#34204755

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12917

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#504×C₇₉	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#504	D₇₉
hypocenter	C₇₉	#12917
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#504

384 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(84,87)(85,93)(86,97)(88,92)(89,106)(90,103)(91,102)(94,112)(95,109)(96,98)(99,113)(100,105)(101,114)(104,110)(107,115)(108,111);(81,83)(84,86,90,100,91,101,108,93,88,98,104,109,89,99,107,112)(85,92,96,110,95,106,113,115,94,87,97,103,105,102,114,111);(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)(84,85,90,96,91,95,108,113,88,94,104,97,89,105,107,114)(86,87,100,103,101,102,93,111,98,92,109,110,99,106,112,115)

Polycyclic presentation:

2^8,79|a^2=f,b^2=f,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^78,e^c=eh

Conjugacy classes & character table

3196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86054);