Abstract group 124500.a: $C_{249}\times D_{250}$

• Label: 124500.a
• Order: \( 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \)
• Exponent: \( 2 \cdot 3 \cdot 5^{3} \cdot 83 \)
• 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.: $213$
• Trans deg.: $62250$
• Rank: $2$

Group information

Description:	$C_{249}\times D_{250}$
Order:	\(124500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \)
Exponent:	\(62250\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \cdot 83 \)
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, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	3	5	6	10	15	25	30	50	75	83	125	150	166	249	250	375	415	498	750	830	1245	2075	2490	4150	6225	10375	12450	20750	31125	62250
Elements	1	251	2	4	502	4	8	20	8	20	40	82	100	40	20582	164	100	200	328	41164	200	328	656	1640	656	1640	3280	8200	3280	8200	16400	16400	124500
Conjugacy classes	1	3	2	2	6	2	4	10	4	10	20	82	50	20	246	164	50	100	164	492	100	164	328	820	328	820	1640	4100	1640	4100	8200	8200	31872
Divisions	1	3	1	1	3	1	1	1	1	1	1	1	1	1	3	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	40
Autjugacy classes	1	2	1	1	2	1	1	1	1	1	1	1	1	1	2	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	36

Minimal presentations

Permutation degree:	$213$
Transitive degree:	$62250$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{62250}=1, b^{a}=b^{499} \rangle$
Permutation group:	Degree $213$ $\langle(87,88)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,124) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 498 \end{array}\right), \left(\begin{array}{rr} 345 & 162 \\ 237 & 345 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{12450}$ . $D_5$	$C_{6225}$ . $D_{10}$	$C_{2490}$ . $D_{25}$	$C_{1245}$ . $D_{50}$	all 28

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

Homology

Abelianization:	$C_{2} \times C_{498} \simeq C_{2}^{2} \times C_{3} \times C_{83}$
Schur multiplier:	$C_{2}$
Commutator length:	not computed

Subgroups

There are 1904 subgroups in 80 conjugacy classes, 44 normal (36 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{498}$	$G/Z \simeq$ $D_{125}$
Commutator:	$G' \simeq$ $C_{125}$	$G/G' \simeq$ $C_2\times C_{498}$
Frattini:	$\Phi \simeq$ $C_{25}$	$G/\Phi \simeq$ $D_5\times C_{498}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{62250}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{249}\times D_{250}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2490}$	$G/\operatorname{soc} \simeq$ $D_{25}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
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

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 124500: $C_{249}\times D_{250}$

Order 62250: $C_{249}\times D_{125}$ x 2, $C_{62250}$

Order 41500: $C_{83}\times D_{250}$

Order 31125: $C_{31125}$

Order 24900: $D_{50}\times C_{249}$

Order 20750: $C_{83}\times D_{125}$ x 2, $C_{20750}$

Order 12450: $D_{25}\times C_{249}$ x 2, $C_{12450}$

Order 10375: $C_{10375}$

Order 8300: $C_{83}\times D_{50}$

Order 6225: $C_{6225}$

Order 4980: $D_5\times C_{498}$

Order 4150: $D_{25}\times C_{83}$ x 2, $C_{4150}$

Order 2490: $D_5\times C_{249}$ x 2, $C_{2490}$

Order 2075: $C_{2075}$

Order 1660: $D_5\times C_{166}$

Order 1500: $C_3\times D_{250}$

Order 1245: $C_{1245}$

Order 996: $C_2\times C_{498}$

Order 830: $D_5\times C_{83}$ x 2, $C_{830}$

Order 750: $C_3\times D_{125}$ x 2, $C_{750}$

Order 500: $D_{250}$

Order 498: $C_{498}$ x 3

Order 415: $C_{415}$

Order 375: $C_{375}$

Order 332: $C_2\times C_{166}$

Order 300: $C_3\times D_{50}$

Order 250: $D_{125}$ x 2, $C_{250}$

Order 249: $C_{249}$

Order 166: $C_{166}$ x 3

Order 150: $C_3\times D_{25}$ x 2, $C_{150}$

Order 125: $C_{125}$

Order 100: $D_{50}$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_3\times D_{10}$

Order 50: $D_{25}$ x 2, $C_{50}$

Order 30: $C_3\times D_5$ x 2, $C_{30}$

Order 25: $C_{25}$

Order 20: $D_{10}$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$

Order 10: $D_5$ x 2, $C_{10}$

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 124500: $C_{249}\times D_{250}$

