#34193710

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1733 ⋊ C₂
Sylow	C₇₉ ⋊ #10766

Subgroups

	subgroup	G/H
center	C₄	#19098418
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1733	C₂
socle	C₂×C₇₉	C₂²⋊#95
prime radical	C₂²×#11929624	C₂²
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1733	D₇₉
hypocenter	C₇₉	#10766
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #1733

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,83)(81,88)(82,92)(84,96)(85,98)(86,97)(87,101)(89,104)(90,95)(91,105)(93,107)(94,108)(99,110)(100,111)(102,109)(103,106);(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)(80,82,85,93,86,94,84,100)(81,87,89,103,90,102,99,91)(83,92,98,107,97,108,96,111)(88,101,104,106,95,109,110,105);(80,81)(82,91)(83,95,97,88)(84,99)(85,89)(86,90)(87,93)(92,106,108,105)(94,103)(96,104,98,110)(100,102)(101,107,109,111)

Polycyclic presentation:

2^8,79|a^2=f,b^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,f^c=fh,i^c=i^78

Conjugacy classes & character table

1774 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 75009);