Abstract group 16600.c: $C_{16600}$

• Label: 16600.c
• Order: \( 2^{3} \cdot 5^{2} \cdot 83 \)
• Exponent: \( 2^{3} \cdot 5^{2} \cdot 83 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{5} \cdot 5 \cdot 41 \)
• Perm deg.: $116$
• Trans deg.: $16600$
• Rank: $1$

Group information

Description:	$C_{16600}$
Order:	\(16600\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 83 \)
Exponent:	\(16600\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 83 \)
Automorphism group:	$C_2^3\times C_{820}$, of order \(6560\)\(\medspace = 2^{5} \cdot 5 \cdot 41 \)
Composition factors:	$C_2$ x 3, $C_5$ x 2, $C_{83}$
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	8	10	20	25	40	50	83	100	166	200	332	415	664	830	1660	2075	3320	4150	8300	16600
Elements	1	1	2	4	4	4	8	20	16	20	82	40	82	80	164	328	328	328	656	1640	1312	1640	3280	6560	16600
Conjugacy classes	1	1	2	4	4	4	8	20	16	20	82	40	82	80	164	328	328	328	656	1640	1312	1640	3280	6560	16600
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:	$116$
Transitive degree:	$16600$
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:	$\langle a \mid a^{16600}=1 \rangle$
Permutation group:	Degree $116$ $\langle(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 360 & 307 \\ 329 & 360 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	$C_8$ $\, \times\, $ $C_{25}$ $\, \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:	$C_{8300}$ . $C_2$	$C_{4150}$ . $C_4$	$C_{3320}$ . $C_5$	$C_{830}$ . $C_{20}$	all 16

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

Homology

Primary decomposition:	$C_{8} \times C_{25} \times C_{83}$
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:	$Z \simeq$ $C_{16600}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{16600}$
Frattini:	$\Phi \simeq$ $C_{20}$	$G/\Phi \simeq$ $C_{830}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{16600}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{16600}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{830}$	$G/\operatorname{soc} \simeq$ $C_{20}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
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 16600: $C_{16600}$

Order 8300: $C_{8300}$

Order 4150: $C_{4150}$

Order 3320: $C_{3320}$

Order 2075: $C_{2075}$

Order 1660: $C_{1660}$

Order 830: $C_{830}$

Order 664: $C_{664}$

Order 415: $C_{415}$

Order 332: $C_{332}$

Order 200: $C_{200}$

Order 166: $C_{166}$

Order 100: $C_{100}$

Order 83: $C_{83}$

Order 50: $C_{50}$

Order 40: $C_{40}$

Order 25: $C_{25}$

Order 20: $C_{20}$

Order 10: $C_{10}$

Order 8: $C_8$

Order 5: $C_5$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 16600: $C_{16600}$

Order 8300: $C_{8300}$

Order 4150: $C_{4150}$

Order 3320: $C_{3320}$

Order 2075: $C_{2075}$

Order 1660: $C_{1660}$

Order 830: $C_{830}$

Order 664: $C_{664}$

Order 415: $C_{415}$

Order 332: $C_{332}$

Order 200: $C_{200}$

Order 166: $C_{166}$

Order 100: $C_{100}$

Order 83: $C_{83}$

Order 50: $C_{50}$

Order 40: $C_{40}$

Order 25: $C_{25}$

Order 20: $C_{20}$

Order 10: $C_{10}$

Order 8: $C_8$

Order 5: $C_5$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16600: $C_{16600}$ ($C_1$)

Order 8300: $C_{8300}$ ($C_2$)

Order 4150: $C_{4150}$ ($C_4$)

Order 3320: $C_{3320}$ ($C_5$)

Order 2075: $C_{2075}$ ($C_8$)

Order 1660: $C_{1660}$ ($C_{10}$)

Order 830: $C_{830}$ ($C_{20}$)

Order 664: $C_{664}$ ($C_{25}$)

Order 415: $C_{415}$ ($C_{40}$)

Order 332: $C_{332}$ ($C_{50}$)

Order 200: $C_{200}$ ($C_{83}$)

Order 166: $C_{166}$ ($C_{100}$)

Order 100: $C_{100}$ ($C_{166}$)

Order 83: $C_{83}$ ($C_{200}$)

Order 50: $C_{50}$ ($C_{332}$)

Order 40: $C_{40}$ ($C_{415}$)

Order 25: $C_{25}$ ($C_{664}$)

Order 20: $C_{20}$ ($C_{830}$)

Order 10: $C_{10}$ ($C_{1660}$)

Order 8: $C_8$ ($C_{2075}$)

Order 5: $C_5$ ($C_{3320}$)

Order 4: $C_4$ ($C_{4150}$)

Order 2: $C_2$ ($C_{8300}$)

Order 1: $C_1$ ($C_{16600}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16600: $C_{16600}$ ($C_1$)

Order 8300: $C_{8300}$ ($C_2$)

Order 4150: $C_{4150}$ ($C_4$)

Order 3320: $C_{3320}$ ($C_5$)

Order 2075: $C_{2075}$ ($C_8$)

Order 1660: $C_{1660}$ ($C_{10}$)

Order 830: $C_{830}$ ($C_{20}$)

Order 664: $C_{664}$ ($C_{25}$)

Order 415: $C_{415}$ ($C_{40}$)

Order 332: $C_{332}$ ($C_{50}$)

Order 200: $C_{200}$ ($C_{83}$)

Order 166: $C_{166}$ ($C_{100}$)

Order 100: $C_{100}$ ($C_{166}$)

Order 83: $C_{83}$ ($C_{200}$)

Order 50: $C_{50}$ ($C_{332}$)

Order 40: $C_{40}$ ($C_{415}$)

Order 25: $C_{25}$ ($C_{664}$)

Order 20: $C_{20}$ ($C_{830}$)

Order 10: $C_{10}$ ($C_{1660}$)

Order 8: $C_8$ ($C_{2075}$)

Order 5: $C_5$ ($C_{3320}$)

Order 4: $C_4$ ($C_{4150}$)

Order 2: $C_2$ ($C_{8300}$)

Order 1: $C_1$ ($C_{16600}$)

Series

Derived series

$C_{16600}$

$\rhd$

$C_1$

Chief series

$C_{16600}$

$\rhd$

$C_{8300}$

$\rhd$

$C_{4150}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{415}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$C_{16600}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{16600}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $16600 \times 16600$ character table is not available for this group.

Rational character table

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