#27392346

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1840 ⋊ C₂
Sylow	C₅₃ ⋊ #21668

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1840	C₂
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1840	D₅₃
hypocenter	C₅₃	#21668
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1840

702 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 219 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) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=ef,c^2=e,d^2=g,g^2=h,b^a=be,c^a=cf,c^b=ch,i^d=i^52

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 177822);