#27295698

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12014

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#12014
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

220 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 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)(54,57,59,58)(55,61,62,67)(56,64,66,65)(60,63,68,69)(72,80)(76,86)(81,92)(82,93)(87,96)(88,97)(94,100)(98,101);(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)(54,56,58,65,59,66,57,64)(55,60,61,63,62,68,67,69)(70,72)(71,76)(73,81)(74,82)(75,80)(77,87)(78,88)(79,86)(83,94)(84,92)(85,93)(89,98)(90,96)(91,97)(95,100)(99,101);(54,55)(56,63)(57,61)(58,67)(59,62)(60,65)(64,68)(66,69)(70,71,73,77,74,78,83,89,75,79,84,90,85,91,95,99)(72,76,81,87,82,88,94,98,80,86,92,96,93,97,100,101)

Polycyclic presentation:

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

Conjugacy classes & character table

2156 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81174);