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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C3×C22⋊C4 — C3×C23⋊C4
 Lower central C1 — C2 — C22 — C3×C23⋊C4
 Upper central C1 — C6 — C22×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 >

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 2A 2B 2C 2D 2E 3A 3B 4A ··· 4E 6A 6B 6C ··· 6H 6I 6J 12A ··· 12J order 1 2 2 2 2 2 3 3 4 ··· 4 6 6 6 ··· 6 6 6 12 ··· 12 size 1 1 2 2 2 4 1 1 4 ··· 4 1 1 2 ··· 2 4 4 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 C23⋊C4 C3×C23⋊C4 kernel C3×C23⋊C4 C3×C22⋊C4 C6×D4 C23⋊C4 C2×C12 C22×C6 C22⋊C4 C2×D4 C2×C4 C23 C2×C6 C22 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 2

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 4 2 0 6 1 3 3 2 5 6 0 3 3 0 6 0
,
 1 1 2 2 2 0 2 2 5 2 1 0 1 1 4 5
,
 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 6 5 5 5 0 4 4 5 6 3 5 4 3 3 2 6
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

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