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G = C62.4C8order 288 = 25·32

2nd non-split extension by C62 of C8 acting via C8/C2=C4

metabelian, soluble, monomial

Aliases: C62.4C8, C326M5(2), (C3×C12).6C8, (C6×C12).3C4, C4.(C322C8), C322C164C2, C324C8.10C4, C22.(C322C8), C324C8.35C22, (C3×C6).23(C2×C8), C4.20(C2×C32⋊C4), (C3×C12).17(C2×C4), (C2×C4).5(C32⋊C4), C2.3(C2×C322C8), (C2×C324C8).20C2, SmallGroup(288,421)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.4C8
C1C32C3×C6C3×C12C324C8C322C16 — C62.4C8
C32C3×C6 — C62.4C8
C1C4C2×C4

Generators and relations for C62.4C8
 G = < a,b,c | a6=b6=1, c8=b3, ab=ba, cac-1=a-1b, cbc-1=a4b >

2C2
2C3
2C3
2C6
2C6
2C6
2C6
2C6
2C6
9C8
9C8
2C12
2C12
2C2×C6
2C2×C6
2C12
2C12
2C3×C6
9C2×C8
9C16
9C16
2C2×C12
2C2×C12
6C3⋊C8
6C3⋊C8
6C3⋊C8
6C3⋊C8
9M5(2)
6C2×C3⋊C8
6C2×C3⋊C8

Smallest permutation representation of C62.4C8
On 48 points
Generators in S48
(1 36 17)(2 26 37 10 18 45)(3 19 38)(4 47 20 12 39 28)(5 40 21)(6 30 41 14 22 33)(7 23 42)(8 35 24 16 43 32)(9 44 25)(11 27 46)(13 48 29)(15 31 34)
(1 9)(2 45 18 10 37 26)(3 11)(4 28 39 12 20 47)(5 13)(6 33 22 14 41 30)(7 15)(8 32 43 16 24 35)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)
(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)| (1,36,17)(2,26,37,10,18,45)(3,19,38)(4,47,20,12,39,28)(5,40,21)(6,30,41,14,22,33)(7,23,42)(8,35,24,16,43,32)(9,44,25)(11,27,46)(13,48,29)(15,31,34), (1,9)(2,45,18,10,37,26)(3,11)(4,28,39,12,20,47)(5,13)(6,33,22,14,41,30)(7,15)(8,32,43,16,24,35)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48), (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( (1,36,17)(2,26,37,10,18,45)(3,19,38)(4,47,20,12,39,28)(5,40,21)(6,30,41,14,22,33)(7,23,42)(8,35,24,16,43,32)(9,44,25)(11,27,46)(13,48,29)(15,31,34), (1,9)(2,45,18,10,37,26)(3,11)(4,28,39,12,20,47)(5,13)(6,33,22,14,41,30)(7,15)(8,32,43,16,24,35)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48), (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([(1,36,17),(2,26,37,10,18,45),(3,19,38),(4,47,20,12,39,28),(5,40,21),(6,30,41,14,22,33),(7,23,42),(8,35,24,16,43,32),(9,44,25),(11,27,46),(13,48,29),(15,31,34)], [(1,9),(2,45,18,10,37,26),(3,11),(4,28,39,12,20,47),(5,13),(6,33,22,14,41,30),(7,15),(8,32,43,16,24,35),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48)], [(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)])

36 conjugacy classes

class 1 2A2B3A3B4A4B4C6A···6F8A8B8C8D8E8F12A···12H16A···16H
order122334446···688888812···1216···16
size112441124···4999918184···418···18

36 irreducible representations

dim1111111244444
type++++-+-
imageC1C2C2C4C4C8C8M5(2)C32⋊C4C322C8C2×C32⋊C4C322C8C62.4C8
kernelC62.4C8C322C16C2×C324C8C324C8C6×C12C3×C12C62C32C2×C4C4C4C22C1
# reps1212244422228

Matrix representation of C62.4C8 in GL4(𝔽5) generated by

1003
0240
0340
4003
,
4000
0410
0220
0004
,
0040
1000
0002
0100
G:=sub<GL(4,GF(5))| [1,0,0,4,0,2,3,0,0,4,4,0,3,0,0,3],[4,0,0,0,0,4,2,0,0,1,2,0,0,0,0,4],[0,1,0,0,0,0,0,1,4,0,0,0,0,0,2,0] >;

C62.4C8 in GAP, Magma, Sage, TeX

C_6^2._4C_8
% in TeX

G:=Group("C6^2.4C8");
// GroupNames label

G:=SmallGroup(288,421);
// by ID

G=gap.SmallGroup(288,421);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,58,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=1,c^8=b^3,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;
// generators/relations

Export

Subgroup lattice of C62.4C8 in TeX

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