#27541796

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₄×Q₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33932

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#16811000	C₂²
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	#33932
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 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)(54,56,57,61)(55,59,58,60);(62,65,68,77)(63,70,72,87)(64,73,67,83)(66,79,81,93)(69,84,82,86)(71,88,89,90)(74,85,80,76)(75,91,92,78);(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)(54,55,57,58)(56,60,61,59)(62,64,68,67)(63,69,72,82)(65,73,77,83)(66,74,81,80)(70,84,87,86)(71,78,89,91)(75,90,92,88)(76,79,85,93);(55,58)(59,60)(62,63,68,72)(64,69,67,82)(65,75,77,92)(66,78,81,91)(70,85,87,76)(71,74,89,80)(73,90,83,88)(79,84,93,86)

Polycyclic presentation:

2^8,53|a^2=h,b^2=eh,c^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=de,i^b=i^52,f^c=fg,g^c=gh

Conjugacy classes & character table

1199 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327272);