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C₂×#888 #5497

order	192
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

C₃ ⋊ C₂×C₄⋊Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 17 points:

(1,2)(3,4,5)(7,9)(10,12,13,17)(11,14,15,16);(6,7,8,9);(4,5)(10,11,13,15)(12,16,17,14);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68799, #68800, #68803, …
maximal normal subgroup of	#68799, #68800, #68803, …
maximal quotient of	C₂×#4647, C₂×#4646, C₂×#4656, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1056);