#34175113

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7143

Subgroups

	subgroup	G/H
center	C₂×C₄	C₇₉⋊#95
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1744	C₂
socle	C₂²×C₇₉	#336
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#7143
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

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

normal subgroups only

This group has 234 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) = 111
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 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)(81,85)(82,88)(87,92)(89,94)(96,98)(97,101,103,108)(99,105)(100,106)(102,104,109,110)(107,111);(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)(80,81,82,86,84,87,89,93,83,85,88,91,90,92,94,95)(96,97)(98,104,106,110)(99,102)(100,103)(101,105,108,111)(107,109)

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,c^2=fh,d^2=g,g^2=h,b^a=bc,c^a=ch,e^a=efh,i^a=i^78,d^b=df

Conjugacy classes & character table

3184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 56412);