Abstract group 297411240.a: $\PSL(2,841)$

• Label: 297411240.a
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2} \cdot 421 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 29 \cdot 421 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2} \cdot 421 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $842$
• Trans deg.: $842$
• Rank: $2$

Group information

Description:	$\PSL(2,841)$
Order:	\(297411240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2} \cdot 421 \)
Exponent:	\(5127780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 29 \cdot 421 \)
Automorphism group:	Group of order \(1189644960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2} \cdot 421 \)
Composition factors:	$\PSL(2,841)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	29	30	35	42	60	70	84	105	140	210	420	421
Elements	1	354061	708122	708122	1416244	708122	2124366	1416244	1416244	2124366	2832488	2832488	4248732	4248732	707280	2832488	8497464	4248732	5664976	8497464	8497464	16994928	16994928	16994928	33989856	148352400	297411240
Conjugacy classes	1	1	1	1	2	1	3	2	2	3	4	4	6	6	2	4	12	6	8	12	12	24	24	24	48	210	423
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	27

Dimension	1	421	840	841	842	1684	2526	3368	5052	6736	10104	20208	40416	176400
Irr. complex chars.	1	2	210	1	209	0	0	0	0	0	0	0	0	0	423
Irr. rational chars.	1	2	0	1	3	3	2	3	3	1	3	3	1	1	27

Minimal presentations

Permutation degree:	$842$
Transitive degree:	$842$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	421	421	421
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSL(2,841)$, $\PSU(2,841)$, $\Omega(3,841)$, $\OmegaMinus(4,29)$, $\PSOMinus(4,29)$, $\POmega(3,841)$, $\POmegaMinus(4,29)$
Permutation group:	Degree $842$ $\langle(3,699,513,791,614,706,136,727,318,296,277,786,259,690,827,423,763,736,783,779,214,95,535,52,837,410,534,163,735,176,399,48,456,152,280,13,161,469,375,184,405,315,710,587,657,672,573,94,431,235,498,76,659,388,299,665,701,83,78,108,261,722,5,460,354,358,210,497,448,213,242,208,802,287,814,585,217,257,282,719,438,211,346,27,702,98,507,811,220,566,234,72,329,182,172,554,790,464,232,734,525,474,289,205,177,716,760,58,540,608,252,47,803,247,253,50,553,319,97,655,138,194,433,4,22,785,571,829,53,403,245,84,415,170,101,368,26,741,476,809,171,264,63,490,378,224,124,609,353,510,130,156,509,463,426,642,677,493,649,68,132,218,327,197,317,688,343,394,178,813,71,165,326,395,560,740,283,119,209,70,612,294,503,462,179,424,416,86,379,567,202,486,796,96,193,753,479,420,515,626,418,816,771,664,309,835,597,459,650,703,805,28,678,732,545,302,147,321,151,266,200,112,576,452,38,373,712,615,676,616,617,449,434,443,541,532,64,251,653,454,708,80,185,441,658,239,485,246,578,201,810,757,6,250,134,730,46,11,23,164,12,569,544,30,561,107,596,572,743,44,158,73,580,574,61,685,117,19,339,159,461,683,618,290,303,174,662,512,275,25,550,483,255,453,222,481,654,831,20,717,647,836,475,523,755,334,828,131,501,494,742,390,589,488,363,116,437,39,312,758,150,226,293,406,21,700,838,696,770,77,199,204,620,830,122,522,605,297,557,263,417,45,577,436,575,93,153,81,630,601,604,238,444,517,281,471,514,262,307,365,473,254,120,33,166,432,65,356,508,349,697,602,624,504,206,447,37,776,286,778,325,377,788,745,670,99,739,316,633,445,478,622,298,189,500,306,51,704,24,340,718,380,724,731,89,41,458,216,766,154,531,638,789,521,361,537,91,495,720,140) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not computed

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,841)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 349672921 subgroups in 111 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\PSL(2,841)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}^2$
421-Sylow subgroup:	$P_{ 421 } \simeq$ $C_{421}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 297411240 (order at least 1) are shown, as well as all normal subgroups of any index.

Order 297411240: $\PSL(2,841)$

Order 353220: $C_{29}^2:C_{420}$

Order 176610: $C_{29}^2:C_{210}$

Order 117740: $C_{29}^2:C_{140}$

Order 88305: $C_{29}^2:C_{105}$

Order 70644: $C_{29}^2:C_{84}$

Order 58870: $C_{29}^2:C_{70}$

Order 50460: $C_{29}^2:C_{60}$

Order 35322: $C_{29}^2:C_{42}$

Order 29435: $C_{29}^2:C_{35}$

Order 25230: $C_{29}^2:C_{30}$

Order 24360: $\PGL(2,29)$ x 2

Order 23548: $C_{29}^2:C_{28}$

Order 17661: $C_{29}^2:C_{21}$

Order 16820: $C_{29}^2:C_{20}$

Order 12615: $C_{29}^2:C_{15}$

Order 12180: $\PSL(2,29)$ x 2

Order 11774: $C_{29}^2:C_{14}$

Order 10092: $C_{29}^2:C_{12}$

Order 8410: $C_{29}^2:C_{10}$

Order 5887: $C_{29}^2:C_7$

Order 5046: $C_{29}^2:C_6$

Order 4205: $C_{29}^2:C_5$

Order 3364: $C_{29}^2:C_4$

Order 2523: $C_{29}^2:C_3$

Order 1682: $C_{29}:D_{29}$

Order 842: $D_{421}$

Order 841: $C_{29}^2$

Order 840: $D_{420}$

Order 812: $F_{29}$ x 2

Order 421: $C_{421}$

Order 420: $D_{210}$ x 2, $C_{420}$

Order 406: $C_{29}:C_{14}$ x 2

Order 280: $D_{140}$

Order 210: $D_{105}$ x 2, $C_{210}$

Order 203: $C_{29}:C_7$ x 2

Order 168: $D_{84}$

Order 140: $D_{70}$ x 2, $C_{140}$

Order 120: $D_{60}$

Order 116: $C_{29}:C_4$ x 2

Order 105: $C_{105}$

Order 84: $D_{42}$ x 2, $C_{84}$

Order 70: $D_{35}$ x 2, $C_{70}$

Order 60: $A_5$ x 2, $D_{30}$ x 2, $C_{60}$

Order 58: $D_{29}$ x 2

Order 56: $D_{28}$

Order 42: $D_{21}$ x 2, $C_{42}$

Order 40: $D_{20}$

Order 35: $C_{35}$

Order 30: $D_{15}$ x 2, $C_{30}$

Order 29: $C_{29}$ x 2

Order 28: $D_{14}$ x 2, $C_{28}$

Order 24: $S_4$ x 2, $D_{12}$

Order 21: $C_{21}$

Order 20: $D_{10}$ x 2, $C_{20}$

Order 15: $C_{15}$

Order 14: $D_7$ x 2, $C_{14}$

Order 12: $D_6$ x 2, $A_4$ x 2, $C_{12}$

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $D_4$

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 297411240: $\PSL(2,841)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,841)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

See the $423 \times 423$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $27 \times 27$ rational character table.