D₅₃×#1404 #27304337

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₅₃×#1404

Splittings

Fitting	C₅₃×#1404 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1404

Subgroups

	subgroup	G/H
center	C₂	AΣL(1,𝔽₁₆)×D₅₃
commutator	C₂×Q₈×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₅₃×#1404	C₂
socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
prime radical	#454×D₅₃	C₂
cosocle	#124×C₅₃	C₂³
hypercenter	#1404	D₅₃
hypocenter	C₅₃	C₂×#1404
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #1404

510 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2)(3,10)(4,13)(5,9)(6,12)(7,11)(8,15)(14,16)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(1,2,3,7)(4,13,14,10)(5,9,12,16)(6,15,8,11);(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89813);