C₂×#20279094 #27553058

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#20279094

Splittings

Sylow

C₅₃ ⋊ C₂×#2493

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄×C₂²⋊C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂×C₄×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂×#2493
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄

2398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(73,76)(77,79);(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)(57,61)(60,63)(66,70)(68,71)(72,74,75,78)(73,77,76,79);(56,58,59,62)(57,60,61,63)(64,66)(65,68)(67,70)(69,71)(73,76)(77,79);(56,57,59,61)(58,60,62,63)(64,65,67,69)(66,68,70,71)(72,73)(74,77)(75,76)(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, 338534);

C2×#20279094 #27553058