C₂×H(5,𝔽₂)×#6157 #28330808

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product	C₂×H(5,𝔽₂)×#6157
unambiguous semidirect product	D₅₃⋊(C₄×H(5,𝔽₂))

Splittings

Fitting	C₂×H(5,𝔽₂)×C₅₃ ⋊ C₄
Sylow	C₅₃ ⋊ C₂×C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²	C₂⁴×#6157
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#6157
Fitting	C₂×H(5,𝔽₂)×C₅₃	C₄
socle	C₂²×C₅₃	C₂⁴×C₄
prime radical	C₂×H(5,𝔽₂)×D₅₃	C₂
cosocle	C₂×D₅₃	C₂⁶
hypercenter	C₂×H(5,𝔽₂)	#6157
hypocenter	C₅₃	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

12378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 6 generators on 63 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,43)(23,34,35,40)(27,32,38,37)(29,41,42,33)(36,44,45,50)(39,46,47,49)(48,51,52,53);(1,2,5,10,19,32,21,8,16,27,39,48,51,44,30,42,28,40,49,45,35,22,13,6,11,20,33,41,25,12,3,7,14,23,36,46,34,17,29,43,50,53,52,47,38,26,15,24,37,31,18,9,4)(57,61)(60,62);(58,63)(60,62);(56,58)(57,60)(59,63)(61,62);(56,57)(58,62)(59,61)(60,63);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1116284);

C2×H(5,𝔽2)×#6157 #28330808