Dic₂₃⋊C₄ #67269

order	368
isoclinism	isoclinic class of Dic₄₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

Dic₂₃⋊C₄

Splittings

Sylow

C₂₃ ⋊ C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₂₃
commutator	C₂×C₂₃	C₂×C₄
Fitting	C₂×C₄×C₂₃	C₂
socle, prime radical, cosocle	C₂²×C₂₃	C₂²
hypercenter	C₂×C₄	D₂₃
hypocenter	C₂₃	C₄⋊C₄
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 368

A minimal faithful permutation representation — 2 generators on 31 points:

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

Polycyclic presentation:

2^4,23|a^2=d,b^2=c,b^a=bd,e^a=e^22

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×Dic₄₆, #10625041, #10625043, …
maximal normal subgroup of	C₄×Dic₄₆, #10625041, #10625043, …
maximal quotient of	#10624989, #10624990, Dic₂₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 11);

Dic23⋊C4 #67269