C₂×#20277030 #27542227

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20277030

Splittings

Fitting	C₂×C₅₃×M₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2263

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈₄₈
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂²×C₅₃	C₂×D₁₆
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	C₂×#2263
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

502 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 155 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) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

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

Conjugacy classes & character table

2132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327703);