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G = C3×C16⋊C4order 192 = 26·3

Direct product of C3 and C16⋊C4

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

Aliases: C3×C16⋊C4, C484C4, C162C12, C42.1C12, C12.35C42, M5(2).2C6, C12.33M4(2), (C4×C12).4C4, (C2×C24).5C4, (C2×C8).2C12, C8⋊C4.4C6, C4.11(C4×C12), C8.19(C2×C12), C24.87(C2×C4), C6.9(C8⋊C4), C4.6(C3×M4(2)), (C3×M5(2)).6C2, (C2×C6).16M4(2), (C2×C24).308C22, C22.4(C3×M4(2)), C2.3(C3×C8⋊C4), (C2×C8).45(C2×C6), (C3×C8⋊C4).9C2, (C2×C4).66(C2×C12), (C2×C12).327(C2×C4), SmallGroup(192,153)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C16⋊C4
C1C2C4C2×C4C2×C8C2×C24C3×M5(2) — C3×C16⋊C4
C1C4 — C3×C16⋊C4
C1C12 — C3×C16⋊C4

Generators and relations for C3×C16⋊C4
 G = < a,b,c | a3=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >

2C2
4C4
2C6
2C8
2C2×C4
4C12
2C2×C12
2C24

Smallest permutation representation of C3×C16⋊C4
On 48 points
Generators in S48
(1 40 28)(2 41 29)(3 42 30)(4 43 31)(5 44 32)(6 45 17)(7 46 18)(8 47 19)(9 48 20)(10 33 21)(11 34 22)(12 35 23)(13 36 24)(14 37 25)(15 38 26)(16 39 27)
(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 6 10 14)(3 11)(4 16 12 8)(7 15)(17 21 25 29)(18 26)(19 31 27 23)(22 30)(33 37 41 45)(34 42)(35 47 43 39)(38 46)

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

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

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

66 conjugacy classes

class 1 2A2B3A3B4A4B4C4D4E6A6B6C6D8A8B8C8D8E8F12A12B12C12D12E12F12G12H12I12J16A···16H24A···24H24I24J24K24L48A···48P
order122334444466668888881212121212121212121216···1624···242424242448···48
size1121111244112222224411112244444···42···244444···4

66 irreducible representations

dim111111111111222244
type+++
imageC1C2C2C3C4C4C4C6C6C12C12C12M4(2)M4(2)C3×M4(2)C3×M4(2)C16⋊C4C3×C16⋊C4
kernelC3×C16⋊C4C3×C8⋊C4C3×M5(2)C16⋊C4C48C4×C12C2×C24C8⋊C4M5(2)C16C42C2×C8C12C2×C6C4C22C3C1
# reps1122822241644224424

Matrix representation of C3×C16⋊C4 in GL4(𝔽97) generated by

61000
06100
00610
00061
,
960159
00049
7522086
2001
,
10038
09600
00751
00022
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[96,0,75,2,0,0,22,0,1,0,0,0,59,49,86,1],[1,0,0,0,0,96,0,0,0,0,75,0,38,0,1,22] >;

C3×C16⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{16}\rtimes C_4
% in TeX

G:=Group("C3xC16:C4");
// GroupNames label

G:=SmallGroup(192,153);
// by ID

G=gap.SmallGroup(192,153);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,701,176,1522,136,4204,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations

Export

Subgroup lattice of C3×C16⋊C4 in TeX

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