#27293330

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1315 ⋊ C₂
Sylow	C₅₃ ⋊ #11469

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	#16811130	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1315	D₅₃
hypocenter	C₅₃	#11469
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1315

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78806);