#34208445

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1397 ⋊ C₂
Sylow	C₇₉ ⋊ C₄×M₆₄

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₉
commutator	C₂×C₇₉	C₂×C₄×C₁₆
Fitting	C₇₉×#1397	C₂
socle	C₂²×C₇₉	C₂²×C₁₆
prime radical	C₂²×D₇₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	#1397	D₇₉
hypocenter	C₇₉	C₄×M₆₄
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

250 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 10112

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

Polycyclic presentation:

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

Conjugacy classes & character table

3280 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 89744);