#34203660

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12670

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2158	C₂
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2158	D₇₉
hypocenter	C₇₉	#12670
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

318 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:

(1,2)(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)(84,87,104,105,91,88,90,89)(85,93,112,113,97,94,96,95)(86,99,102,100,103,101,107,106)(92,108,110,98,111,109,115,114);(80,81,82,83)(84,86)(85,92)(87,99)(88,101)(89,106)(90,107)(91,103)(93,108)(94,109)(95,114)(96,115)(97,111)(98,113)(100,105)(102,104)(110,112);(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)(84,85,89,95,90,96,88,94,91,97,105,113,104,112,87,93)(86,98,101,92,102,109,106,110,103,114,99,111,107,108,100,115)

Polycyclic presentation:

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

Conjugacy classes & character table

1804 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84959);