C₆₇×#13047 #30692298

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₇×#13047

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×D₄	C₂³×C₆₇
prime radical	C₂×#453×C₆₇	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×D₄×C₆₇ ◁ G

888 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 79 points:

(2,4)(5,6)(7,10,12,9)(8,11)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(1,2,3,4)(7,12);(6,9)(7,12)(10,11)

Conjugacy classes & character table

2680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6035);