C₇×Dic₄₇₅ #27210848

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×Dic₄₇₅
unambiguous semidirect product	C₁₉⋊(C₇×Dic₂₅)

Splittings

derived, Hall	C₁₉×C₂₅ ⋊ C₄×C₇
Hall	C₇×C₁₉×C₂₅ ⋊ C₄
Hall	C₇×C₂₅ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ C₇×Dic₁₉
Sylow	C₁₉ ⋊ C₇×Dic₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₄₇₅
commutator, hypocenter	C₁₉×C₂₅	C₄×C₇
Frattini	C₂×C₅	C₇×D₉₅
Fitting	C₂×C₇×C₁₉×C₂₅	C₂
socle, prime radical	C₂×C₅×C₇×C₁₉	D₅
cosocle	C₂×C₁₉×C₂₅	C₂×C₇
LCS	G ▷ C₁₉×C₂₅
UCS	C₁ ◁ C₂×C₇

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 13300

A minimal faithful permutation representation — 2 generators on 55 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,23)(22,26)(24,29)(25,32)(27,35)(28,34)(30,37)(31,39)(33,41)(36,42)(38,43)(40,44)(45,48,47,46)(49,50,51,52,53,54,55);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,25,40,39,23,22,33,43,29,28,27,42,37,30,36,35,34,24,38,41,26,21,31,44,32)

Conjugacy classes & character table

3346 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 6);

C7×Dic475 #27210848