#34195002

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #11013

Subgroups

	subgroup	G/H
center	C₂²	#19098439
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#410×C₇₉	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂²×#41×C₇₉	C₂²
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#410	D₇₉
hypocenter	C₇₉	#11013
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#410

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 5056

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

Polycyclic presentation:

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

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, 76301);