C₂×#347×D₅₃ #27536744

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#347×D₅₃

Splittings

Fitting	C₂×C₅₃×#347 ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#347

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#347	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#347	D₅₃
hypocenter	C₅₃	C₂²×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

782 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)(57,61,63,64)(58,65)(62,70,71,69)(66,68);(56,57,58,62,59,64,66,70,60,63,65,71,67,61,68,69);(54,55)

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322220);

C2×#347×D53 #27536744