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C₂²×D₃×Q₈ #5958

order	192
automorphism	Aut(G) order 442368
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×D₃×Q₈

Splittings

Fitting	C₂²×C₃×Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂³×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁵
Frattini	C₂	C₂⁴×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂³×C₃	C₂³
prime radical	C₂³×D₃	C₂²
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂³×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#70113, #70266, #76499, …
maximal normal subgroup of	#70113, #70266, #76499, …
maximal quotient of	C₂²×#887, C₂²×#889, C₂²×#888, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1517);