C₆₇×#12631 #30691882

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#12631

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₆₇
prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

464 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 103
transitive degree	μ_t(G) = 4288

A minimal faithful permutation representation — 3 generators on 103 points:

(1,2,4,3)(5,6)(7,15)(8,19)(9,14)(10,12)(11,16)(13,21)(17,33)(18,29)(20,28)(22,27)(23,26)(24,32)(25,34)(30,35)(31,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(5,7,9,17,10,18,23,35)(6,11,14,25,12,24,26,36)(8,16,20,31,21,32,22,34)(13,15,27,30,19,29,28,33);(1,3,4,2)(6,12)(11,24)(13,19)(14,26)(15,29)(25,36)(27,28)(30,33)

Conjugacy classes & character table

3082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5619);