C₉₇×Dic₃₃ #27165371

order	12804
isoclinism	isoclinic class of D₃₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₉₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉₇×Dic₃₃
unambiguous semidirect product	C₃⋊(Dic₁₁×C₉₇)
unambiguous semidirect product	C₁₁⋊(Dic₃×C₉₇)

Splittings

derived, Hall	C₃×C₁₁ ⋊ C₄×C₉₇
Hall	C₃×C₁₁×C₉₇ ⋊ C₄
Hall	C₁₁×C₉₇ ⋊ Dic₃
Hall	C₃×C₉₇ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₃×C₉₇
Sylow	C₃ ⋊ Dic₁₁×C₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉₇	D₃₃
commutator, hypocenter	C₃×C₁₁	C₄×C₉₇
Frattini	C₂	D₃₃×C₉₇
Fitting, socle, prime radical	C₂×C₃×C₁₁×C₉₇	C₂
cosocle	C₂×C₃×C₁₁	C₂×C₉₇
LCS	G ▷ C₃×C₁₁
UCS	C₁ ◁ C₂×C₉₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 12804

A minimal faithful permutation representation — 2 generators on 115 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(13,14)(15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,4,6,8,10,11,9,7,5,3)(12,13,14)

Conjugacy classes & character table

3492 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉₇×Dic₉₉, Dic₃₃×C₉₇⋊C₃, C₉₇×C₃⋊Dic₃₃, …
maximal normal subgroup of	Dic₃₃×C₉₇⋊C₃, C₃×C₉₇×Dic₃₃
maximal quotient of	C₉₇×Dic₉₉, C₉₇×C₃⋊Dic₃₃, C₃×C₉₇×Dic₃₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12804, 8);

C97×Dic33 #27165371