Abstract group 124500.c: $C_{125}:C_{996}$

• Label: 124500.c
• Order: \( 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 83 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 5^{5} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5^{2} \cdot 41 \)
• Perm deg.: $215$
• Trans deg.: $124500$
• Rank: $2$

Group information

Description:	$C_{125}:C_{996}$
Order:	\(124500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \)
Exponent:	\(124500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \)
Automorphism group:	$C_2^2\times C_{82}\times (C_{25}.C_{25}.C_5):C_4$, of order \(4100000\)\(\medspace = 2^{5} \cdot 5^{5} \cdot 41 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$ x 3, $C_{83}$
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	3	4	5	6	10	12	15	25	30	50	75	83	125	150	166	249	250	332	375	415	498	750	830	996	1245	2075	2490	4150	6225	10375	12450	20750	31125	62250
Elements	1	1	2	250	4	2	4	500	8	20	8	20	40	82	100	40	82	164	100	20500	200	328	164	200	328	41000	656	1640	656	1640	3280	8200	3280	8200	16400	16400	124500
Conjugacy classes	1	1	2	2	2	2	2	4	4	10	4	10	20	82	50	20	82	164	50	164	100	164	164	100	164	328	328	820	328	820	1640	4100	1640	4100	8200	8200	31872
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	36
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	36

Minimal presentations

Permutation degree:	$215$
Transitive degree:	$124500$
Rank:	$2$
Inequivalent generating pairs:	$2016$

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

Constructions

Presentation:	$\langle a, b \mid a^{996}=b^{125}=1, b^{a}=b^{124} \rangle$
Permutation group:	Degree $215$ $\langle(2,4)(3,5)(6,11)(7,12)(8,10)(9,13)(14,22)(15,23)(16,21)(17,24)(18,20)(19,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 436 & 74 \\ 132 & 63 \end{array}\right), \left(\begin{array}{rr} 345 & 162 \\ 237 & 345 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	$C_3$ $\, \times\, $ $C_{83}$ $\, \times\, $ $(C_{125}:C_4)$
Semidirect product:	$C_{31125}$ $\,\rtimes\,$ $C_4$	$C_{375}$ $\,\rtimes\,$ $C_{332}$	$C_{125}$ $\,\rtimes\,$ $C_{996}$	$C_{10375}$ $\,\rtimes\,$ $C_{12}$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12450}$ . $D_5$	$C_{2490}$ . $D_{25}$	$C_{498}$ . $D_{125}$	$C_{62250}$ . $C_2$	all 24

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

Homology

Abelianization:	$C_{996} \simeq C_{4} \times C_{3} \times C_{83}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 656 subgroups in 48 conjugacy classes, 36 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{498}$	$G/Z \simeq$ $D_{125}$
Commutator:	$G' \simeq$ $C_{125}$	$G/G' \simeq$ $C_{996}$
Frattini:	$\Phi \simeq$ $C_{50}$	$G/\Phi \simeq$ $D_5\times C_{249}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{62250}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{125}:C_{996}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2490}$	$G/\operatorname{soc} \simeq$ $D_{25}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{125}$
83-Sylow subgroup:	$P_{ 83 } \simeq$ $C_{83}$

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 124500: $C_{125}:C_{996}$

Order 62250: $C_{62250}$

Order 41500: $C_{125}:C_{332}$

Order 31125: $C_{31125}$

Order 24900: $C_{25}:C_{996}$

Order 20750: $C_{20750}$

Order 12450: $C_{12450}$

Order 10375: $C_{10375}$

Order 8300: $C_{25}:C_{332}$

Order 6225: $C_{6225}$

Order 4980: $C_5:C_{996}$

Order 4150: $C_{4150}$

Order 2490: $C_{2490}$

Order 2075: $C_{2075}$

Order 1660: $C_5:C_{332}$

Order 1500: $C_{125}:C_{12}$

Order 1245: $C_{1245}$

Order 996: $C_{996}$

Order 830: $C_{830}$

Order 750: $C_{750}$

Order 500: $C_{125}:C_4$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 375: $C_{375}$

Order 332: $C_{332}$

Order 300: $C_{25}:C_{12}$

Order 250: $C_{250}$

Order 249: $C_{249}$

Order 166: $C_{166}$

Order 150: $C_{150}$

Order 125: $C_{125}$

Order 100: $C_{25}:C_4$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_5:C_{12}$

Order 50: $C_{50}$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_5:C_4$

Order 15: $C_{15}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 124500: $C_{125}:C_{996}$

Order 62250: $C_{62250}$

Order 41500: $C_{125}:C_{332}$

Order 31125: $C_{31125}$

Order 24900: $C_{25}:C_{996}$

Order 20750: $C_{20750}$

Order 12450: $C_{12450}$

Order 10375: $C_{10375}$

Order 8300: $C_{25}:C_{332}$

Order 6225: $C_{6225}$

Order 4980: $C_5:C_{996}$

Order 4150: $C_{4150}$

Order 2490: $C_{2490}$

Order 2075: $C_{2075}$

Order 1660: $C_5:C_{332}$

Order 1500: $C_{125}:C_{12}$

Order 1245: $C_{1245}$

Order 996: $C_{996}$

Order 830: $C_{830}$

Order 750: $C_{750}$

Order 500: $C_{125}:C_4$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 375: $C_{375}$

Order 332: $C_{332}$

Order 300: $C_{25}:C_{12}$

Order 250: $C_{250}$

Order 249: $C_{249}$

Order 166: $C_{166}$

Order 150: $C_{150}$

Order 125: $C_{125}$

Order 100: $C_{25}:C_4$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_5:C_{12}$

