C₂×#350×D₅₃ #27536787

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#350×D₅₃

Splittings

Fitting	C₂×C₅₃×#350 ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#350

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#350	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×D₅₃	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#350	D₅₃
hypocenter	C₅₃	C₂²×#350
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 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)(56,58)(57,63)(59,72)(60,74)(61,69)(62,71)(64,80)(65,82)(66,68)(67,79)(70,76)(73,86)(75,77)(78,84)(81,85)(83,87);(56,57,60,65,61,66,75,83,62,67,76,84,59,64,73,81)(58,68,70,85,69,79,86,82,71,80,74,87,72,63,77,78);(54,55)

Conjugacy classes & character table

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322263);

C2×#350×D53 #27536787