C₂×#20273113 #27350586

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

Aliases

direct product

C₂×#20273113

Splittings

Fitting	C₂²×C₄⋊C₈×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1480

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#16811173	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂×#1480
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 79 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(66,72)(69,75)(73,78)(76,79);(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)(57,60)(58,63)(61,62)(64,66)(65,69)(67,73)(68,72)(70,76)(71,75)(74,78)(77,79);(56,57)(58,60)(59,62)(61,63)(64,65,67,70,68,71,74,77)(66,69,73,76,72,75,78,79);(54,55)

Conjugacy classes & character table

2156 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 136062);

C2×#20273113 #27350586