#34178786

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8154

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₇₉
commutator	C₂³×C₇₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1280	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1280	D₇₉
hypocenter	C₇₉	#8154
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1280

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 107 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)(80,82,83,81)(93,98)(97,103)(99,105)(104,107);(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)(84,86)(85,88,89,90)(87,91)(92,94,95,100,96,101,102,106)(93,97,99,104,98,103,105,107);(80,81,83,82)(84,85)(86,90)(87,89)(88,91)(92,93)(94,97)(95,99)(96,98)(100,104)(101,103)(102,105)(106,107)

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=e,f^2=h,b^a=bd,c^a=ch,f^a=fg,d^b=dg,i^c=i^78

Conjugacy classes & character table

1780 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 60085);