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## G = C3×C42⋊6C4order 192 = 26·3

### Direct product of C3 and C42⋊6C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C42⋊6C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C12 — C3×C42⋊C2 — C3×C42⋊6C4
 Lower central C1 — C2 — C4 — C3×C42⋊6C4
 Upper central C1 — C2×C12 — C22×C12 — C3×C42⋊6C4

Generators and relations for C3×C426C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >

Subgroups: 170 in 110 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C426C4, C2×C4×C12, C3×C42⋊C2, C6×M4(2), C3×C426C4
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≀C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C426C4, C3×C2.C42, C3×C4≀C2, C3×C426C4

Smallest permutation representation of C3×C426C4
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 44 40)(26 41 37)(27 42 38)(28 43 39)(29 35 48)(30 36 45)(31 33 46)(32 34 47)
(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 6 2 5)(3 8 4 7)(9 14 10 13)(11 16 12 15)(17 22 18 21)(19 24 20 23)(25 28 27 26)(29 32 31 30)(33 36 35 34)(37 40 39 38)(41 44 43 42)(45 48 47 46)
(1 35 3 25)(2 33 4 27)(5 34 7 28)(6 36 8 26)(9 29 11 40)(10 31 12 38)(13 32 15 39)(14 30 16 37)(17 48 19 44)(18 46 20 42)(21 47 23 43)(22 45 24 41)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A ··· 12H 12I ··· 12AB 12AC ··· 12AJ 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 1 1 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C12 C12 C12 D4 Q8 D4 C3×D4 C3×Q8 C3×D4 C4≀C2 C3×C4≀C2 kernel C3×C42⋊6C4 C2×C4×C12 C3×C42⋊C2 C6×M4(2) C42⋊6C4 C4×C12 C3×C4⋊C4 C3×M4(2) C2×C42 C42⋊C2 C2×M4(2) C42 C4⋊C4 M4(2) C2×C12 C2×C12 C22×C6 C2×C4 C2×C4 C23 C6 C2 # reps 1 1 1 1 2 4 4 4 2 2 2 8 8 8 2 1 1 4 2 2 8 16

Matrix representation of C3×C426C4 in GL3(𝔽73) generated by

 64 0 0 0 8 0 0 0 8
,
 72 0 0 0 1 0 0 13 27
,
 1 0 0 0 27 0 0 46 46
,
 27 0 0 0 72 71 0 0 1
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[72,0,0,0,1,13,0,0,27],[1,0,0,0,27,46,0,0,46],[27,0,0,0,72,0,0,71,1] >;

C3×C426C4 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_6C_4
% in TeX

G:=Group("C3xC4^2:6C4");
// GroupNames label

G:=SmallGroup(192,145);
// by ID

G=gap.SmallGroup(192,145);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,248,6053]);
// 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*b*d^-1=b*c=c*b,d*c*d^-1=c^-1>;
// generators/relations

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