#27299221

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2110 ⋊ C₂
Sylow	C₅₃ ⋊ #12618

Subgroups

	subgroup	G/H
center	C₄	C₄×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	#1752
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#12618
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₃₂ ◁ #2110

382 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)(54,56,57,65,58,64,68,77,59,66,67,80,69,78,81,79)(55,61,62,74,63,75,76,85,60,70,71,82,72,73,83,84);(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)(55,60)(56,64,66,78)(57,67)(61,73,70,75)(63,72)(65,79,80,77)(68,81)(74,85,82,84);(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)(54,55,57,62,58,63,68,76,59,60,67,71,69,72,81,83)(56,61,65,74,64,75,77,85,66,70,80,82,78,73,79,84)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=e,e^2=g,g^2=h,b^a=bd,d^a=dh,i^a=i^52,c^b=cg,e^b=eh,i^b=i^52,f^c=fh

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84697);