C₂²×#353×C₆₇ #30712603

order	17152
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂²×#353×C₆₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	#99
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₇
prime radical	C₂²×#127×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₆₇ ◁ C₂³×D₄×C₆₇ ◁ G

2062 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 2062 conjugacy classes of subgroups — too many to draw. Only its 382 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 — 4 generators on 79 points:

(1,2)(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);(3,4);(5,6)(7,10,12,9)(8,11);(6,9)(7,12)(10,11)

Conjugacy classes & character table

3484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 26340);