#34180104

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1458 ⋊ C₂
Sylow	C₇₉ ⋊ #8409

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1458	C₂
socle	C₂²×C₇₉	#379
prime radical	#23776733	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1458	D₇₉
hypocenter	C₇₉	#8409
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 119 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)(88,91)(89,95)(90,99)(92,106)(93,103)(94,108)(96,102)(97,113)(98,114)(100,115)(101,117)(104,107)(105,116)(109,119)(110,111)(112,118);(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)(81,85)(82,83)(86,87)(88,90)(89,94)(91,99)(92,105)(93,100)(95,108)(96,98)(97,112)(101,107)(102,114)(103,115)(104,117)(106,116)(109,111)(110,119)(113,118);(80,81,83,85,84,87,82,86)(88,89,92,97,93,96,107,111)(90,98,101,112,100,94,105,109)(91,102,104,113,103,95,106,110)(99,114,116,119,115,108,117,118)

Polycyclic presentation:

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

Conjugacy classes & character table

1762 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 61403);