Abstract group 7550.a: $C_{7550}$

• Label: 7550.a
• Order: \( 2 \cdot 5^{2} \cdot 151 \)
• Exponent: \( 2 \cdot 5^{2} \cdot 151 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 5^{3} \)
• Perm deg.: $178$
• Trans deg.: $7550$
• Rank: $1$

Group information

Description:	$C_{7550}$
Order:	\(7550\)\(\medspace = 2 \cdot 5^{2} \cdot 151 \)
Exponent:	\(7550\)\(\medspace = 2 \cdot 5^{2} \cdot 151 \)
Automorphism group:	$C_{10}\times C_{300}$, of order \(3000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \)
Composition factors:	$C_2$, $C_5$ x 2, $C_{151}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,151$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	5	10	25	50	151	302	755	1510	3775	7550
Elements	1	1	4	4	20	20	150	150	600	600	3000	3000	7550
Conjugacy classes	1	1	4	4	20	20	150	150	600	600	3000	3000	7550
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:	$178$
Transitive degree:	$7550$
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:	$\langle a \mid a^{7550}=1 \rangle$
Permutation group:	Degree $178$ $\langle(1,2)(3,15,27,10,22,5,17,24,12,19,7,14,26,9,21,4,16,23,11,18,6,13,25,8,20) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 65 & 0 \\ 0 & 65 \end{array}\right), \left(\begin{array}{rr} 150 & 0 \\ 0 & 150 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{151})$
Direct product:	$C_2$ $\, \times\, $ $C_{25}$ $\, \times\, $ $C_{151}$
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:	$C_{1510}$ . $C_5$	$C_{755}$ . $C_{10}$	$C_{10}$ . $C_{755}$	$C_5$ . $C_{1510}$	more information

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

Homology

Primary decomposition:	$C_{2} \times C_{25} \times C_{151}$
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:	$Z \simeq$ $C_{7550}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{7550}$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $C_{1510}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{7550}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{7550}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1510}$	$G/\operatorname{soc} \simeq$ $C_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
151-Sylow subgroup:	$P_{ 151 } \simeq$ $C_{151}$

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 7550: $C_{7550}$

Order 3775: $C_{3775}$

Order 1510: $C_{1510}$

Order 755: $C_{755}$

Order 302: $C_{302}$

Order 151: $C_{151}$

Order 50: $C_{50}$

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 7550: $C_{7550}$

Order 3775: $C_{3775}$

Order 1510: $C_{1510}$

Order 755: $C_{755}$

Order 302: $C_{302}$

Order 151: $C_{151}$

Order 50: $C_{50}$

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 7550: $C_{7550}$ ($C_1$)

Order 3775: $C_{3775}$ ($C_2$)

Order 1510: $C_{1510}$ ($C_5$)

Order 755: $C_{755}$ ($C_{10}$)

Order 302: $C_{302}$ ($C_{25}$)

Order 151: $C_{151}$ ($C_{50}$)

Order 50: $C_{50}$ ($C_{151}$)

Order 25: $C_{25}$ ($C_{302}$)

Order 10: $C_{10}$ ($C_{755}$)

Order 5: $C_5$ ($C_{1510}$)

Order 2: $C_2$ ($C_{3775}$)

Order 1: $C_1$ ($C_{7550}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7550: $C_{7550}$ ($C_1$)

Order 3775: $C_{3775}$ ($C_2$)

Order 1510: $C_{1510}$ ($C_5$)

Order 755: $C_{755}$ ($C_{10}$)

Order 302: $C_{302}$ ($C_{25}$)

Order 151: $C_{151}$ ($C_{50}$)

Order 50: $C_{50}$ ($C_{151}$)

Order 25: $C_{25}$ ($C_{302}$)

Order 10: $C_{10}$ ($C_{755}$)

Order 5: $C_5$ ($C_{1510}$)

Order 2: $C_2$ ($C_{3775}$)

Order 1: $C_1$ ($C_{7550}$)

Series

Derived series	$C_{7550}$	$\rhd$	$C_1$
Chief series	$C_{7550}$	$\rhd$	$C_{3775}$	$\rhd$	$C_{755}$	$\rhd$	$C_{151}$	$\rhd$	$C_1$
Lower central series	$C_{7550}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{7550}$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $7550 \times 7550$ character table is not available for this group.

Rational character table

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