#27295794

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×D₄⋊C₁₆ ⋊ C₂
Sylow	C₅₃ ⋊ C₁₆×D₈

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×D₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#11719922	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	D₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₁₆×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

258 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 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)(62,64,65,71,66,72,73,77,67,63,68,69,74,70,75,76);(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)(55,58)(56,61)(59,60);(54,55)(56,58)(57,60)(59,61)(62,63,65,69,66,70,73,76,67,64,68,71,74,72,75,77)

Polycyclic presentation:

2^8,53|a^2=e,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,d^b=df,i^c=i^52

Conjugacy classes & character table

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81270);