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## G = C3×D4⋊4D4order 192 = 26·3

### Direct product of C3 and D4⋊4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×D4 — C3×C8⋊C22 — C3×D4⋊4D4
 Lower central C1 — C2 — C2×C4 — C3×D4⋊4D4
 Upper central C1 — C6 — C2×C12 — C3×D4⋊4D4

Generators and relations for C3×D44D4
G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C22×C6, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C4×C12, C3×M4(2), C3×D8, C3×SD16, C6×D4, C6×D4, C3×C4○D4, C3×C4○D4, D44D4, C3×C4.D4, C3×C4≀C2, C3×C41D4, C3×C8⋊C22, C3×2+ 1+4, C3×D44D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, D44D4, C3×C22≀C2, C3×D44D4

Permutation representations of C3×D44D4
On 24 points - transitive group 24T350
Generators in S24
(1 11 7)(2 12 8)(3 9 5)(4 10 6)(13 21 17)(14 22 18)(15 23 19)(16 24 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)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 20)(8 19)(9 22)(10 21)(11 24)(12 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3)(5 7)(9 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,350);

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E ··· 6L 6M 6N 8A 8B 12A 12B 12C 12D 12E ··· 12L 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 6 6 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 ··· 12 24 24 24 24 size 1 1 2 4 4 4 4 8 1 1 2 2 4 4 4 4 1 1 2 2 4 ··· 4 8 8 8 8 2 2 2 2 4 ··· 4 8 8 8 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 C3×D4 C3×D4 C3×D4 D4⋊4D4 C3×D4⋊4D4 kernel C3×D4⋊4D4 C3×C4.D4 C3×C4≀C2 C3×C4⋊1D4 C3×C8⋊C22 C3×2+ 1+4 D4⋊4D4 C4.D4 C4≀C2 C4⋊1D4 C8⋊C22 2+ 1+4 C3×D4 C3×Q8 C22×C6 D4 Q8 C23 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 2 4 4 4 2 4

Matrix representation of C3×D44D4 in GL4(𝔽7) generated by

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

C3×D44D4 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_4D_4
% in TeX

G:=Group("C3xD4:4D4");
// GroupNames label

G:=SmallGroup(192,886);
// by ID

G=gap.SmallGroup(192,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,4204,2111,1068,172,3036]);
// Polycyclic

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

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