Dic₅₅₉₃ #36375384

order	22372
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₅₉₃
unambiguous semidirect product	C₇⋊Dic₇₉₉
unambiguous semidirect product	C₄₇⋊Dic₁₁₉

Splittings

derived, Hall	C₇×C₁₇×C₄₇ ⋊ C₄
Hall	C₁₇×C₄₇ ⋊ Dic₇
Hall	C₇×C₄₇ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₃₂₉
Sylow	C₇ ⋊ Dic₇₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₅₉₃
commutator, hypocenter	C₇×C₁₇×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₄₇	C₂
LCS	G ▷ C₇×C₁₇×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 22372

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

Polycyclic presentation:

2,7,17,47,2|a^2=e,b^a=b^6,c^a=c^16,d^a=d^46

Conjugacy classes & character table

5596 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₁₈₆
maximal normal subgroup of	Dic₁₁₁₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22372, 8);

Dic5593 #36375384