#34199514

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #11929

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3147	C₂
socle	C₂²×C₇₉	C₂×#102
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#3147	D₇₉
hypocenter	C₇₉	#11929
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

414 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:

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

Polycyclic presentation:

2^8,79|a^2=e,c^2=h,d^2=gh,e^2=fgh,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,i^a=i^78,d^b=dg,e^b=efgh,f^b=fh,g^b=gh

Conjugacy classes & character table

1774 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 80813);