C₂×#20277533 #27543262

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂×#20277533

Splittings

Fitting	C₂×C₅₃×#386 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×D₄×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#386	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×D₅₃	C₂
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#386	D₅₃
hypocenter	C₅₃	C₂×D₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#386

8774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 67 points:

(1,2)(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)(65,67);(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)(61,63);(64,65)(66,67);(56,57,58,59)(60,61)(62,63)(65,67);(54,55)

Conjugacy classes & character table

1556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 328738);

C2×#20277533 #27543262