#34197259

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #11401

Subgroups

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
prime radical	C₂³×C₇₉	C₄×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#11401
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆

298 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)(89,95)(94,104)(96,107)(97,108)(105,114)(106,115)(109,117)(116,119);(81,84)(85,87)(88,90,91,98,92,99,101,110,93,100,102,111,103,112,113,118)(89,94,96,105,97,106,109,116,95,104,107,114,108,115,117,119);(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)(90,94)(91,96)(92,97)(93,95)(98,105)(99,106)(100,104)(101,109)(102,107)(103,108)(110,116)(111,114)(112,115)(113,117)(118,119)

Polycyclic presentation:

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

Conjugacy classes & character table

5104 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 78558);