#27168964

order	12928
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #1276

Subgroups

	subgroup	G/H
center	C₄²	C₂²⋊D₁₀₁
commutator	C₄×C₁₀₁	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₀₁
Fitting	#431×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#95
prime radical	C₂³×C₁₀₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₀₁	C₂²
hypercenter	#431	D₁₀₁
hypocenter	C₁₀₁	#1276
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

208 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 117
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 2 generators on 117 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)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(111,114,116,115);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(102,103,104,106,105,107,108,109)(110,111)(112,114)(113,116)(115,117)

Polycyclic presentation:

2^7,101|a^2=d,b^2=e,c^2=f,d^2=g,b^a=bc,c^a=cf,e^a=ef,h^b=h^100

Conjugacy classes & character table

2056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 2342);