Abstract group 8300.a: $C_{83}\times D_{50}$

• Label: 8300.a
• Order: \( 2^{2} \cdot 5^{2} \cdot 83 \)
• Exponent: \( 2 \cdot 5^{2} \cdot 83 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 83 \)
• $\card{Z(G)}$: \( 2 \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5^{3} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \cdot 41 \)
• Perm deg.: $110$
• Trans deg.: $4150$
• Rank: $2$

Group information

Description:	$C_{83}\times D_{50}$
Order:	\(8300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 83 \)
Exponent:	\(4150\)\(\medspace = 2 \cdot 5^{2} \cdot 83 \)
Automorphism group:	$C_2\times C_{82}\times D_{25}.C_{10}$, of order \(82000\)\(\medspace = 2^{4} \cdot 5^{3} \cdot 41 \)
Composition factors:	$C_2$ x 2, $C_5$ x 2, $C_{83}$
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	5	10	25	50	83	166	415	830	2075	4150
Elements	1	51	4	4	20	20	82	4182	328	328	1640	1640	8300
Conjugacy classes	1	3	2	2	10	10	82	246	164	164	820	820	2324
Divisions	1	3	1	1	1	1	1	3	1	1	1	1	16
Autjugacy classes	1	2	1	1	1	1	1	2	1	1	1	1	14

Minimal presentations

Permutation degree:	$110$
Transitive degree:	$4150$
Rank:	$2$
Inequivalent generating pairs:	$252$

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:	$\langle a, b \mid a^{2}=b^{4150}=1, b^{a}=b^{499} \rangle$
Permutation group:	Degree $110$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 498 \end{array}\right), \left(\begin{array}{rr} 130 & 399 \\ 57 & 130 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	$C_2$ $\, \times\, $ $C_{83}$ $\, \times\, $ $D_{25}$
Semidirect product:	$C_{4150}$ $\,\rtimes\,$ $C_2$	$C_{50}$ $\,\rtimes\,$ $C_{166}$	$C_{2075}$ $\,\rtimes\,$ $C_2^2$	$C_{25}$ $\,\rtimes\,$ $(C_2\times C_{166})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{830}$ . $D_5$	$C_{415}$ . $D_{10}$	$C_{10}$ . $(D_5\times C_{83})$	$C_5$ . $(D_5\times C_{166})$	more information

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

Homology

Abelianization:	$C_{2} \times C_{166} \simeq C_{2}^{2} \times C_{83}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 198 subgroups in 30 conjugacy classes, 18 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{166}$	$G/Z \simeq$ $D_{25}$
Commutator:	$G' \simeq$ $C_{25}$	$G/G' \simeq$ $C_2\times C_{166}$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $D_5\times C_{166}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4150}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{83}\times D_{50}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{830}$	$G/\operatorname{soc} \simeq$ $D_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
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 8300: $C_{83}\times D_{50}$

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

Order 2075: $C_{2075}$

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

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

Order 415: $C_{415}$

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

Order 166: $C_{166}$ x 3

Order 100: $D_{50}$

Order 83: $C_{83}$

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

Order 25: $C_{25}$

Order 20: $D_{10}$

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

Order 5: $C_5$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

Order 2075: $C_{2075}$

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

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

Order 415: $C_{415}$

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

Order 166: $C_{166}$ x 2

Order 100: $D_{50}$

Order 83: $C_{83}$

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

Order 25: $C_{25}$

Order 20: $D_{10}$

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

Order 5: $C_5$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 8300: $C_{83}\times D_{50}$ ($C_1$)

Order 4150: $D_{25}\times C_{83}$ ($C_2$) x 2, $C_{4150}$ ($C_2$)

Order 2075: $C_{2075}$ ($C_2^2$)

Order 830: $C_{830}$ ($D_5$)

Order 415: $C_{415}$ ($D_{10}$)

Order 166: $C_{166}$ ($D_{25}$)

Order 100: $D_{50}$ ($C_{83}$)

Order 83: $C_{83}$ ($D_{50}$)

Order 50: $D_{25}$ ($C_{166}$) x 2, $C_{50}$ ($C_{166}$)

Order 25: $C_{25}$ ($C_2\times C_{166}$)

Order 10: $C_{10}$ ($D_5\times C_{83}$)

Order 5: $C_5$ ($D_5\times C_{166}$)

Order 2: $C_2$ ($D_{25}\times C_{83}$)

Order 1: $C_1$ ($C_{83}\times D_{50}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8300: $C_{83}\times D_{50}$ ($C_1$)

Order 4150: $D_{25}\times C_{83}$ ($C_2$), $C_{4150}$ ($C_2$)

Order 2075: $C_{2075}$ ($C_2^2$)

Order 830: $C_{830}$ ($D_5$)

Order 415: $C_{415}$ ($D_{10}$)

Order 166: $C_{166}$ ($D_{25}$)

Order 100: $D_{50}$ ($C_{83}$)

Order 83: $C_{83}$ ($D_{50}$)

Order 50: $C_{50}$ ($C_{166}$), $D_{25}$ ($C_{166}$)

Order 25: $C_{25}$ ($C_2\times C_{166}$)

Order 10: $C_{10}$ ($D_5\times C_{83}$)

Order 5: $C_5$ ($D_5\times C_{166}$)

Order 2: $C_2$ ($D_{25}\times C_{83}$)

Order 1: $C_1$ ($C_{83}\times D_{50}$)

Series

Derived series

$C_{83}\times D_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_1$

Chief series

$C_{83}\times D_{50}$

$\rhd$

$D_{25}\times C_{83}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{415}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$C_{83}\times D_{50}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_{166}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

The $2324 \times 2324$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.