#34199815

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1367 ⋊ C₂
Sylow	C₇₉ ⋊ #11997

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1367	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₈
prime radical	C₂²×D₇₉	C₄⋊C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1367	D₇₉
hypocenter	C₇₉	#11997
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 10112

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

Polycyclic presentation:

2^8,79|a^2=e,b^2=d,d^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,i^c=i^78

Conjugacy classes & character table

2260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 81114);