Abstract group 250.2: $C_{250}$

• Label: 250.2
• Order: \( 2 \cdot 5^{3} \)
• Exponent: \( 2 \cdot 5^{3} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{2} \cdot 5^{2} \)
• Perm deg.: $127$
• Trans deg.: $250$
• Rank: $1$

Group information

Description:	$C_{250}$
Order:	\(250\)\(\medspace = 2 \cdot 5^{3} \)
Exponent:	\(250\)\(\medspace = 2 \cdot 5^{3} \)
Automorphism group:	$C_{100}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Composition factors:	$C_2$, $C_5$ x 3
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	5	10	25	50	125	250
Elements	1	1	4	4	20	20	100	100	250
Conjugacy classes	1	1	4	4	20	20	100	100	250
Divisions	1	1	1	1	1	1	1	1	8
Autjugacy classes	1	1	1	1	1	1	1	1	8

Dimension	1	4	20	100
Irr. complex chars.	250	0	0	0	250
Irr. rational chars.	2	2	2	2	8

Minimal presentations

Permutation degree:	$127$
Transitive degree:	$250$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	2	100
Arbitrary	1	2	100

Constructions

Presentation:	$\langle a \mid a^{250}=1 \rangle$
Permutation group:	Degree $127$ $\langle(1,2), (3,127,102,77,52,27,122,97,72,47,22,117,92,67,42,17,112,87,62,37,12,107,82,57,32,7,126,101,76,51,26,121,96,71,46,21,116,91,66,41,16,111,86,61,36,11,106,81,56,31,6,125,100,75,50,25,120,95,70,45,20,115,90,65,40,15,110,85,60,35,10,105,80,55,30,5,124,99,74,49,24,119,94,69,44,19,114,89,64,39,14,109,84,59,34,9,104,79,54,29,4,123,98,73,48,23,118,93,68,43,18,113,88,63,38,13,108,83,58,33,8,103,78,53,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 55 & 0 \\ 0 & 64 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{251})$
Direct product:	$C_2$ $\, \times\, $ $C_{125}$
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:	$C_{50}$ . $C_5$	$C_5$ . $C_{50}$	$C_{25}$ . $C_{10}$	$C_{10}$ . $C_{25}$	more information
Aut. group:	$\Aut(C_{251})$	$\Aut(C_{502})$

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

Homology

Primary decomposition:	$C_{2} \times C_{125}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 8 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{250}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{250}$
Frattini:	$\Phi \simeq$ $C_{25}$	$G/\Phi \simeq$ $C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{250}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{250}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_{25}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{125}$

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 250: $C_{250}$

Order 125: $C_{125}$

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 250: $C_{250}$

Order 125: $C_{125}$

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 250: $C_{250}$ ($C_1$)

Order 125: $C_{125}$ ($C_2$)

Order 50: $C_{50}$ ($C_5$)

Order 25: $C_{25}$ ($C_{10}$)

Order 10: $C_{10}$ ($C_{25}$)

Order 5: $C_5$ ($C_{50}$)

Order 2: $C_2$ ($C_{125}$)

Order 1: $C_1$ ($C_{250}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 250: $C_{250}$ ($C_1$)

Order 125: $C_{125}$ ($C_2$)

Order 50: $C_{50}$ ($C_5$)

Order 25: $C_{25}$ ($C_{10}$)

Order 10: $C_{10}$ ($C_{25}$)

Order 5: $C_5$ ($C_{50}$)

Order 2: $C_2$ ($C_{125}$)

Order 1: $C_1$ ($C_{250}$)

Series

Derived series	$C_{250}$	$\rhd$	$C_1$
Chief series	$C_{250}$	$\rhd$	$C_{125}$	$\rhd$	$C_{25}$	$\rhd$	$C_5$	$\rhd$	$C_1$
Lower central series	$C_{250}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{250}$

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $250 \times 250$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	5A	10A	25A	50A	125A	250A
	Size	1	1	4	4	20	20	100	100
	2 P	1A	1A	5A	5A	25A	25A	125A	125A
	5 P	1A	2A	1A	2A	5A	10A	25A	50A
250.2.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
250.2.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
250.2.1c		$4$	$4$	$4$	$4$	$4$	$4$	$-1$	$-1$
250.2.1d		$4$	$-4$	$4$	$-4$	$4$	$-4$	$-1$	$1$
250.2.1e		$20$	$20$	$20$	$20$	$-5$	$-5$	$0$	$0$
250.2.1f		$20$	$-20$	$20$	$-20$	$-5$	$5$	$0$	$0$
250.2.1g		$100$	$100$	$-25$	$-25$	$0$	$0$	$0$	$0$
250.2.1h		$100$	$-100$	$-25$	$25$	$0$	$0$	$0$	$0$