C₇₉⋊#7047 #34174877

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

unambiguous semidirect product

C₇₉⋊#7047

Splittings

Sylow

C₇₉ ⋊ #7047

Subgroups

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₂⁴×C₄	C₂×D₇₉
Fitting	C₇₉×#1779	C₂
socle	C₂³×C₇₉	#95
prime radical	C₂⁵×C₇₉	Q₈
cosocle	C₂⁴×C₄×C₇₉	C₂²
hypercenter	#1779	D₇₉
hypocenter	C₇₉	#7047
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ #1779

608 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 99 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:

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

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,c^2=fh,d^2=fh,e^2=fh,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh,i^b=i^78

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 56176);