#34203315

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12602

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#349×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#349	D₇₉
hypocenter	C₇₉	#12602
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

490 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)(80,82,83,81)(85,91)(90,94)(92,98)(93,104)(102,106)(103,107)(105,111)(113,114);(84,86,87,96,88,97,99,110,89,95,100,108,101,109,112,115)(85,90,92,102,93,103,105,113,91,94,98,106,104,107,111,114);(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)(80,81,83,82)(84,85)(86,94)(87,98)(88,93)(89,91)(90,95)(92,100)(96,102)(97,107)(99,111)(101,104)(103,109)(105,112)(106,108)(110,113)(114,115)

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ch,f^a=fh,i^a=i^78,d^b=dh,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, 84614);