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## G = C3×C4.9C42order 192 = 26·3

### Direct product of C3 and C4.9C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C4.9C42
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C12 — C3×C42⋊C2 — C3×C4.9C42
 Lower central C1 — C4 — C3×C4.9C42
 Upper central C1 — C12 — C3×C4.9C42

Generators and relations for C3×C4.9C42
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 154 in 94 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22×C12, C4.9C42, C3×C42⋊C2, C6×M4(2), C3×C4.9C42
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, C4.9C42, C3×C2.C42, C3×C4.9C42

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 4F ··· 4M 6A 6B 6C ··· 6H 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K ··· 12Z 24A ··· 24H order 1 2 2 2 2 3 3 4 4 4 4 4 4 ··· 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 2 2 2 1 1 1 1 2 2 2 4 ··· 4 1 1 2 ··· 2 4 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 Q8 D4 C3×D4 C3×Q8 C3×D4 C4.9C42 C3×C4.9C42 kernel C3×C4.9C42 C3×C42⋊C2 C6×M4(2) C4.9C42 C4×C12 C2×C24 C42⋊C2 C2×M4(2) C42 C2×C8 C2×C12 C2×C12 C22×C6 C2×C4 C2×C4 C23 C3 C1 # reps 1 2 1 2 8 4 4 2 16 8 2 1 1 4 2 2 2 4

Matrix representation of C3×C4.9C42 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 46 0 0 0 0 46 0 0 0 0 46 0 0 0 0 46
,
 27 0 1 0 46 0 0 0 1 1 0 0 71 0 0 46
,
 1 0 0 14 0 72 0 13 0 0 46 46 0 0 0 27
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[27,46,1,71,0,0,1,0,1,0,0,0,0,0,0,46],[1,0,0,0,0,72,0,0,0,0,46,0,14,13,46,27] >;

C3×C4.9C42 in GAP, Magma, Sage, TeX

C_3\times C_4._9C_4^2
% in TeX

G:=Group("C3xC4.9C4^2");
// GroupNames label

G:=SmallGroup(192,143);
// by ID

G=gap.SmallGroup(192,143);
# by ID

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

G:=Group<a,b,c,d|a^3=b^4=c^4=d^4=1,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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