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G = C3×C23⋊C4order 96 = 25·3

Direct product of C3 and C23⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×C23⋊C4, C23⋊C12, (C2×C4)⋊C12, (C2×C12)⋊2C4, C22⋊C41C6, (C22×C6)⋊1C4, (C6×D4).7C2, (C2×D4).1C6, (C2×C6).21D4, C23.1(C2×C6), C22.2(C3×D4), C22.2(C2×C12), C6.21(C22⋊C4), (C22×C6).1C22, (C3×C22⋊C4)⋊2C2, (C2×C6).19(C2×C4), C2.3(C3×C22⋊C4), SmallGroup(96,49)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C23⋊C4
C1C2C22C23C22×C6C3×C22⋊C4 — C3×C23⋊C4
C1C2C22 — C3×C23⋊C4
C1C6C22×C6 — C3×C23⋊C4

Generators and relations for C3×C23⋊C4
 G = < a,b,c,d,e | a3=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

2C2
2C2
2C2
4C2
2C22
2C4
4C22
4C4
4C22
4C4
2C6
2C6
2C6
4C6
2D4
2C2×C4
2D4
2C2×C4
2C12
2C2×C6
4C2×C6
4C12
4C12
4C2×C6
2C3×D4
2C2×C12
2C2×C12
2C3×D4

Permutation representations of C3×C23⋊C4
On 24 points - transitive group 24T91
Generators in S24
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
(2 12)(3 9)(5 16)(8 15)(17 21)(20 24)
(2 12)(4 10)(6 13)(8 15)(18 22)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(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)| (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(3,9)(5,16)(8,15)(17,21)(20,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (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( (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(3,9)(5,16)(8,15)(17,21)(20,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (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([(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,21),(10,13,22),(11,14,23),(12,15,24)], [(2,12),(3,9),(5,16),(8,15),(17,21),(20,24)], [(2,12),(4,10),(6,13),(8,15),(18,22),(20,24)], [(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(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(24,91);

On 24 points - transitive group 24T93
Generators in S24
(1 9 8)(2 10 7)(3 12 5)(4 11 6)(13 20 23)(14 17 24)(15 18 21)(16 19 22)
(1 21)(2 22)(3 24)(4 23)(5 17)(6 20)(7 19)(8 18)(9 15)(10 16)(11 13)(12 14)
(1 4)(6 8)(9 11)(13 15)(18 20)(21 23)
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(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)| (1,9,8)(2,10,7)(3,12,5)(4,11,6)(13,20,23)(14,17,24)(15,18,21)(16,19,22), (1,21)(2,22)(3,24)(4,23)(5,17)(6,20)(7,19)(8,18)(9,15)(10,16)(11,13)(12,14), (1,4)(6,8)(9,11)(13,15)(18,20)(21,23), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (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( (1,9,8)(2,10,7)(3,12,5)(4,11,6)(13,20,23)(14,17,24)(15,18,21)(16,19,22), (1,21)(2,22)(3,24)(4,23)(5,17)(6,20)(7,19)(8,18)(9,15)(10,16)(11,13)(12,14), (1,4)(6,8)(9,11)(13,15)(18,20)(21,23), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (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([(1,9,8),(2,10,7),(3,12,5),(4,11,6),(13,20,23),(14,17,24),(15,18,21),(16,19,22)], [(1,21),(2,22),(3,24),(4,23),(5,17),(6,20),(7,19),(8,18),(9,15),(10,16),(11,13),(12,14)], [(1,4),(6,8),(9,11),(13,15),(18,20),(21,23)], [(1,4),(2,3),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(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(24,93);

On 24 points - transitive group 24T115
Generators in S24
(1 14 21)(2 15 22)(3 16 23)(4 13 24)(5 19 9)(6 20 10)(7 17 11)(8 18 12)
(1 2)(3 7)(4 6)(5 8)(9 12)(10 24)(11 23)(13 20)(14 15)(16 17)(18 19)(21 22)
(1 3)(2 7)(4 5)(6 8)(9 24)(10 12)(11 22)(13 19)(14 16)(15 17)(18 20)(21 23)
(1 8)(2 5)(3 6)(4 7)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 20)
(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)| (1,14,21)(2,15,22)(3,16,23)(4,13,24)(5,19,9)(6,20,10)(7,17,11)(8,18,12), (1,2)(3,7)(4,6)(5,8)(9,12)(10,24)(11,23)(13,20)(14,15)(16,17)(18,19)(21,22), (1,3)(2,7)(4,5)(6,8)(9,24)(10,12)(11,22)(13,19)(14,16)(15,17)(18,20)(21,23), (1,8)(2,5)(3,6)(4,7)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20), (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( (1,14,21)(2,15,22)(3,16,23)(4,13,24)(5,19,9)(6,20,10)(7,17,11)(8,18,12), (1,2)(3,7)(4,6)(5,8)(9,12)(10,24)(11,23)(13,20)(14,15)(16,17)(18,19)(21,22), (1,3)(2,7)(4,5)(6,8)(9,24)(10,12)(11,22)(13,19)(14,16)(15,17)(18,20)(21,23), (1,8)(2,5)(3,6)(4,7)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20), (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([(1,14,21),(2,15,22),(3,16,23),(4,13,24),(5,19,9),(6,20,10),(7,17,11),(8,18,12)], [(1,2),(3,7),(4,6),(5,8),(9,12),(10,24),(11,23),(13,20),(14,15),(16,17),(18,19),(21,22)], [(1,3),(2,7),(4,5),(6,8),(9,24),(10,12),(11,22),(13,19),(14,16),(15,17),(18,20),(21,23)], [(1,8),(2,5),(3,6),(4,7),(9,22),(10,23),(11,24),(12,21),(13,17),(14,18),(15,19),(16,20)], [(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(24,115);

C3×C23⋊C4 is a maximal subgroup of   C3⋊C2≀C4  (C2×D4).D6  C23.D12  C23.2D12  C23⋊C45S3  C23⋊D12  C23.5D12

33 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4E6A6B6C···6H6I6J12A···12J
order122222334···4666···66612···12
size112224114···4112···2444···4

33 irreducible representations

dim11111111112244
type+++++
imageC1C2C2C3C4C4C6C6C12C12D4C3×D4C23⋊C4C3×C23⋊C4
kernelC3×C23⋊C4C3×C22⋊C4C6×D4C23⋊C4C2×C12C22×C6C22⋊C4C2×D4C2×C4C23C2×C6C22C3C1
# reps12122242442412

Matrix representation of C3×C23⋊C4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
4206
1332
5603
3060
,
1122
2022
5210
1145
,
6000
0600
0060
0006
,
6555
0445
6354
3326
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,1,5,3,2,3,6,0,0,3,0,6,6,2,3,0],[1,2,5,1,1,0,2,1,2,2,1,4,2,2,0,5],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[6,0,6,3,5,4,3,3,5,4,5,2,5,5,4,6] >;

C3×C23⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes C_4
% in TeX

G:=Group("C3xC2^3:C4");
// GroupNames label

G:=SmallGroup(96,49);
// by ID

G=gap.SmallGroup(96,49);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,1090]);
// Polycyclic

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

Export

Subgroup lattice of C3×C23⋊C4 in TeX

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