#34193240

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #10659

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1955	C₂
socle	C₂²×C₇₉	#399
prime radical	C₂²×D₄×C₇₉	Q₈
hypercenter	#1955	D₇₉
hypocenter	C₇₉	#10659
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

356 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 5056

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

Polycyclic presentation:

2^8,79|a^2=d,b^2=de,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fgh,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh

Conjugacy classes & character table

1756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 74539);