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G = C3×C4.10C42order 192 = 26·3

Direct product of C3 and C4.10C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C4.10C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C12 — C6×M4(2) — C3×C4.10C42
 Lower central C1 — C4 — C3×C4.10C42
 Upper central C1 — C12 — C3×C4.10C42

Generators and relations for C3×C4.10C42
G = < a,b,c,d | a3=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 122 in 86 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C24, C2×C12, C22×C6, C2×M4(2), C2×C24, C3×M4(2), C22×C12, C4.10C42, C6×M4(2), C3×C4.10C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C4.10C42, C3×C2.C42, C3×C4.10C42

Smallest permutation representation of C3×C4.10C42
On 48 points
Generators in S48
(1 39 19)(2 40 20)(3 33 21)(4 34 22)(5 35 23)(6 36 24)(7 37 17)(8 38 18)(9 46 25)(10 47 26)(11 48 27)(12 41 28)(13 42 29)(14 43 30)(15 44 31)(16 45 32)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 41 7 43 5 45 3 47)(2 44 4 42 6 48 8 46)(9 20 15 22 13 24 11 18)(10 19 12 17 14 23 16 21)(25 40 31 34 29 36 27 38)(26 39 28 37 30 35 32 33)

G:=sub<Sym(48)| (1,39,19)(2,40,20)(3,33,21)(4,34,22)(5,35,23)(6,36,24)(7,37,17)(8,38,18)(9,46,25)(10,47,26)(11,48,27)(12,41,28)(13,42,29)(14,43,30)(15,44,31)(16,45,32), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,41,7,43,5,45,3,47)(2,44,4,42,6,48,8,46)(9,20,15,22,13,24,11,18)(10,19,12,17,14,23,16,21)(25,40,31,34,29,36,27,38)(26,39,28,37,30,35,32,33)>;

G:=Group( (1,39,19)(2,40,20)(3,33,21)(4,34,22)(5,35,23)(6,36,24)(7,37,17)(8,38,18)(9,46,25)(10,47,26)(11,48,27)(12,41,28)(13,42,29)(14,43,30)(15,44,31)(16,45,32), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,41,7,43,5,45,3,47)(2,44,4,42,6,48,8,46)(9,20,15,22,13,24,11,18)(10,19,12,17,14,23,16,21)(25,40,31,34,29,36,27,38)(26,39,28,37,30,35,32,33) );

G=PermutationGroup([[(1,39,19),(2,40,20),(3,33,21),(4,34,22),(5,35,23),(6,36,24),(7,37,17),(8,38,18),(9,46,25),(10,47,26),(11,48,27),(12,41,28),(13,42,29),(14,43,30),(15,44,31),(16,45,32)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,47,45,43),(42,48,46,44)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,41,7,43,5,45,3,47),(2,44,4,42,6,48,8,46),(9,20,15,22,13,24,11,18),(10,19,12,17,14,23,16,21),(25,40,31,34,29,36,27,38),(26,39,28,37,30,35,32,33)]])

66 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 8A ··· 8L 12A 12B 12C 12D 12E ··· 12J 24A ··· 24X order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 8 ··· 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 2 ··· 2 4 ··· 4 1 1 1 1 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + - image C1 C2 C3 C4 C6 C12 D4 Q8 C3×D4 C3×Q8 C4.10C42 C3×C4.10C42 kernel C3×C4.10C42 C6×M4(2) C4.10C42 C2×C24 C2×M4(2) C2×C8 C2×C12 C22×C6 C2×C4 C23 C3 C1 # reps 1 3 2 12 6 24 3 1 6 2 2 4

Matrix representation of C3×C4.10C42 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 46 0 0 0 0 46 0 0 0 0 46 0 0 0 0 46
,
 0 0 1 0 0 0 0 1 46 0 0 0 0 27 0 0
,
 0 1 0 0 27 0 0 0 0 0 0 46 0 0 1 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,0,46,0,0,0,0,27,1,0,0,0,0,1,0,0],[0,27,0,0,1,0,0,0,0,0,0,1,0,0,46,0] >;

C3×C4.10C42 in GAP, Magma, Sage, TeX

C_3\times C_4._{10}C_4^2
% in TeX

G:=Group("C3xC4.10C4^2");
// GroupNames label

G:=SmallGroup(192,144);
// by ID

G=gap.SmallGroup(192,144);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,248,2111,172,6053,124]);
// Polycyclic

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

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