Abstract group 41500.b: $C_{41500}$

• Label: 41500.b
• Order: \( 2^{2} \cdot 5^{3} \cdot 83 \)
• Exponent: \( 2^{2} \cdot 5^{3} \cdot 83 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5^{2} \cdot 41 \)
• Perm deg.: $212$
• Trans deg.: $41500$
• Rank: $1$

Group information

Description:	$C_{41500}$
Order:	\(41500\)\(\medspace = 2^{2} \cdot 5^{3} \cdot 83 \)
Exponent:	\(41500\)\(\medspace = 2^{2} \cdot 5^{3} \cdot 83 \)
Automorphism group:	$C_2^2\times C_{4100}$, of order \(16400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 41 \)
Composition factors:	$C_2$ x 2, $C_5$ x 3, $C_{83}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,83$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	4	5	10	20	25	50	83	100	125	166	250	332	415	500	830	1660	2075	4150	8300	10375	20750	41500
Elements	1	1	2	4	4	8	20	20	82	40	100	82	100	164	328	200	328	656	1640	1640	3280	8200	8200	16400	41500
Conjugacy classes	1	1	2	4	4	8	20	20	82	40	100	82	100	164	328	200	328	656	1640	1640	3280	8200	8200	16400	41500
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	24
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	24

Minimal presentations

Permutation degree:	$212$
Transitive degree:	$41500$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	not computed	not computed
Arbitrary	1	not computed	not computed

Constructions

Presentation:	$\langle a \mid a^{41500}=1 \rangle$
Permutation group:	Degree $212$ $\langle(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 242 & 450 \\ 492 & 242 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	$C_4$ $\, \times\, $ $C_{125}$ $\, \times\, $ $C_{83}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{8300}$ . $C_5$	$C_{830}$ . $C_{50}$	$C_{50}$ . $C_{830}$	$C_5$ . $C_{8300}$	all 16

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Primary decomposition:	$C_{4} \times C_{125} \times C_{83}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 24 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{41500}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{41500}$
Frattini:	$\Phi \simeq$ $C_{50}$	$G/\Phi \simeq$ $C_{830}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{41500}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{41500}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{830}$	$G/\operatorname{soc} \simeq$ $C_{50}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{125}$
83-Sylow subgroup:	$P_{ 83 } \simeq$ $C_{83}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 41500: $C_{41500}$

Order 20750: $C_{20750}$

Order 10375: $C_{10375}$

Order 8300: $C_{8300}$

Order 4150: $C_{4150}$

Order 2075: $C_{2075}$

Order 1660: $C_{1660}$

Order 830: $C_{830}$

Order 500: $C_{500}$

Order 415: $C_{415}$

Order 332: $C_{332}$

Order 250: $C_{250}$

Order 166: $C_{166}$

Order 125: $C_{125}$

Order 100: $C_{100}$

Order 83: $C_{83}$

Order 50: $C_{50}$

Order 25: $C_{25}$

Order 20: $C_{20}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 41500: $C_{41500}$

Order 20750: $C_{20750}$

Order 10375: $C_{10375}$

Order 8300: $C_{8300}$

Order 4150: $C_{4150}$

Order 2075: $C_{2075}$

Order 1660: $C_{1660}$

Order 830: $C_{830}$

Order 500: $C_{500}$

Order 415: $C_{415}$

Order 332: $C_{332}$

Order 250: $C_{250}$

Order 166: $C_{166}$

Order 125: $C_{125}$

Order 100: $C_{100}$

Order 83: $C_{83}$

Order 50: $C_{50}$

Order 25: $C_{25}$

Order 20: $C_{20}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 41500: $C_{41500}$ ($C_1$)

Order 20750: $C_{20750}$ ($C_2$)

Order 10375: $C_{10375}$ ($C_4$)

Order 8300: $C_{8300}$ ($C_5$)

Order 4150: $C_{4150}$ ($C_{10}$)

Order 2075: $C_{2075}$ ($C_{20}$)

Order 1660: $C_{1660}$ ($C_{25}$)

Order 830: $C_{830}$ ($C_{50}$)

Order 500: $C_{500}$ ($C_{83}$)

Order 415: $C_{415}$ ($C_{100}$)

Order 332: $C_{332}$ ($C_{125}$)

Order 250: $C_{250}$ ($C_{166}$)

Order 166: $C_{166}$ ($C_{250}$)

Order 125: $C_{125}$ ($C_{332}$)

Order 100: $C_{100}$ ($C_{415}$)

Order 83: $C_{83}$ ($C_{500}$)

Order 50: $C_{50}$ ($C_{830}$)

Order 25: $C_{25}$ ($C_{1660}$)

Order 20: $C_{20}$ ($C_{2075}$)

Order 10: $C_{10}$ ($C_{4150}$)

Order 5: $C_5$ ($C_{8300}$)

Order 4: $C_4$ ($C_{10375}$)

Order 2: $C_2$ ($C_{20750}$)

Order 1: $C_1$ ($C_{41500}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 41500: $C_{41500}$ ($C_1$)

Order 20750: $C_{20750}$ ($C_2$)

Order 10375: $C_{10375}$ ($C_4$)

Order 8300: $C_{8300}$ ($C_5$)

Order 4150: $C_{4150}$ ($C_{10}$)

Order 2075: $C_{2075}$ ($C_{20}$)

Order 1660: $C_{1660}$ ($C_{25}$)

Order 830: $C_{830}$ ($C_{50}$)

Order 500: $C_{500}$ ($C_{83}$)

Order 415: $C_{415}$ ($C_{100}$)

Order 332: $C_{332}$ ($C_{125}$)

Order 250: $C_{250}$ ($C_{166}$)

Order 166: $C_{166}$ ($C_{250}$)

Order 125: $C_{125}$ ($C_{332}$)

Order 100: $C_{100}$ ($C_{415}$)

Order 83: $C_{83}$ ($C_{500}$)

Order 50: $C_{50}$ ($C_{830}$)

Order 25: $C_{25}$ ($C_{1660}$)

Order 20: $C_{20}$ ($C_{2075}$)

Order 10: $C_{10}$ ($C_{4150}$)

Order 5: $C_5$ ($C_{8300}$)

Order 4: $C_4$ ($C_{10375}$)

Order 2: $C_2$ ($C_{20750}$)

Order 1: $C_1$ ($C_{41500}$)

Series

Derived series

$C_{41500}$

$\rhd$

$C_1$

Chief series

$C_{41500}$

$\rhd$

$C_{20750}$

$\rhd$

$C_{10375}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{415}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$C_{41500}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{41500}$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $41500 \times 41500$ character table is not available for this group.

Rational character table

The $24 \times 24$ rational character table is not available for this group.