C₂×#20278159 #27552041

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20278159

Splittings

Fitting	C₂×C₅₃×#396 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2417

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#396	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×C₂²⋊D₅₃	C₂²
hypercenter	C₂×#396	D₅₃
hypocenter	C₅₃	C₂×#2417
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#396

3830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 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)(64,67,68,66)(65,70,71,69);(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)(57,59)(61,63);(60,61)(62,63)(64,66,68,67)(65,69,71,70);(56,57,58,59)(64,65,68,71)(66,70,67,69);(54,55)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 337517);

C2×#20278159 #27552041