C₂²×#16811458 #28297984

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#16811458

Splittings

Fitting	C₂²×H(5,𝔽₂)×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#585

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂²×H(5,𝔽₂)×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂²×#585
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

60366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 7 generators on 73 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)(58,61,62,69)(59,65,64,71)(60,68,67,73)(63,70,66,72);(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)(59,64)(63,66)(65,71)(70,72);(60,67)(63,66)(68,73)(70,72);(58,60)(59,63)(61,68)(62,67)(64,66)(65,70)(69,73)(71,72);(58,59)(60,66)(61,65)(62,64)(63,67)(68,72)(69,71)(70,73);(56,57);(54,55)

Conjugacy classes & character table

1904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1083460);

C22×#16811458 #28297984