#27271302

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7282

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133131
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂×C₄×D₄	C₂×D₅₃
Fitting	C₄×#99×C₅₃	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×C₄×D₄×C₅₃	C₂²
hypercenter	C₄×#99	D₅₃
hypocenter	C₅₃	#7282
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ C₄×#99

388 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 69
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 69 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)(54,55)(56,60,61,58)(57,59)(63,67,69,68);(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)(55,58)(59,60)(62,63,65,68,66,69,64,67)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=f,d^2=f,b^a=bc,c^a=cf,e^a=efg,i^a=i^52,c^b=cg,g^b=gh,e^c=eh

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56778);