#34178656

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1307 ⋊ C₂
Sylow	C₇₉ ⋊ #8119

Subgroups

	subgroup	G/H
center	C₂²	#95×D₇₉
commutator	C₂³×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1307	C₂
socle	C₂²×C₇₉	C₂×#95
prime radical	C₂³×D₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1307	D₇₉
hypocenter	C₇₉	#8119
LCS	G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

566 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 155 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:

(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)(88,91,93,92)(89,95,97,96)(90,99,100,101)(94,102,98,103);(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)(80,82,83,85,84,86,87,81)(88,90)(89,94,97,98)(91,99)(92,101)(93,100)(95,102,96,103);(80,81,83,86,84,85,87,82)(88,89,92,96,93,97,91,95)(90,98,101,102,100,94,99,103)

Polycyclic presentation:

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

Conjugacy classes & character table

1066 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 59955);