#34203186

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1386 ⋊ C₂
Sylow	C₇₉ ⋊ #12567

Subgroups

	subgroup	G/H
center	C₂²	#102×D₇₉
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1386	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×D₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1386	D₇₉
hypocenter	C₇₉	#12567
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

586 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)(84,87,91,90)(85,93,96,107)(86,98,102,101)(88,103,106,89)(92,108,110,113)(94,104,112,95)(97,109,111,115)(99,100,105,114);(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)(84,86,89,100,90,101,106,99,91,102,103,114,87,98,88,105)(85,92,95,109,93,108,94,111,96,110,104,115,107,113,112,97);(80,81,82,83)(84,85)(86,97)(87,93)(88,104)(89,94)(90,107)(91,96)(92,99)(95,106)(98,109)(100,108)(101,115)(102,111)(103,112)(105,110)(113,114)

Polycyclic presentation:

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

Conjugacy classes & character table

1312 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84485);