1-111-1    linear of order 2
ρ711-1-11-1-1111-1-11111-1-1-111-1-1-1    linear of order 2
ρ81111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ91111-1-1-1-111-i-ii-ii-iii111111    linear of order 4
ρ101-11-111-1-111-iii-i-iii-i-1-1-111-1    linear of order 4
ρ111-11-111-1-111i-i-iii-i-ii-1-1-111-1    linear of order 4
ρ121111-1-1-1-111ii-ii-ii-i-i111111    linear of order 4
ρ131-1-111-11-111-ii-iii-ii-i1-1-1-1-11    linear of order 4
ρ1411-1-1-111-111-i-i-ii-iiii-111-1-1-1    linear of order 4
ρ1511-1-1-111-111iii-ii-i-i-i-111-1-1-1    linear of order 4
ρ161-1-111-11-111i-ii-i-ii-ii1-1-1-1-11    linear of order 4
ρ174-4-4400001-20000000012-12-1-2    orthogonal lifted from C2×C32⋊C4
ρ18444400001-2000000001-21-21-2    orthogonal lifted from C32⋊C4
ρ1944-4-400001-200000000-1-212-12    orthogonal lifted from C2×C32⋊C4
ρ204-44-400001-200000000-12-1-212    orthogonal lifted from C2×C32⋊C4
ρ2144-4-40000-210000000021-2-12-1    orthogonal lifted from C2×C32⋊C4
ρ224-44-40000-21000000002-121-2-1    orthogonal lifted from C2×C32⋊C4
ρ234-4-440000-2100000000-2-12-121    orthogonal lifted from C2×C32⋊C4
ρ2444440000-2100000000-21-21-21    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)

10000
012000
001200
000120
000012
,
120000
01000
00100
00010
00001
,
10000
01000
00100
00001
0001212
,
10000
0121200
01000
00001
0001212
,
50000
000120
000012
012000
01100

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

Export

Character table of C22×C32⋊C4 in TeX

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