C₂×#20276712 #27538108

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

Aliases

direct product

C₂×#20276712

Splittings

Fitting	C₂×C₅₃×#352 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2198

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#352	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	C₂×#16810793	C₂
hypercenter	C₂×#352	D₅₃
hypocenter	C₅₃	C₂×#2198
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#352

1244 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:

(3,6,8,5)(4,10,12,13)(7,14,16,15)(9,11,17,18)(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)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70);(3,5,8,6)(4,9,12,17)(7,14,16,15)(10,11,13,18);(3,4)(5,13)(6,10)(7,11)(8,12)(9,14)(15,17)(16,18)(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);(1,2)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323584);

C2×#20276712 #27538108