#27270679

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7038

Subgroups

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₅₃×#1797	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	#1797	D₅₃
hypocenter	C₅₃	#7038
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

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

normal subgroups only

This group has 358 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 — 2 generators on 77 points:

(1,2)(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)(54,56)(55,58,59,60)(57,61)(62,64,65,67,66,68,69,63)(70,71)(72,76)(73,75)(74,77);(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)(54,55)(56,60)(57,59)(58,61)(62,63,65,68,66,67,69,64)(71,74,75,76)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=fh,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=efgh,i^a=i^52,c^b=cg,d^b=dh,i^b=i^52

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56155);