C₂²×#16810456 #27542197

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#16810456

Splittings

Fitting	C₂²×C₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×M₆₄

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂²×C₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₁₆
prime radical	C₂³×D₅₃	C₁₆
cosocle	C₁₆×C₅₃	C₂⁴
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	C₂²×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

502 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 291 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 points:

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

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327673);