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G = C3×C8.C8order 192 = 26·3

Direct product of C3 and C8.C8

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

Aliases: C3×C8.C8, C24.7C8, C8.1C24, C24.22Q8, C24.109D4, C42.8C12, M5(2).4C6, C8.6(C3×Q8), (C4×C8).10C6, C4.8(C2×C24), (C2×C8).9C12, C8.29(C3×D4), C6.15(C4⋊C8), (C4×C24).21C2, (C2×C24).20C4, C12.48(C2×C8), (C4×C12).23C4, C12.68(C4⋊C4), (C3×M5(2)).8C2, (C2×C6).17M4(2), (C2×C24).443C22, C22.5(C3×M4(2)), C2.5(C3×C4⋊C8), C4.19(C3×C4⋊C4), (C2×C8).97(C2×C6), (C2×C4).68(C2×C12), (C2×C12).329(C2×C4), SmallGroup(192,170)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C8.C8
C1C2C4C8C2×C8C2×C24C3×M5(2) — C3×C8.C8
C1C2C4 — C3×C8.C8
C1C24C2×C24 — C3×C8.C8

Generators and relations for C3×C8.C8
 G = < a,b,c | a3=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >

2C2
2C4
2C4
2C6
2C2×C4
2C12
2C12
2C16
2C16
2C2×C12
2C48
2C48

Smallest permutation representation of C3×C8.C8
On 48 points
Generators in S48
(1 44 27)(2 45 28)(3 46 29)(4 47 30)(5 48 31)(6 33 32)(7 34 17)(8 35 18)(9 36 19)(10 37 20)(11 38 21)(12 39 22)(13 40 23)(14 41 24)(15 42 25)(16 43 26)
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)(17 23 29 19 25 31 21 27)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 40 46 36 42 48 38 44)
(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)

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

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

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

84 conjugacy classes

class 1 2A2B3A3B4A4B4C···4G6A6B6C6D8A8B8C8D8E···8J12A12B12C12D12E···12N16A···16H24A···24H24I···24T48A···48P
order12233444···4666688888···81212121212···1216···1624···2424···2448···48
size11211112···2112211112···211112···24···41···12···24···4

84 irreducible representations

dim11111111111122222222
type++++-
imageC1C2C2C3C4C4C6C6C8C12C12C24D4Q8M4(2)C3×D4C3×Q8C3×M4(2)C8.C8C3×C8.C8
kernelC3×C8.C8C4×C24C3×M5(2)C8.C8C4×C12C2×C24C4×C8M5(2)C24C42C2×C8C8C24C24C2×C6C8C8C22C3C1
# reps1122222484416112224816

Matrix representation of C3×C8.C8 in GL3(𝔽97) generated by

6100
010
001
,
100
0500
0064
,
4700
001
0640
G:=sub<GL(3,GF(97))| [61,0,0,0,1,0,0,0,1],[1,0,0,0,50,0,0,0,64],[47,0,0,0,0,64,0,1,0] >;

C3×C8.C8 in GAP, Magma, Sage, TeX

C_3\times C_8.C_8
% in TeX

G:=Group("C3xC8.C8");
// GroupNames label

G:=SmallGroup(192,170);
// by ID

G=gap.SmallGroup(192,170);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,3027,248,102,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^8=1,c^8=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C3×C8.C8 in TeX

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