C₂²×#16810419 #27536913

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#16810419

Splittings

Fitting	C₂²×#101×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#355

Subgroups

	subgroup	G/H
center	C₂³	#13133131
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₂²×#101×C₅₃	C₂
socle	C₂³×C₅₃	#99
prime radical	C₂³×D₂₁₂	C₄
cosocle	C₂×Q₈×C₅₃	C₂⁴
hypercenter	C₂²×#101	D₅₃
hypocenter	C₅₃	C₂²×#355
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ C₂²×#101

1382 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1382 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 73 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)(59,63)(60,67)(64,66)(65,72)(68,69)(71,73);(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)(58,59,61,65,62,66,70,73)(60,64,68,72,67,63,69,71);(56,57);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322389);