C₂×#20272857 #27304299

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272857

Splittings

Sylow

C₅₃ ⋊ C₂×#1401

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂²×C₅₃	C₂²⋊C₁₆
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	C₂×#1401
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6784

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

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, 89775);

C2×#20272857 #27304299