C₂×#20273014 #27349715

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#20273014

Splittings

Fitting	C₂×C₈²×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1454

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	C₂×#1454
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

802 conjugacy classes of subgroups; only the 379 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 379 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) = 6784

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

Conjugacy classes & character table

3440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135191);