D₅₃×#2244 #27539402

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

D₅₃×#2244

Splittings

Fitting	C₅₃×#2244 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2244

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2244	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#493×D₅₃	C₂
hypercenter	#2244	D₅₃
hypocenter	C₅₃	C₂×#2244
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2244

686 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 points:

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

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, 324878);