C₂×#20276732 #27538245

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276732

Splittings

Fitting	C₂×C₅₃×H(3,𝔽₄) ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2202

Subgroups

	subgroup	G/H
center	C₂²	#16810810
commutator, cosocle	C₂×Q₈×C₅₃	C₂⁴
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₂×C₅₃×H(3,𝔽₄)	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	C₂×H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	C₂×#2202
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×H(3,𝔽₄)

1284 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 171 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) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 71 points:

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

Conjugacy classes & character table

1022 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323721);