#432×C₁₉₇ #27160643

order	12608
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#432×C₁₉₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉₇	C₂³
commutator	C₂²	C₂²×C₄×C₁₉₇
Frattini, cosocle	C₂×C₄	C₂³×C₁₉₇
socle	C₂²×C₁₉₇	C₂²×C₄
prime radical	C₂³×C₁₉₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉₇ ◁ G

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 213
transitive degree	μ_t(G) = 6304

A minimal faithful permutation representation — 3 generators on 213 points:

(1,2)(3,6)(4,5)(7,8)(9,10,11,14,12,13,15,16)(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);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,13)(14,16)

Conjugacy classes & character table

5516 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#38740567, #38740569, #38740573, …
maximal subgroup of	#1285×C₁₉₇, #1287×C₁₉₇, #1288×C₁₉₇, …
maximal normal subgroup of	#1285×C₁₉₇, #1287×C₁₉₇, #1288×C₁₉₇, …
maximal quotient of	#1745×C₁₉₇, #1749×C₁₉₇, #1767×C₁₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12608, 113);

#432×C197 #27160643