Abstract group 62750.e: $C_{125}\times D_{251}$

• Label: 62750.e
• Order: \( 2 \cdot 5^{3} \cdot 251 \)
• Exponent: \( 2 \cdot 5^{3} \cdot 251 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5^{3} \)
• $\card{Z(G)}$: \( 5^{3} \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 5^{5} \cdot 251 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5^{5} \)
• Perm deg.: $376$
• Trans deg.: $31375$
• Rank: $2$

Group information

Description:	$C_{125}\times D_{251}$
Order:	\(62750\)\(\medspace = 2 \cdot 5^{3} \cdot 251 \)
Exponent:	\(62750\)\(\medspace = 2 \cdot 5^{3} \cdot 251 \)
Automorphism group:	$C_{251}.C_5^2.C_5^2.C_{20}.C_2$, of order \(6275000\)\(\medspace = 2^{3} \cdot 5^{5} \cdot 251 \)
Composition factors:	$C_2$, $C_5$ x 3, $C_{251}$
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	125	250	251	1255	6275	31375
Elements	1	251	4	1004	20	5020	100	25100	250	1000	5000	25000	62750
Conjugacy classes	1	1	4	4	20	20	100	100	125	500	2500	12500	15875
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:	$376$
Transitive degree:	$31375$
Rank:	$2$
Inequivalent generating pairs:	$450$

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b \mid a^{250}=b^{251}=1, b^{a}=b^{250} \rangle$
Permutation group:	Degree $376$ $\langle(1,2,4,6,8,10,12,14,16,18,20,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,88,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,144,147,150,146,152,154,156,158,160,162,164,166,168,170,172,174,176,178,180,182,184,186,188,190,192,194,196,198,200,202,204,207,210,206,212,214,216,218,220,222,224,226,228,230,232,234,236,238,240,242,244,246,248,250,251,249,247,245,243,241,239,237,235,233,231,229,227,225,223,221,219,217,215,213,209,211,208,205,203,201,199,197,195,193,191,189,187,185,183,181,179,177,175,173,171,169,167,165,163,161,159,157,155,153,149,151,148,145,143,141,139,137,135,133,131,129,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,90,91,89,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,59,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,19,17,15,13,11,9,7,5,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 139 & 0 \\ 0 & 112 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{251})$
Direct product:	$C_{125}$ $\, \times\, $ $D_{251}$
Semidirect product:	$C_{31375}$ $\,\rtimes\,$ $C_2$	$C_{251}$ $\,\rtimes\,$ $C_{250}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{6275}$ . $C_{10}$	$C_{1255}$ . $C_{50}$	$(C_{25}\times D_{251})$ . $C_5$	$(C_5\times D_{251})$ . $C_{25}$	all 6

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

Homology

Abelianization:	$C_{250} \simeq C_{2} \times C_{125}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 1016 subgroups in 16 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{125}$	$G/Z \simeq$ $D_{251}$
Commutator:	$G' \simeq$ $C_{251}$	$G/G' \simeq$ $C_{250}$
Frattini:	$\Phi \simeq$ $C_{25}$	$G/\Phi \simeq$ $C_5\times D_{251}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{31375}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{125}\times D_{251}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1255}$	$G/\operatorname{soc} \simeq$ $C_{50}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{125}$
251-Sylow subgroup:	$P_{ 251 } \simeq$ $C_{251}$

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 62750: $C_{125}\times D_{251}$

Order 31375: $C_{31375}$

Order 12550: $C_{25}\times D_{251}$

Order 6275: $C_{6275}$

Order 2510: $C_5\times D_{251}$

Order 1255: $C_{1255}$

Order 502: $D_{251}$

Order 251: $C_{251}$

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 62750: $C_{125}\times D_{251}$

Order 31375: $C_{31375}$

Order 12550: $C_{25}\times D_{251}$

Order 6275: $C_{6275}$

Order 2510: $C_5\times D_{251}$

Order 1255: $C_{1255}$

Order 502: $D_{251}$

Order 251: $C_{251}$

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 62750: $C_{125}\times D_{251}$ ($C_1$)

Order 31375: $C_{31375}$ ($C_2$)

Order 12550: $C_{25}\times D_{251}$ ($C_5$)

Order 6275: $C_{6275}$ ($C_{10}$)

Order 2510: $C_5\times D_{251}$ ($C_{25}$)

Order 1255: $C_{1255}$ ($C_{50}$)

Order 502: $D_{251}$ ($C_{125}$)

Order 251: $C_{251}$ ($C_{250}$)

Order 125: $C_{125}$ ($D_{251}$)

Order 25: $C_{25}$ ($C_5\times D_{251}$)

Order 5: $C_5$ ($C_{25}\times D_{251}$)

Order 1: $C_1$ ($C_{125}\times D_{251}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 62750: $C_{125}\times D_{251}$ ($C_1$)

Order 31375: $C_{31375}$ ($C_2$)

Order 12550: $C_{25}\times D_{251}$ ($C_5$)

Order 6275: $C_{6275}$ ($C_{10}$)

Order 2510: $C_5\times D_{251}$ ($C_{25}$)

Order 1255: $C_{1255}$ ($C_{50}$)

Order 502: $D_{251}$ ($C_{125}$)

Order 251: $C_{251}$ ($C_{250}$)

Order 125: $C_{125}$ ($D_{251}$)

Order 25: $C_{25}$ ($C_5\times D_{251}$)

Order 5: $C_5$ ($C_{25}\times D_{251}$)

Order 1: $C_1$ ($C_{125}\times D_{251}$)

Series

Derived series

$C_{125}\times D_{251}$

$\rhd$

$C_{251}$

$\rhd$

$C_1$

Chief series

$C_{125}\times D_{251}$

$\rhd$

$C_{31375}$

$\rhd$

$C_{6275}$

$\rhd$

$C_{1255}$

$\rhd$

$C_{251}$

$\rhd$

$C_1$

Lower central series

$C_{125}\times D_{251}$

$\rhd$

$C_{251}$

Upper central series

$C_1$

$\lhd$

$C_{125}$

Supergroups

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

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

Character theory

Complex character table

The $15875 \times 15875$ character table is not available for this group.

Rational character table

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