Dic₅₉₅₇ #37558563

order	23828
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₉₅₇
unambiguous semidirect product	C₇⋊Dic₈₅₁
unambiguous semidirect product	C₂₃⋊Dic₂₅₉

Splittings

derived, Hall	C₇×C₂₃×C₃₇ ⋊ C₄
Hall	C₂₃×C₃₇ ⋊ Dic₇
Hall	C₇×C₃₇ ⋊ Dic₂₃
Hall	C₇×C₂₃ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₁₆₁
Sylow	C₂₃ ⋊ Dic₂₅₉
Sylow	C₇ ⋊ Dic₈₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₉₅₇
commutator, hypocenter	C₇×C₂₃×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₃×C₃₇	C₂
LCS	G ▷ C₇×C₂₃×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 23828

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

Polycyclic presentation:

2,7,23,37,2|a^2=e,b^a=b^6,c^a=c^22,d^a=d^36

Conjugacy classes & character table

5960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₉₁₄
maximal normal subgroup of	Dic₁₁₉₁₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23828, 8);

Dic5957 #37558563