C₆₇×#12123 #30691374

order	17152
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#12123

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₆₇
prime radical	C₆₇×#2021	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₆₇ ◁ C₂×C₂²⋊C₄×C₆₇ ◁ G

570 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 91
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 91 points:

(2,5)(3,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,2,4,6)(3,8,7,5)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(11,18)(12,20)(14,21)(17,23)

Conjugacy classes & character table

1675 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5111);