C₄×#15692312 #27186856

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×#15692312

Splittings

Sylow

C₁₀₁ ⋊ C₄×C₂²⋊C₈

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#87938
commutator	C₂×C₁₀₁	C₂×C₄×C₈
Frattini	C₂³	C₂²×#87938
Fitting	C₂×C₄²×C₁₀₁	C₄
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂³×D₁₀₁	C₂×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂×C₄²	#87938
hypocenter	C₁₀₁	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

436 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 6464

A minimal faithful permutation representation — 3 generators on 117 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20234);