#27272002

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×C₁₂₈ ⋊ C₂
Sylow	C₅₃ ⋊ M₂₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₅₃
commutator, socle	C₂×C₅₃	C₂×C₆₄
Fitting	C₅₃×C₁₂₈	C₂
prime radical	C₂×D₅₃	C₆₄
cosocle	C₅₃×C₆₄	C₂²
hypercenter	C₁₂₈	D₅₃
hypocenter	C₅₃	M₂₅₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 181
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 2 generators on 181 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)(55,62)(63,83)(64,84)(65,85)(66,86)(67,87)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(128,153)(129,154)(130,155)(131,156)(132,157)(133,158)(134,159)(135,160)(136,161)(137,162)(163,174)(164,175)(165,176)(166,177)(167,178)(179,181);(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)(54,55,56,63,57,64,68,88,58,65,69,89,73,92,98,128,59,66,70,90,74,93,99,129,77,95,102,131,108,134,138,163,60,67,71,91,75,94,100,130,78,96,103,132,109,135,139,164,80,97,105,133,111,136,141,165,114,137,144,166,148,167,168,179,61,62,72,83,76,84,101,118,79,85,104,119,110,122,140,153,81,86,106,120,112,123,142,154,115,125,145,156,149,159,169,174,82,87,107,121,113,124,143,155,116,126,146,157,150,160,170,175,117,127,147,158,151,161,171,176,152,162,172,177,173,178,180,181)

Polycyclic presentation:

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

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57478);