C₂×#23758379 #34117246

order	20100
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product

C₂×#23758379

Splittings

Fitting, socle	C₂×C₅²×C₆₇ ⋊ C₂×C₃
derived, Hall	C₅²×C₆₇ ⋊ C₂²×C₃
Hall	C₂×C₅⋊D₃₃₅ ⋊ C₃
Hall	C₆₇×C₅²⋊C₃ ⋊ C₂²
Hall	C₅²⋊C₃ ⋊ C₂×D₆₇
Sylow	C₆₇ ⋊ C₂×(C₅⋊D₅)⋊C₃
Sylow	C₅² ⋊ C₂×C₃×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#23758379
commutator, cosocle, hypocenter	C₅²×C₆₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₆₇	C₂×C₃
LCS	G ▷ C₅²×C₆₇
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 2010

A minimal faithful permutation representation — 2 generators on 84 points:

(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69,71)(70,80,76,73,74,82)(72,77,81,78,79,75);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(68,70,75,81,73)(69,72,74,80,78)(71,76,77,79,82)(83,84)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 102);

C2×#23758379 #34117246