C₆₇×#12763 #30692014

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₆₇×#12763

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×H(3,𝔽₄)×C₆₇	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×D₄×C₆₇ ◁ G

992 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 87
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 87 points:

(1,2)(3,4)(5,6)(7,13,14,20)(8,16,18,11)(9,12)(10,15)(17,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(5,7,9,14)(6,10,12,19)(8,17,18,15)(11,20,16,13);(2,4)(7,14)(10,19)(11,16)(15,17)

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5751);