C₂×#365×D₅₃ #27537097

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#365×D₅₃

Splittings

Fitting	C₂×C₅₃×#365 ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#365

Subgroups

	subgroup	G/H
center	C₂²	#106×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×D₅₃	SD₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂²×#365
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1696

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322573);

C2×#365×D53 #27537097