C₁₇×#10624548 #27136712

order	12393
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₁₇ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₁₇×#10624548

Subgroups

	subgroup	G/H
center, socle	C₃³×C₁₇	C₃³
commutator	C₃²	C₃²×C₉×C₁₇
Frattini, cosocle	C₃³	C₃³×C₁₇
prime radical	C₃⁴×C₁₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₁₇ ◁ G

554 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 224 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) = 44
transitive degree	μ_t(G) = 4131

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,4,3,5,7,6,8,9)(10,11,13)(12,15,17)(14,18,16)(20,22,25)(23,27,26)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(19,20,23,21,22,26,24,25,27);(11,14,15)(13,17,18)

Conjugacy classes & character table

2601 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12393, 108);