Order 50: $C_{50}$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_5:C_4$

Order 15: $C_{15}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 124500: $C_{125}:C_{996}$ ($C_1$)

Order 62250: $C_{62250}$ ($C_2$)

Order 41500: $C_{125}:C_{332}$ ($C_3$)

Order 31125: $C_{31125}$ ($C_4$)

Order 20750: $C_{20750}$ ($C_6$)

Order 12450: $C_{12450}$ ($D_5$)

Order 10375: $C_{10375}$ ($C_{12}$)

Order 6225: $C_{6225}$ ($C_5:C_4$)

Order 4150: $C_{4150}$ ($C_3\times D_5$)

Order 2490: $C_{2490}$ ($D_{25}$)

Order 2075: $C_{2075}$ ($C_5:C_{12}$)

Order 1500: $C_{125}:C_{12}$ ($C_{83}$)

Order 1245: $C_{1245}$ ($C_{25}:C_4$)

Order 830: $C_{830}$ ($C_3\times D_{25}$)

Order 750: $C_{750}$ ($C_{166}$)

Order 500: $C_{125}:C_4$ ($C_{249}$)

Order 498: $C_{498}$ ($D_{125}$)

Order 415: $C_{415}$ ($C_{25}:C_{12}$)

Order 375: $C_{375}$ ($C_{332}$)

Order 250: $C_{250}$ ($C_{498}$)

Order 249: $C_{249}$ ($C_{125}:C_4$)

Order 166: $C_{166}$ ($C_3\times D_{125}$)

Order 150: $C_{150}$ ($D_5\times C_{83}$)

Order 125: $C_{125}$ ($C_{996}$)

Order 83: $C_{83}$ ($C_{125}:C_{12}$)

Order 75: $C_{75}$ ($C_5:C_{332}$)

Order 50: $C_{50}$ ($D_5\times C_{249}$)

Order 30: $C_{30}$ ($D_{25}\times C_{83}$)

Order 25: $C_{25}$ ($C_5:C_{996}$)

Order 15: $C_{15}$ ($C_{25}:C_{332}$)

Order 10: $C_{10}$ ($D_{25}\times C_{249}$)

Order 6: $C_6$ ($C_{83}\times D_{125}$)

Order 5: $C_5$ ($C_{25}:C_{996}$)

Order 3: $C_3$ ($C_{125}:C_{332}$)

Order 2: $C_2$ ($C_{249}\times D_{125}$)

Order 1: $C_1$ ($C_{125}:C_{996}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 124500: $C_{125}:C_{996}$ ($C_1$)

Order 62250: $C_{62250}$ ($C_2$)

Order 41500: $C_{125}:C_{332}$ ($C_3$)

Order 31125: $C_{31125}$ ($C_4$)

Order 20750: $C_{20750}$ ($C_6$)

Order 12450: $C_{12450}$ ($D_5$)

Order 10375: $C_{10375}$ ($C_{12}$)

Order 6225: $C_{6225}$ ($C_5:C_4$)

Order 4150: $C_{4150}$ ($C_3\times D_5$)

Order 2490: $C_{2490}$ ($D_{25}$)

Order 2075: $C_{2075}$ ($C_5:C_{12}$)

Order 1500: $C_{125}:C_{12}$ ($C_{83}$)

Order 1245: $C_{1245}$ ($C_{25}:C_4$)

Order 830: $C_{830}$ ($C_3\times D_{25}$)

Order 750: $C_{750}$ ($C_{166}$)

Order 500: $C_{125}:C_4$ ($C_{249}$)

Order 498: $C_{498}$ ($D_{125}$)

Order 415: $C_{415}$ ($C_{25}:C_{12}$)

Order 375: $C_{375}$ ($C_{332}$)

Order 250: $C_{250}$ ($C_{498}$)

Order 249: $C_{249}$ ($C_{125}:C_4$)

Order 166: $C_{166}$ ($C_3\times D_{125}$)

Order 150: $C_{150}$ ($D_5\times C_{83}$)

Order 125: $C_{125}$ ($C_{996}$)

Order 83: $C_{83}$ ($C_{125}:C_{12}$)

Order 75: $C_{75}$ ($C_5:C_{332}$)

Order 50: $C_{50}$ ($D_5\times C_{249}$)

Order 30: $C_{30}$ ($D_{25}\times C_{83}$)

Order 25: $C_{25}$ ($C_5:C_{996}$)

Order 15: $C_{15}$ ($C_{25}:C_{332}$)

Order 10: $C_{10}$ ($D_{25}\times C_{249}$)

Order 6: $C_6$ ($C_{83}\times D_{125}$)

Order 5: $C_5$ ($C_{25}:C_{996}$)

Order 3: $C_3$ ($C_{125}:C_{332}$)

Order 2: $C_2$ ($C_{249}\times D_{125}$)

Order 1: $C_1$ ($C_{125}:C_{996}$)

Series

Derived series

$C_{125}:C_{996}$

$\rhd$

$C_{125}$

$\rhd$

$C_1$

Chief series

$C_{125}:C_{996}$

$\rhd$

$C_{62250}$

$\rhd$

$C_{31125}$

$\rhd$

$C_{10375}$

$\rhd$

$C_{125}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{125}:C_{996}$

$\rhd$

$C_{125}$

Upper central series

$C_1$

$\lhd$

$C_{498}$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $31872 \times 31872$ character table is not available for this group.

Rational character table

The $36 \times 36$ rational character table is not available for this group.