Abstract group 1850.4: $C_{1850}$

• Label: 1850.4
• Order: \( 2 \cdot 5^{2} \cdot 37 \)
• Exponent: \( 2 \cdot 5^{2} \cdot 37 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Perm deg.: $64$
• Trans deg.: $1850$
• Rank: $1$

Group information

Description:	$C_{1850}$
Order:	\(1850\)\(\medspace = 2 \cdot 5^{2} \cdot 37 \)
Exponent:	\(1850\)\(\medspace = 2 \cdot 5^{2} \cdot 37 \)
Automorphism group:	$C_4\times C_{180}$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$, $C_5$ x 2, $C_{37}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,37$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	5	10	25	37	50	74	185	370	925	1850
Elements	1	1	4	4	20	36	20	36	144	144	720	720	1850
Conjugacy classes	1	1	4	4	20	36	20	36	144	144	720	720	1850
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	12
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	12

Minimal presentations

Permutation degree:	$64$
Transitive degree:	$1850$
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:	$\langle a \mid a^{1850}=1 \rangle$
Permutation group:	Degree $64$ $\langle(1,2), (3,27,22,17,12,7,26,21,16,11,6,25,20,15,10,5,24,19,14,9,4,23,18,13,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 142 & 35 \\ 92 & 142 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{149})$
Direct product:	$C_2$ $\, \times\, $ $C_{25}$ $\, \times\, $ $C_{37}$
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:	$C_{370}$ . $C_5$	$C_5$ . $C_{370}$	$C_{185}$ . $C_{10}$	$C_{10}$ . $C_{185}$	more information

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

Homology

Primary decomposition:	$C_{2} \times C_{25} \times C_{37}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 12 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{1850}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{1850}$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $C_{370}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{1850}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{1850}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{370}$	$G/\operatorname{soc} \simeq$ $C_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$

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 1850: $C_{1850}$

Order 925: $C_{925}$

Order 370: $C_{370}$

Order 185: $C_{185}$

Order 74: $C_{74}$

Order 50: $C_{50}$

Order 37: $C_{37}$

Order 25: $C_{25}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1850: $C_{1850}$

Order 925: $C_{925}$

Order 370: $C_{370}$

Order 185: $C_{185}$

Order 74: $C_{74}$

Order 50: $C_{50}$

Order 37: $C_{37}$

Order 25: $C_{25}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1850: $C_{1850}$ ($C_1$)

Order 925: $C_{925}$ ($C_2$)

Order 370: $C_{370}$ ($C_5$)

Order 185: $C_{185}$ ($C_{10}$)

Order 74: $C_{74}$ ($C_{25}$)

Order 50: $C_{50}$ ($C_{37}$)

Order 37: $C_{37}$ ($C_{50}$)

Order 25: $C_{25}$ ($C_{74}$)

Order 10: $C_{10}$ ($C_{185}$)

Order 5: $C_5$ ($C_{370}$)

Order 2: $C_2$ ($C_{925}$)

Order 1: $C_1$ ($C_{1850}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1850: $C_{1850}$ ($C_1$)

Order 925: $C_{925}$ ($C_2$)

Order 370: $C_{370}$ ($C_5$)

Order 185: $C_{185}$ ($C_{10}$)

Order 74: $C_{74}$ ($C_{25}$)

Order 50: $C_{50}$ ($C_{37}$)

Order 37: $C_{37}$ ($C_{50}$)

Order 25: $C_{25}$ ($C_{74}$)

Order 10: $C_{10}$ ($C_{185}$)

Order 5: $C_5$ ($C_{370}$)

Order 2: $C_2$ ($C_{925}$)

Order 1: $C_1$ ($C_{1850}$)

Series

Derived series	$C_{1850}$	$\rhd$	$C_1$
Chief series	$C_{1850}$	$\rhd$	$C_{925}$	$\rhd$	$C_{185}$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Lower central series	$C_{1850}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{1850}$

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 $1850 \times 1850$ character table is not available for this group.

Rational character table

The $12 \times 12$ rational character table is not available for this group.