#34206511

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#359×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #13209

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#359×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	#23773824	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#359	D₇₉
hypocenter	C₇₉	#13209
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

482 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 87810);