C₆₇×#7201 #30686452

order	17152
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#7201

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#95
commutator	C₂³	C₄×C₈×C₆₇
Frattini, cosocle	C₂³×C₈	C₂²×C₆₇
socle	C₂²×C₆₇	#336
prime radical	C₂⁴×C₆₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

286 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 4288

A minimal faithful permutation representation — 2 generators on 91 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,18,15,11,16,20,14,17,13,23,22,19,24,21)(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);(1,3)(2,5)(4,8)(6,7)(9,11,13,20,15,17,24,23,14,19,12,21,22,10,16,18)

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, 189);