D₅₃×#1327 #27293275

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₅₃×#1327

Splittings

Fitting	C₅₃×#1327 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1327

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1327	C₂
socle	C₂×C₅₃	C₂×#324
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1327	D₅₃
hypocenter	C₅₃	C₂×#1327
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1327

410 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 91 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:

(2,7,10,8)(4,12)(9,14,16,15)(11,13)(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);(1,2,4,9,5,8,13,15,6,10,12,16,3,7,11,14);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78751);