C₂⁴×C₄×#6157 #28330797

order	13568
isoclinism	isoclinic class of #6157
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product	C₂⁴×C₄×#6157
unambiguous semidirect product	D₅₃⋊(C₂³×C₄²)

Splittings

Fitting	C₂⁴×C₄×C₅₃ ⋊ C₄
derived, Sylow	C₅₃ ⋊ C₂⁴×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#6157
commutator, hypocenter	C₅₃	C₂⁴×C₄²
Frattini	C₂	C₂⁵×#6157
Fitting	C₂⁴×C₄×C₅₃	C₄
socle	C₂⁵×C₅₃	C₂×C₄
prime radical	C₂⁵×D₅₃	C₂²
cosocle	C₂×D₅₃	C₂⁶
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄

24030 conjugacy classes of subgroups; only the 12696 normal subgroups are shown.

normal subgroups only

This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 12696 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) = 65
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 6 generators on 65 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,43)(23,34,35,40)(27,32,38,37)(29,41,42,33)(36,44,45,50)(39,46,47,49)(48,51,52,53);(1,2,5,10,19,32,21,8,16,27,39,48,51,44,30,42,28,40,49,45,35,22,13,6,11,20,33,41,25,12,3,7,14,23,36,46,34,17,29,43,50,53,52,47,38,26,15,24,37,31,18,9,4)(62,63,64,65);(60,61);(58,59);(56,57);(54,55)

Conjugacy classes & character table

1088 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1116273);