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## G = C3×M4(2).C4order 192 = 26·3

### Direct product of C3 and M4(2).C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×M4(2).C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C8.C4 — C3×M4(2).C4
 Lower central C1 — C2 — C4 — C3×M4(2).C4
 Upper central C1 — C12 — C22×C12 — C3×M4(2).C4

Generators and relations for C3×M4(2).C4
G = < a,b,c,d | a3=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >

Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, C8.C4, C2×M4(2), C2×M4(2), C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, M4(2).C4, C3×C8.C4, C6×M4(2), C6×M4(2), C3×M4(2).C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, M4(2).C4, C6×C4⋊C4, C3×M4(2).C4

Smallest permutation representation of C3×M4(2).C4
On 48 points
Generators in S48
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 25 17)(6 26 18)(7 27 19)(8 28 20)(9 39 44)(10 40 45)(11 33 46)(12 34 47)(13 35 48)(14 36 41)(15 37 42)(16 38 43)
(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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)
(1 41 7 43 5 45 3 47)(2 48 8 42 6 44 4 46)(9 32 11 30 13 28 15 26)(10 31 12 29 14 27 16 25)(17 40 23 34 21 36 19 38)(18 39 24 33 22 35 20 37)

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

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

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

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 1 1 2 2 2 2 2 2 4 4 type + + + + - - image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 Q8 C3×D4 C3×Q8 C3×Q8 M4(2).C4 C3×M4(2).C4 kernel C3×M4(2).C4 C3×C8.C4 C6×M4(2) M4(2).C4 C3×M4(2) C8.C4 C2×M4(2) M4(2) C2×C12 C2×C12 C22×C6 C2×C4 C2×C4 C23 C3 C1 # reps 1 4 3 2 8 8 6 16 2 1 1 4 2 2 2 4

Matrix representation of C3×M4(2).C4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 46 54 0 0 59 27 0 0 59 27 0 1 46 27 27 0
,
 72 0 0 0 72 1 0 0 72 0 1 0 0 0 0 72
,
 1 0 71 0 0 0 72 1 60 0 72 0 60 27 72 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,59,59,46,54,27,27,27,0,0,0,27,0,0,1,0],[72,72,72,0,0,1,0,0,0,0,1,0,0,0,0,72],[1,0,60,60,0,0,0,27,71,72,72,72,0,1,0,0] >;

C3×M4(2).C4 in GAP, Magma, Sage, TeX

C_3\times M_4(2).C_4
% in TeX

G:=Group("C3xM4(2).C4");
// GroupNames label

G:=SmallGroup(192,863);
// by ID

G=gap.SmallGroup(192,863);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1059,4204,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations

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