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## G = C3×2+ 1+4order 96 = 25·3

### Direct product of C3 and 2+ 1+4

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

Aliases: C3×2+ 1+4, C6.19C24, C12.51C23, D4(C3×D4), Q8(C3×Q8), C4○D45C6, D44(C2×C6), (C2×D4)⋊6C6, Q85(C2×C6), (C6×D4)⋊15C2, C232(C2×C6), (C2×C12)⋊9C22, (C2×C6).7C23, C2.4(C23×C6), C4.9(C22×C6), (C3×D4)⋊13C22, (C22×C6)⋊2C22, (C3×Q8)⋊12C22, C22.2(C22×C6), (C3×D4)(C3×D4), (C3×Q8)(C3×Q8), (C2×C4)⋊2(C2×C6), (C3×C4○D4)⋊8C2, SmallGroup(96,224)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×2+ 1+4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C6×D4 — C3×2+ 1+4
 Lower central C1 — C2 — C3×2+ 1+4
 Upper central C1 — C6 — C3×2+ 1+4

Generators and relations for C3×2+ 1+4
G = < a,b,c,d,e | a3=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C2×D4, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, 2+ 1+4, C6×D4, C3×C4○D4, C3×2+ 1+4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2+ 1+4, C23×C6, C3×2+ 1+4

Permutation representations of C3×2+ 1+4
On 24 points - transitive group 24T92
Generators in S24
(1 15 9)(2 16 10)(3 13 11)(4 14 12)(5 19 22)(6 20 23)(7 17 24)(8 18 21)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)
(1 22 3 24)(2 23 4 21)(5 13 7 15)(6 14 8 16)(9 19 11 17)(10 20 12 18)
(5 7)(6 8)(17 19)(18 20)(21 23)(22 24)

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

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

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

G:=TransitiveGroup(24,92);

C3×2+ 1+4 is a maximal subgroup of
2+ 1+46S3  2+ 1+4.4S3  2+ 1+4.5S3  2+ 1+47S3  D12.32C23  D12.33C23  D6.C24  2+ 1+4⋊C9  2+ 1+42C9
C3×2+ 1+4 is a maximal quotient of
C3×D42  C3×Q82

51 conjugacy classes

 class 1 2A 2B ··· 2J 3A 3B 4A ··· 4F 6A 6B 6C ··· 6T 12A ··· 12L order 1 2 2 ··· 2 3 3 4 ··· 4 6 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 2 ··· 2 2 ··· 2

51 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C3 C6 C6 2+ 1+4 C3×2+ 1+4 kernel C3×2+ 1+4 C6×D4 C3×C4○D4 2+ 1+4 C2×D4 C4○D4 C3 C1 # reps 1 9 6 2 18 12 1 2

Matrix representation of C3×2+ 1+4 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 6 1 2 0 0 5 3 1 6 5 3 3 3 1 4 0
,
 0 3 1 0 3 2 5 4 6 1 6 2 5 1 5 6
,
 5 4 1 1 6 3 0 1 5 2 1 0 1 1 1 5
,
 4 2 0 6 1 3 3 2 5 6 0 3 3 0 6 0
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[6,0,6,3,1,5,5,1,2,3,3,4,0,1,3,0],[0,3,6,5,3,2,1,1,1,5,6,5,0,4,2,6],[5,6,5,1,4,3,2,1,1,0,1,1,1,1,0,5],[4,1,5,3,2,3,6,0,0,3,0,6,6,2,3,0] >;

C3×2+ 1+4 in GAP, Magma, Sage, TeX

C_3\times 2_+^{1+4}
% in TeX

G:=Group("C3xES+(2,2)");
// GroupNames label

G:=SmallGroup(96,224);
// by ID

G=gap.SmallGroup(96,224);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601,476,1347]);
// Polycyclic

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

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