#27293879

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11602

Subgroups

	subgroup	G/H
center	C₈	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂×C₅₃	C₂×D₄⋊C₈
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#11602
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#443

380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 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)(56,67)(61,76)(66,82)(68,83)(69,70)(72,84)(77,85)(78,81);(54,56)(55,61)(57,72)(58,68)(59,70)(60,67)(62,77)(63,81)(64,78)(65,76)(66,71)(69,73)(74,84)(75,83)(79,82)(80,85);(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)(54,55,58,62,59,64,74,79,60,65,75,80,73,63,57,71)(56,66,68,61,70,77,84,78,67,82,83,76,69,85,72,81)

Polycyclic presentation:

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

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79355);