#34175751

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1304 ⋊ C₂
Sylow	C₇₉ ⋊ #7394

Subgroups

	subgroup	G/H
center	C₂	#23761396
commutator	C₂×C₂²⋊C₄×C₇₉	C₂×C₄
Frattini	#379	C₂×D₇₉
Fitting	C₇₉×#1304	C₂
socle	C₂×C₇₉	#1343
prime radical	#23764526	C₂
cosocle	#379×C₇₉	C₂²
hypercenter	#1304	D₇₉
hypocenter	C₇₉	#7394
LCS	G ▷ C₂×C₂²⋊C₄×C₇₉ ▷ C₂⁴×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴ ◁ #379 ◁ #1304

418 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 95
transitive degree	μ_t(G) = 1264

A minimal faithful permutation representation — 2 generators on 95 points:

(1,3)(2,6)(4,10)(5,11)(7,15)(8,14)(9,12)(13,16)(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)(92,93)(94,95);(1,2)(3,9)(4,12)(5,8)(6,10,14,16)(7,13)(11,15)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19)

Polycyclic presentation:

2^8,79|a^2=d,c^2=f,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=def,g^b=gh,i^b=i^78,d^c=dg,e^c=eh

Conjugacy classes & character table

1039 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57050);