C₂×#20273270 #27351425

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20273270

Splittings

Fitting	C₂×C₅₃×#328 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1516

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#328	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#16811136	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#328	D₅₃
hypocenter	C₅₃	C₂×#1516
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#328

1498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 79 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)(58,63)(60,62)(65,70)(69,77)(71,78)(72,79);(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)(56,58)(57,60)(59,63)(61,62)(64,66)(65,69)(67,75)(68,74)(70,77)(71,79)(72,78)(73,76);(56,57)(58,62,63,60)(59,61)(64,65)(66,72,74,79)(67,71)(68,70)(69,73,77,75)(76,78);(54,55)

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 136901);

C2×#20273270 #27351425