Dic₄₈₂₃ #34064371

order	19292
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₈₂₃
unambiguous semidirect product	C₇⋊Dic₆₈₉

Splittings

derived, Hall	C₇×C₁₃×C₅₃ ⋊ C₄
Hall	C₁₃×C₅₃ ⋊ Dic₇
Hall	C₇×C₅₃ ⋊ Dic₁₃
Hall	C₇×C₁₃ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₉₁
Sylow	C₁₃ ⋊ Dic₃₇₁
Sylow	C₇ ⋊ Dic₆₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₈₂₃
commutator, hypocenter	C₇×C₁₃×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₃×C₅₃	C₂
LCS	G ▷ C₇×C₁₃×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 19292

A minimal faithful permutation representation — 2 generators on 77 points:

(1,2)(3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,3,2,4)(6,11)(7,10)(8,9)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,7,13,53,2|a^2=e,b^a=b^6,c^a=c^12,d^a=d^52

Conjugacy classes & character table

4826 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₆₄₆, Dic₁₄₄₆₉
maximal normal subgroup of	Dic₉₆₄₆
maximal quotient of	Dic₁₄₄₆₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19292, 9);

Dic4823 #34064371