#34193878

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #10796

Subgroups

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₇₉
commutator	C₂³×C₇₉	C₂×C₄²
Fitting	C₂×#376×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#376	D₇₉
hypocenter	C₇₉	#10796
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 107 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(85,87)(88,90,91,89)(94,100)(97,103)(101,106)(104,107);(80,81,82,83)(88,89,91,90)(92,94)(93,97)(95,101)(96,100)(98,104)(99,103)(102,106)(105,107);(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)(70,71)(72,73)(74,75)(76,77)(78,79)(84,85)(86,87)(89,90)(92,93,95,98,96,99,102,105)(94,97,101,104,100,103,106,107)

Polycyclic presentation:

2^8,79|a^2=f,b^2=g,c^2=d,f^2=h,b^a=bd,c^a=ce,i^a=i^78,c^b=ch

Conjugacy classes & character table

3208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 75177);