#34175608

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7334

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₉
commutator	C₂²×C₇₉	C₂×C₃₂
Frattini	C₂²×C₁₆	C₂×D₇₉
Fitting	C₂×C₂²⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²⋊C₁₆
prime radical	C₂⁴×C₇₉	C₁₆
cosocle	C₂²×C₁₆×C₇₉	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₇₉
hypocenter	C₇₉	#7334
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₂²⋊C₁₆

182 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 119
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 119 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)(80,82)(81,84)(83,87)(85,86);(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)(82,86,87,84)(83,85)(88,89,90,94,91,95,98,104,92,96,99,105,101,107,110,114,93,97,100,106,102,108,111,115,103,109,112,116,113,117,118,119)

Polycyclic presentation:

2^8,79|a^2=d,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,i^a=i^78,d^b=de

Conjugacy classes & character table

3208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 56907);