D₁₄₇₇₅ #44443569

order	29550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₄₇₇₅
unambiguous semidirect product	C₃⋊D₄₉₂₅
unambiguous semidirect product	C₂₅⋊D₅₉₁
unambiguous semidirect product	C₁₉₇⋊D₇₅

Splittings

Fitting, derived, Hall	C₃×C₂₅×C₁₉₇ ⋊ C₂
Hall	C₂₅×C₁₉₇ ⋊ D₃
Hall	C₃×C₁₉₇ ⋊ D₂₅
Sylow	C₁₉₇ ⋊ D₇₅
Hall	C₃×C₂₅ ⋊ D₁₉₇
Sylow	C₂₅ ⋊ D₅₉₁
Sylow	C₃ ⋊ D₄₉₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₂₅×C₁₉₇	C₂
Frattini	C₅	D₂₉₅₅
socle	C₃×C₅×C₁₉₇	D₅
LCS	G ▷ C₃×C₂₅×C₁₉₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 225
transitive degree	μ_t(G) = 14775

A minimal faithful permutation representation — 2 generators on 225 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225);(2,3)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(30,225)(31,224)(32,223)(33,222)(34,221)(35,220)(36,219)(37,218)(38,217)(39,216)(40,215)(41,214)(42,213)(43,212)(44,211)(45,210)(46,209)(47,208)(48,207)(49,206)(50,205)(51,204)(52,203)(53,202)(54,201)(55,200)(56,199)(57,198)(58,197)(59,196)(60,195)(61,194)(62,193)(63,192)(64,191)(65,190)(66,189)(67,188)(68,187)(69,186)(70,185)(71,184)(72,183)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,176)(80,175)(81,174)(82,173)(83,172)(84,171)(85,170)(86,169)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)

Polycyclic presentation:

2,3,5,197,5|c^5=e^4,b^a=b^2,c^a=c^4e,d^a=d^196,e^a=e^4

Conjugacy classes & character table

7389 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₄₇₇₅
maximal normal subgroup of	C₂×D₁₄₇₇₅
maximal quotient of	Dic₁₄₇₇₅, C₂×D₁₄₇₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29550, 8);

D14775 #44443569