#27489983

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #29656

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2405	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂³×D₄×C₅₃	C₂²
hypercenter	#2405	D₅₃
hypocenter	C₅₃	#29656
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #2405

1428 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 1696

A minimal faithful permutation representation — 4 generators on 77 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)(65,73)(68,75)(71,77)(74,76);(58,61)(62,65)(63,68)(64,71)(66,73)(67,74)(69,75)(70,76)(72,77);(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)(54,56)(55,58)(57,60)(59,61)(62,64,66,72)(63,67,69,70)(65,71,73,77)(68,74,75,76);(54,55)(56,58)(57,59)(60,61)(62,63,66,69)(64,70,72,67)(65,68,73,75)(71,76,77,74)

Polycyclic presentation:

2^8,53|a^2=e,b^2=e,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,i^b=i^52,d^c=de

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 275459);