C₂×#20276703 #27538098

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

Aliases

direct product

C₂×#20276703

Splittings

Sylow

C₅₃ ⋊ C₂×#2197

Subgroups

	subgroup	G/H
center	C₂²	#16810809
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×B(4,𝔽₂)	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	C₂×B(4,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂×#2197
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×B(4,𝔽₂)

1320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 71 points:

(6,14)(12,18)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21);(3,5,7,13)(4,8,10,15)(6,12,14,18)(9,11,16,17)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71);(3,4)(5,11)(6,9)(7,10)(8,12)(13,17)(14,16)(15,18);(1,2)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323574);

C2×#20276703 #27538098