H(3,ℤ/4)×D₁₀₁ #27170831

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

H(3,ℤ/4)×D₁₀₁

Splittings

Fitting	H(3,ℤ/4)×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ C₂×H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₄	C₄²×D₁₀₁
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	H(3,ℤ/4)×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂×#95
prime radical	C₂³×D₁₀₁	D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	H(3,ℤ/4)	D₁₀₁
hypocenter	C₁₀₁	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

404 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 3 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);(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)(102,104)(103,107,109,114)(105,111)(106,112)(108,110,115,116)(113,117);(102,103)(104,110,112,116)(105,108)(106,109)(107,111,114,117)(113,115)

Conjugacy classes & character table

1144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 4209);