C₂×#72214186 #72229326

order	16224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#72214186

Splittings

Hall	C₂×#72214177 ⋊ C₃
Hall	#92788 ⋊ C₂×C₂²⋊C₄
Sylow	C₁₃² ⋊ C₂×C₃×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#72210302
commutator	C₂×C₁₃²	C₂²×C₃×C₄
Frattini	C₂	C₂×#72210302
Fitting, socle	C₂²×C₁₃²	C₂×C₃×C₄
prime radical	C₂²×#11759815⋊C₂	C₂
cosocle	C₂×C₁₃⋊D₁₃	C₂³×C₃
hypocenter	C₁₃²	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂²

852 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 32
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 32 points:

(4,6)(8,13,17,14,10,18,11,19,15,9,16,12)(20,21,24,27,29,28,22,30,25,23,31,32);(1,2)(7,15,19,10,11,12,14,9,18,8,16,13,17);(3,4)(5,6)(21,23)(22,25)(24,27)(26,32)(28,31)(29,30)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214210, #72259776
maximal normal subgroup of	C₂×#72214210, #72259776
maximal quotient of	#72259595

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,6)(8,13,17,14,10,18,11,19,15,9,16,12)(20,21,24,27,29,28,22,30,25,23,31,32), (1,2)(7,15,19,10,11,12,14,9,18,8,16,13,17), (3,4)(5,6)(21,23)(22,25)(24,27)(26,32)(28,31)(29,30));