#27295806

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×D₄⋊C₁₆ ⋊ C₂
Sylow	C₅₃ ⋊ #12039

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
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	#12039
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

242 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 85 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)(62,64,66,73,67,68,71,76,65,72,74,77,75,63,69,70);(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)(63,68)(64,72)(70,76)(73,77);(54,55)(56,58)(57,60)(59,61)(62,63,66,70,67,64,71,73,65,68,74,76,75,72,69,77)

Polycyclic presentation:

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

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81282);