#27537950

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2138 ⋊ C₂
Sylow	C₅₃ ⋊ #33525

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2138	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811437	C₂
hypercenter	#2138	D₅₃
hypocenter	C₅₃	#33525
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 121 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)(61,75)(66,89)(71,99)(76,103)(77,104)(86,112)(90,114)(91,108)(97,119)(100,111)(101,109)(105,118)(106,121)(107,115)(113,117)(116,120);(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)(55,57)(58,61)(59,66)(60,71)(62,76)(63,77)(64,75)(65,86)(67,90)(68,91)(69,89)(70,97)(72,100)(73,101)(74,99)(78,105)(79,106)(80,104)(81,107)(82,103)(83,108)(84,109)(85,111)(87,113)(88,112)(92,115)(93,116)(94,114)(95,117)(96,118)(98,119)(102,120)(110,121);(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)(58,60,64,74)(59,65,69,88)(61,71,75,99)(62,79,82,110)(63,84,80,73)(66,86,89,112)(67,93,94,102)(68,95,83,87)(70,96,98,78)(72,92,85,81)(76,106,103,121)(77,109,104,101)(90,116,114,120)(91,117,108,113)(97,118,119,105)(100,115,111,107);(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)(58,59,64,69)(60,70,74,98)(61,66,75,89)(62,78,82,96)(63,83,80,68)(65,85,88,72)(67,92,94,81)(71,97,99,119)(73,102,84,93)(76,105,103,118)(77,108,104,91)(79,95,110,87)(86,111,112,100)(90,115,114,107)(101,120,109,116)(106,117,121,113)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,i^a=i^52,e^b=eg,g^b=gh,i^b=i^52,d^c=dh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323426);