#27185942

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×H(5,𝔽₂)×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ H(7,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₁₀₁
commutator, socle, cosocle	C₂×C₁₀₁	C₂⁶
Fitting	C₂×H(5,𝔽₂)×C₁₀₁	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₀₁
hypocenter	C₁₀₁	H(7,𝔽₂)
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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) = 1616

A minimal faithful permutation representation — 6 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)(103,109)(107,110)(108,115)(114,116);(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)(104,112)(107,110)(111,117)(114,116);(105,113)(108,115)(111,117)(114,116);(102,105)(103,108)(104,111)(106,113)(107,114)(109,115)(110,116)(112,117);(102,104)(103,107)(105,111)(106,112)(108,114)(109,110)(113,117)(115,116);(102,103)(104,110)(105,108)(106,109)(107,112)(111,116)(113,115)(114,117)

Polycyclic presentation:

2^7,101|b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^f=h^100

Conjugacy classes & character table

1765 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 19320);