#27293423

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1315 ⋊ C₂
Sylow	C₅₃ ⋊ #11492

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1315	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#16811169	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1315	D₅₃
hypocenter	C₅₃	#11492
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1315

360 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78899);