#27299186

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12612

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#12612
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2)(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)(62,64,65,72,66,70,71,77,67,73,74,76,75,63,68,69);(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)(54,55,56,57)(59,61);(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)(58,59)(60,61)(62,63,65,69,66,64,71,72,67,70,74,77,75,73,68,76)

Polycyclic presentation:

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

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84662);