Order 62250: $C_{62250}$

Order 41500: $C_{83}\times D_{250}$

Order 31125: $C_{31125}$

Order 24900: $D_{50}\times C_{249}$

Order 10375: $C_{10375}$

Order 8300: $C_{83}\times D_{50}$

Order 6225: $C_{6225}$

Order 4980: $D_5\times C_{498}$

Order 2075: $C_{2075}$

Order 1660: $D_5\times C_{166}$

Order 1500: $C_3\times D_{250}$

Order 1245: $C_{1245}$

Order 996: $C_2\times C_{498}$

Order 500: $D_{250}$

Order 415: $C_{415}$

Order 375: $C_{375}$

Order 332: $C_2\times C_{166}$

Order 300: $C_3\times D_{50}$

Order 249: $C_{249}$

Order 125: $C_{125}$

Order 100: $D_{50}$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_3\times D_{10}$

Order 25: $C_{25}$

Order 20: $D_{10}$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 124500: $C_{249}\times D_{250}$ ($C_1$)

Order 62250: $C_{249}\times D_{125}$ ($C_2$) x 2, $C_{62250}$ ($C_2$)

Order 41500: $C_{83}\times D_{250}$ ($C_3$)

Order 31125: $C_{31125}$ ($C_2^2$)

Order 20750: $C_{83}\times D_{125}$ ($C_6$) x 2, $C_{20750}$ ($C_6$)

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

Order 10375: $C_{10375}$ ($C_2\times C_6$)

Order 6225: $C_{6225}$ ($D_{10}$)

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

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

Order 2075: $C_{2075}$ ($C_3\times D_{10}$)

Order 1500: $C_3\times D_{250}$ ($C_{83}$)

Order 1245: $C_{1245}$ ($D_{50}$)

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

Order 750: $C_3\times D_{125}$ ($C_{166}$) x 2, $C_{750}$ ($C_{166}$)

Order 500: $D_{250}$ ($C_{249}$)

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

Order 415: $C_{415}$ ($C_3\times D_{50}$)

Order 375: $C_{375}$ ($C_2\times C_{166}$)

Order 250: $D_{125}$ ($C_{498}$) x 2, $C_{250}$ ($C_{498}$)

Order 249: $C_{249}$ ($D_{250}$)

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

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

Order 125: $C_{125}$ ($C_2\times C_{498}$)

Order 83: $C_{83}$ ($C_3\times D_{250}$)

Order 75: $C_{75}$ ($D_5\times C_{166}$)

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

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

Order 25: $C_{25}$ ($D_5\times C_{498}$)

Order 15: $C_{15}$ ($C_{83}\times D_{50}$)

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

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

Order 5: $C_5$ ($D_{50}\times C_{249}$)

Order 3: $C_3$ ($C_{83}\times D_{250}$)

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

Order 1: $C_1$ ($C_{249}\times D_{250}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 124500: $C_{249}\times D_{250}$ ($C_1$)

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

Order 41500: $C_{83}\times D_{250}$ ($C_3$)

Order 31125: $C_{31125}$ ($C_2^2$)

Order 10375: $C_{10375}$ ($C_2\times C_6$)

Order 6225: $C_{6225}$ ($D_{10}$)

Order 2075: $C_{2075}$ ($C_3\times D_{10}$)

Order 1500: $C_3\times D_{250}$ ($C_{83}$)

Order 1245: $C_{1245}$ ($D_{50}$)

Order 500: $D_{250}$ ($C_{249}$)

Order 415: $C_{415}$ ($C_3\times D_{50}$)

Order 375: $C_{375}$ ($C_2\times C_{166}$)

Order 249: $C_{249}$ ($D_{250}$)

Order 125: $C_{125}$ ($C_2\times C_{498}$)

Order 83: $C_{83}$ ($C_3\times D_{250}$)

Order 75: $C_{75}$ ($D_5\times C_{166}$)

Order 25: $C_{25}$ ($D_5\times C_{498}$)

Order 15: $C_{15}$ ($C_{83}\times D_{50}$)

Order 5: $C_5$ ($D_{50}\times C_{249}$)

Order 3: $C_3$ ($C_{83}\times D_{250}$)

Order 1: $C_1$ ($C_{249}\times D_{250}$)

Series

Derived series

$C_{249}\times D_{250}$

$\rhd$

$C_{125}$

$\rhd$

$C_1$

Chief series

$C_{249}\times D_{250}$

$\rhd$

$C_{249}\times D_{125}$

$\rhd$

$C_{31125}$

$\rhd$

$C_{10375}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{415}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$C_{249}\times D_{250}$

$\rhd$

$C_{125}$

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 2 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 $40 \times 40$ rational character table is not available for this group.