C₂×#23767645 #34455873

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#23767645

Splittings

Sylow

C₇₉ ⋊ C₂×#2480

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator, cosocle	C₂³×C₇₉	C₂⁵
Frattini	C₂³	C₂⁴×D₇₉
Fitting	C₂×#379×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
prime radical	C₂⁶×C₇₉	C₂²
hypercenter	C₂×#379	D₇₉
hypocenter	C₇₉	C₂×#2480
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#379

4102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 5 generators on 101 points:

(8,11)(12,13)(15,18,19,17)(16,22,21,20)(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)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100);(4,6)(7,9,10,14)(8,12,11,13)(16,21)(20,22)(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101);(3,4)(5,6)(8,11)(9,14)(15,17,19,18)(16,20,21,22)(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101);(7,8)(9,13)(10,11)(12,14)(15,16)(17,20)(18,22)(19,21);(1,2)

Conjugacy classes & character table

2260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 337172);

C2×#23767645 #34455873