#27305116

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13732

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈₄₈
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#13732
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 125 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)(62,64,75,101,67,78,102,119,82,105,121,120,96,117,80,91,65,76,94,118,81,104,85,107,68,79,103,124,84,106,66,77)(63,70,83,99,72,92,112,109,87,115,114,125,111,122,89,97,69,88,110,123,90,116,95,100,73,93,113,108,86,98,71,74);(54,56,57,61)(55,58,59,60)(63,69)(70,88)(71,89)(72,90)(73,87)(74,97)(83,110)(86,111)(92,116)(93,115)(95,112)(98,122)(99,123)(100,109)(108,125)(113,114);(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)(54,55,57,59)(56,60,61,58)(62,63,65,69)(64,74,76,97)(66,83,80,110)(67,86,81,111)(68,87,82,73)(70,91,88,77)(71,94,89,75)(72,96,90,84)(78,108,104,125)(79,109,105,100)(85,114,102,113)(92,120,116,124)(93,107,115,119)(95,103,112,121)(98,118,122,101)(99,117,123,106)

Polycyclic presentation:

2^8,53|a^2=d,b^2=dh,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^52

Conjugacy classes & character table

3398 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90592);