C₂²⋊C₄×C₁₅₇ #14578917

order	2512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₅₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂²⋊C₄×C₁₅₇
unambiguous semidirect product	C₂²⋊(C₄×C₁₅₇)

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₇	C₂²
commutator	C₂	C₂×C₄×C₁₅₇
Frattini, cosocle	C₂²	C₂²×C₁₅₇
prime radical	C₂³×C₁₅₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ G

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 165
transitive degree	μ_t(G) = 1256

A minimal faithful permutation representation — 2 generators on 165 points:

(1,2,4,6)(3,5,7,8)(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);(2,5)(6,8)

Conjugacy classes & character table

1570 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#99×C₁₅₇, C₂×C₂²⋊C₄×C₁₅₇, #117×C₁₅₇, …
maximal normal subgroup of	#99×C₁₅₇, C₂×C₂²⋊C₄×C₁₅₇, #117×C₁₅₇, …
maximal quotient of	#95×C₁₅₇, C₂²⋊C₈×C₁₅₇, #99×C₁₅₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2512, 3);

C22⋊C4×C157 #14578917