#34175265

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7203

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579075
commutator	C₂³×C₇₉	C₄×C₈
Frattini	C₂×C₄×D₄	C₂×D₇₉
Fitting	C₄×#100×C₇₉	C₂
socle	C₂²×C₇₉	#342
prime radical	C₂²×D₄×C₇₉	C₂×C₄
cosocle	C₂×C₄×D₄×C₇₉	C₂²
hypercenter	C₄×#100	D₇₉
hypocenter	C₇₉	#7203
LCS	G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ C₄×#100

364 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 95
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 95 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)(80,82,84,87,85,86,83,81)(89,92)(93,94);(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)(80,81,83,86,85,87,84,82)(88,89)(90,94,95,92)(91,93)

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,d^2=g,e^2=g,b^a=bc,c^a=cf,f^a=fh,i^a=i^78,d^b=df,i^b=i^78,d^c=dh

Conjugacy classes & character table

1768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 56564);