C₄×SD₃₂×C₂₂₇ #44396129

order	29056
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×SD₃₂×C₂₂₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₂₇	D₈
commutator	C₈	C₂²×C₄×C₂₂₇
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₇
socle	C₂²×C₂₂₇	C₂×D₈
prime radical	C₂×D₈×C₂₂₇	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₇ ◁ C₄²×C₂₂₇ ◁ C₄×C₈×C₂₂₇ ◁ G

162 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 247
transitive degree	μ_t(G) = 14528

A minimal faithful permutation representation — 3 generators on 247 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247);(5,6,9,13)(7,10,16,19)(8,17,15,12)(11,20,18,14);(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18)

Conjugacy classes & character table

9988 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29056, 905);