#27270758

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1367 ⋊ C₂
Sylow	C₅₃ ⋊ #7074

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133137
commutator	C₂×C₈×C₅₃	C₂×C₈
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1367	C₂
socle	C₂²×C₅₃	D₄⋊C₈
prime radical	C₂×D₄₂₄	C₈
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1367	D₅₃
hypocenter	C₅₃	#7074
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

148 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 2 generators on 85 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)(55,59)(56,62)(61,66)(63,68)(71,75)(72,81)(73,84)(76,77)(78,85)(79,80)(82,83);(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)(54,55,56,60,58,61,63,67,57,59,62,65,64,66,68,69)(70,71,73,77,74,79,84,78)(72,80,82,76,83,75,81,85)

Polycyclic presentation:

2^8,53|a^2=d,c^2=eh,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cegh,d^b=dgh,e^b=eg,i^b=i^52

Conjugacy classes & character table

1514 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56234);