#34205019

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1867 ⋊ C₂
Sylow	C₇₉ ⋊ #12958

Subgroups

	subgroup	G/H
center	C₂²	#19098749
commutator	C₂×D₄×C₇₉	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₇₉
Fitting	C₇₉×#1867	C₂
socle	C₂²×C₇₉	B(4,𝔽₂)
prime radical	C₂×#14579129	C₂²
cosocle	C₂²×D₄×C₇₉	C₂³
hypercenter	#1867	D₇₉
hypocenter	C₇₉	#12958
LCS	G ▷ C₂×D₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #1867

728 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 91
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 91 points:

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

Polycyclic presentation:

2^8,79|a^2=f,b^2=f,b^a=bd,c^a=ce,g^a=gh,e^b=egh,d^c=dg,i^c=i^78,e^d=eh

Conjugacy classes & character table

1288 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86318);