#34197612

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #11495

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₉
commutator, prime radical	C₂³×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#446×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#446	D₇₉
hypocenter	C₇₉	#11495
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 111 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)(80,83,84,91)(81,86,87,93)(82,89,90,95)(85,92,88,94)(96,99,98,108,101,109,102,100)(97,104,107,111,105,110,103,106);(80,82,84,90)(81,85,87,88)(83,89,91,95)(86,92,93,94)(96,98,101,102)(97,103,105,107)(99,100,109,108)(104,111,110,106);(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)(82,88,90,85)(83,86)(84,87)(89,94,95,92)(91,93)(96,97,100,106,102,103,109,110,101,105,108,111,98,107,99,104)

Polycyclic presentation:

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

Conjugacy classes & character table

2248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 78911);