Abstract group 7550.d: $C_{25}\times D_{151}$

• Label: 7550.d
• Order: \( 2 \cdot 5^{2} \cdot 151 \)
• Exponent: \( 2 \cdot 5^{2} \cdot 151 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5^{2} \)
• $\card{Z(G)}$: \( 5^{2} \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 5^{3} \cdot 151 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 5^{3} \)
• Perm deg.: $176$
• Trans deg.: $3775$
• Rank: $2$

Group information

Description:	$C_{25}\times D_{151}$
Order:	\(7550\)\(\medspace = 2 \cdot 5^{2} \cdot 151 \)
Exponent:	\(7550\)\(\medspace = 2 \cdot 5^{2} \cdot 151 \)
Automorphism group:	$C_{151}.C_5^2.C_{60}.C_2$, of order \(453000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \cdot 151 \)
Composition factors:	$C_2$, $C_5$ x 2, $C_{151}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	5	10	25	50	151	755	3775
Elements	1	151	4	604	20	3020	150	600	3000	7550
Conjugacy classes	1	1	4	4	20	20	75	300	1500	1925
Divisions	1	1	1	1	1	1	1	1	1	9
Autjugacy classes	1	1	1	1	1	1	1	1	1	9

Minimal presentations

Permutation degree:	$176$
Transitive degree:	$3775$
Rank:	$2$
Inequivalent generating pairs:	$90$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{3775}=1, b^{a}=b^{301} \rangle$
Permutation group:	Degree $176$ $\langle(2,4)(3,5)(6,11)(7,12)(8,10)(9,13)(14,22)(15,23)(16,21)(17,24)(18,20)(19,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 122 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{151})$
Direct product:	$C_{25}$ $\, \times\, $ $D_{151}$
Semidirect product:	$C_{3775}$ $\,\rtimes\,$ $C_2$	$C_{151}$ $\,\rtimes\,$ $C_{50}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{755}$ . $C_{10}$	$(C_5\times D_{151})$ . $C_5$	$C_5$ . $(C_5\times D_{151})$		more information

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

Homology

Abelianization:	$C_{50} \simeq C_{2} \times C_{25}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 462 subgroups in 12 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{25}$	$G/Z \simeq$ $D_{151}$
Commutator:	$G' \simeq$ $C_{151}$	$G/G' \simeq$ $C_{50}$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $C_5\times D_{151}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{3775}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{25}\times D_{151}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{755}$	$G/\operatorname{soc} \simeq$ $C_{10}$
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: $C_{25}\times D_{151}$

Order 3775: $C_{3775}$

Order 1510: $C_5\times D_{151}$

Order 755: $C_{755}$

Order 302: $D_{151}$

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: $C_{25}\times D_{151}$

Order 3775: $C_{3775}$

Order 1510: $C_5\times D_{151}$

Order 755: $C_{755}$

Order 302: $D_{151}$

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: $C_{25}\times D_{151}$ ($C_1$)

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

Order 1510: $C_5\times D_{151}$ ($C_5$)

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

Order 302: $D_{151}$ ($C_{25}$)

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

Order 25: $C_{25}$ ($D_{151}$)

Order 5: $C_5$ ($C_5\times D_{151}$)

Order 1: $C_1$ ($C_{25}\times D_{151}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7550: $C_{25}\times D_{151}$ ($C_1$)

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

Order 1510: $C_5\times D_{151}$ ($C_5$)

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

Order 302: $D_{151}$ ($C_{25}$)

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

Order 25: $C_{25}$ ($D_{151}$)

Order 5: $C_5$ ($C_5\times D_{151}$)

Order 1: $C_1$ ($C_{25}\times D_{151}$)

Series

Derived series	$C_{25}\times D_{151}$	$\rhd$	$C_{151}$	$\rhd$	$C_1$
Chief series	$C_{25}\times D_{151}$	$\rhd$	$C_{3775}$	$\rhd$	$C_{755}$	$\rhd$	$C_{151}$	$\rhd$	$C_1$
Lower central series	$C_{25}\times D_{151}$	$\rhd$	$C_{151}$
Upper central series	$C_1$	$\lhd$	$C_{25}$

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 $1925 \times 1925$ character table is not available for this group.

Rational character table

The $9 \times 9$ rational character table is not available for this group.