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## G = C3×C16⋊C22order 192 = 26·3

### Direct product of C3 and C16⋊C22

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×C16⋊C22
 Chief series C1 — C2 — C4 — C8 — C24 — C3×D8 — C3×D16 — C3×C16⋊C22
 Lower central C1 — C2 — C4 — C8 — C3×C16⋊C22
 Upper central C1 — C6 — C2×C12 — C2×C24 — C3×C16⋊C22

Generators and relations for C3×C16⋊C22
G = < a,b,c,d | a3=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >

Subgroups: 226 in 90 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, D8, SD16, Q16, C2×D4, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, M5(2), D16, SD32, C2×D8, C4○D8, C48, C2×C24, C3×D8, C3×D8, C3×D8, C3×SD16, C3×Q16, C6×D4, C3×C4○D4, C16⋊C22, C3×M5(2), C3×D16, C3×SD32, C6×D8, C3×C4○D8, C3×C16⋊C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C2×D8, C3×D8, C6×D4, C16⋊C22, C6×D8, C3×C16⋊C22

Smallest permutation representation of C3×C16⋊C22
On 48 points
Generators in S48
(1 19 43)(2 20 44)(3 21 45)(4 22 46)(5 23 47)(6 24 48)(7 25 33)(8 26 34)(9 27 35)(10 28 36)(11 29 37)(12 30 38)(13 31 39)(14 32 40)(15 17 41)(16 18 42)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 21)(18 28)(20 26)(22 24)(23 31)(25 29)(30 32)(33 37)(34 44)(36 42)(38 40)(39 47)(41 45)(46 48)
(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 6A 6B 6C 6D 6E ··· 6J 8A 8B 8C 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 2 2 3 3 4 4 4 6 6 6 6 6 ··· 6 8 8 8 12 12 12 12 12 12 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 8 8 8 1 1 2 2 8 1 1 2 2 8 ··· 8 2 2 4 2 2 2 2 8 8 4 4 4 4 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D8 D8 C3×D4 C3×D4 C3×D8 C3×D8 C16⋊C22 C3×C16⋊C22 kernel C3×C16⋊C22 C3×M5(2) C3×D16 C3×SD32 C6×D8 C3×C4○D8 C16⋊C22 M5(2) D16 SD32 C2×D8 C4○D8 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 2 2 2 2 4 4 2 4

Matrix representation of C3×C16⋊C22 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 1 6 4 2 4 6 5 1 6 0 5 0 6 1 3 2
,
 1 0 0 0 0 6 0 0 0 0 0 4 0 0 2 0
,
 1 0 6 3 0 2 3 2 0 3 2 2 0 1 1 2
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,4,6,6,6,6,0,1,4,5,5,3,2,1,0,2],[1,0,0,0,0,6,0,0,0,0,0,2,0,0,4,0],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2] >;

C3×C16⋊C22 in GAP, Magma, Sage, TeX

C_3\times C_{16}\rtimes C_2^2
% in TeX

G:=Group("C3xC16:C2^2");
// GroupNames label

G:=SmallGroup(192,942);
// by ID

G=gap.SmallGroup(192,942);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,2102,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
// generators/relations

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