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## G = C3×C42⋊C22order 192 = 26·3

### Direct product of C3 and C42⋊C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C42⋊C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4≀C2 — C3×C42⋊C22
 Lower central C1 — C2 — C4 — C3×C42⋊C22
 Upper central C1 — C12 — C22×C12 — C3×C42⋊C22

Generators and relations for C3×C42⋊C22
G = < a,b,c,d,e | a3=b4=c4=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c, ebe=bc2, cd=dc, ce=ec, de=ed >

Subgroups: 258 in 154 conjugacy classes, 78 normal (46 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C42⋊C22, C3×C4≀C2, C3×C42⋊C2, C6×M4(2), C6×C4○D4, C3×C42⋊C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, C42⋊C22, C6×C22⋊C4, C3×C42⋊C22

Smallest permutation representation of C3×C42⋊C22
On 48 points
Generators in S48
(1 21 17)(2 22 18)(3 13 23)(4 14 24)(5 11 15)(6 12 16)(7 20 10)(8 19 9)(25 39 41)(26 40 42)(27 37 43)(28 38 44)(29 46 36)(30 47 33)(31 48 34)(32 45 35)
(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 13 5 7)(2 14 6 8)(3 15 10 17)(4 16 9 18)(11 20 21 23)(12 19 22 24)(25 30 27 32)(26 31 28 29)(33 43 35 41)(34 44 36 42)(37 45 39 47)(38 46 40 48)
(1 47)(2 40)(3 27)(4 31)(5 45)(6 38)(7 39)(8 46)(9 29)(10 25)(11 35)(12 44)(13 37)(14 48)(15 32)(16 28)(17 30)(18 26)(19 36)(20 41)(21 33)(22 42)(23 43)(24 34)
(1 14)(2 7)(3 16)(4 17)(5 8)(6 13)(9 15)(10 18)(11 19)(12 23)(20 22)(21 24)(25 26)(27 28)(29 32)(30 31)(33 34)(35 36)(37 38)(39 40)(41 42)(43 44)(45 46)(47 48)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F ··· 4K 6A 6B 6C ··· 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K ··· 12V 24A ··· 24H order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 4 ··· 4 6 6 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 2 2 2 4 4 1 1 1 1 2 2 2 4 ··· 4 1 1 2 ··· 2 4 4 4 4 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 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C12 C12 C12 D4 D4 C3×D4 C3×D4 C42⋊C22 C3×C42⋊C22 kernel C3×C42⋊C22 C3×C4≀C2 C3×C42⋊C2 C6×M4(2) C6×C4○D4 C42⋊C22 C6×D4 C6×Q8 C3×C4○D4 C4≀C2 C42⋊C2 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C12 C22×C6 C2×C4 C23 C3 C1 # reps 1 4 1 1 1 2 2 2 4 8 2 2 2 4 4 8 3 1 6 2 2 4

Matrix representation of C3×C42⋊C22 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 0 72 32 63 0 0 1 0 27 63 0 0 0 0 41 23 0 0 0 0 19 32
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 46
,
 72 71 0 0 0 0 0 1 0 0 0 0 0 0 0 12 1 0 0 0 0 1 0 0 0 0 1 61 0 0 0 0 0 71 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 61 35 0 0 72 0 72 0 0 0 0 0 12 38 0 0 0 0 2 61

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,32,27,41,19,0,0,63,63,23,32],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,0,0,1,0,0,0,12,1,61,71,0,0,1,0,0,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,61,72,12,2,0,0,35,0,38,61] >;

C3×C42⋊C22 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_2^2
% in TeX

G:=Group("C3xC4^2:C2^2");
// GroupNames label

G:=SmallGroup(192,854);
// by ID

G=gap.SmallGroup(192,854);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1059,4204,2111,172,124]);
// Polycyclic

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

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