Abstract group 498000.a: $C_{1000}.C_{498}$

• Label: 498000.a
• Order: \( 2^{4} \cdot 3 \cdot 5^{3} \cdot 83 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5^{3} \cdot 83 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 83 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 5^{5} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5^{2} \cdot 41 \)
• Perm deg.: $219$
• Trans deg.: $249000$
• Rank: $2$

Group information

Description:	$C_{1000}.C_{498}$
Order:	\(498000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{3} \cdot 83 \)
Exponent:	\(249000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \cdot 83 \)
Automorphism group:	Group of order \(32800000\)\(\medspace = 2^{8} \cdot 5^{5} \cdot 41 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$ x 3, $C_{83}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$. Whether it is rational has not been computed.

Group statistics

Statistics about orders of elements in this group have not been computed.

Minimal presentations

Permutation degree:	$219$
Transitive degree:	$249000$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Groups of Lie type:	$\GOrthMinus(2,499)$
Presentation:	$\langle a, b \mid a^{498}=b^{1000}=1, b^{a}=b^{499} \rangle$
Permutation group:	Degree $219$ $\langle(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 194 & 446 \\ 135 & 194 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 498 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
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:	$C_{62250}$ . $D_4$	$C_{49800}$ . $D_5$	$C_{9960}$ . $D_{25}$	$C_{4980}$ . $D_{50}$	all 64

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:	$C_1$
Commutator length:	not computed

Subgroups

There are 6928 subgroups in 160 conjugacy classes, 76 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{498}$	$G/Z \simeq$ $D_{500}$
Commutator:	$G' \simeq$ $C_{500}$	$G/G' \simeq$ $C_2\times C_{498}$
Frattini:	$\Phi \simeq$ $C_{100}$	$G/\Phi \simeq$ $D_5\times C_{498}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{249000}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{1000}.C_{498}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2490}$	$G/\operatorname{soc} \simeq$ $D_{100}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
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 498000: $C_{1000}.C_{498}$

Order 249000: $C_{249000}$, $C_{31125}:Q_8$, $C_{249}\times D_{500}$

Order 166000: $C_{1000}.C_{166}$

Order 124500: $C_{125}:C_{996}$, $C_{124500}$, $C_{249}\times D_{250}$

Order 99600: $C_{200}.C_{498}$

Order 83000: $C_{83000}$, $C_{10375}:Q_8$, $C_{83}\times D_{500}$

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

Order 49800: $C_{49800}$, $C_{6225}:Q_8$, $D_{100}\times C_{249}$

Order 41500: $C_{125}:C_{332}$, $C_{41500}$, $C_{83}\times D_{250}$

Order 33200: $C_{16600}:C_2$

Order 31125: $C_{31125}$

Order 24900: $C_{25}:C_{996}$, $C_{24900}$, $D_{50}\times C_{249}$

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

Order 19920: $C_{120}:C_{166}$

Order 16600: $C_{16600}$, $C_{2075}:Q_8$, $C_{83}\times D_{100}$

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

Order 10375: $C_{10375}$

Order 9960: $C_{1245}:Q_8$, $D_{20}\times C_{249}$, $C_{9960}$

Order 8300: $C_{25}:C_{332}$, $C_{8300}$, $C_{83}\times D_{50}$

Order 6640: $C_{40}:C_{166}$

Order 6225: $C_{6225}$

Order 6000: $C_{1000}:C_6$

Order 4980: $C_5:C_{996}$, $D_5\times C_{498}$, $C_{4980}$

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

Order 3984: $\SD_{16}\times C_{249}$

Order 3320: $C_{415}:Q_8$, $D_{20}\times C_{83}$, $C_{3320}$

Order 3000: $C_3\times D_{500}$, $C_{375}:Q_8$, $C_{3000}$

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

Order 2075: $C_{2075}$

Order 2000: $C_{1000}:C_2$

Order 1992: $C_{1992}$, $Q_8\times C_{249}$, $D_4\times C_{249}$

Order 1660: $D_5\times C_{166}$, $C_{1660}$, $C_5:C_{332}$

Order 1500: $C_3\times D_{250}$, $C_{1500}$, $C_{125}:C_{12}$

Order 1328: $\SD_{16}\times C_{83}$

Order 1245: $C_{1245}$

Order 1200: $C_{200}:C_6$

Order 1000: $D_{500}$, $C_{125}:Q_8$, $C_{1000}$

Order 996: $C_{996}$ x 2, $C_2\times C_{498}$

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

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

Order 664: $D_4\times C_{83}$, $C_{664}$, $Q_8\times C_{83}$

Order 600: $C_{600}$, $C_3\times D_{100}$, $C_{75}:Q_8$

Order 500: $D_{250}$, $C_{500}$, $C_{125}:C_4$

Order 498: $C_{498}$ x 2

Order 415: $C_{415}$

Order 400: $C_{200}:C_2$

Order 375: $C_{375}$

Order 332: $C_{332}$ x 2, $C_2\times C_{166}$

Order 300: $C_3\times D_{50}$, $C_{300}$, $C_{25}:C_{12}$

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

Order 249: $C_{249}$

Order 240: $C_{120}:C_2$

Order 200: $D_{100}$, $C_{25}:Q_8$, $C_{200}$

Order 166: $C_{166}$ x 2

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

Order 125: $C_{125}$

Order 120: $C_{120}$, $C_3\times D_{20}$, $C_{15}:Q_8$

Order 100: $D_{50}$, $C_{100}$, $C_{25}:C_4$

Order 83: $C_{83}$

Order 80: $C_{40}:C_2$

Order 75: $C_{75}$

Order 60: $C_{60}$, $C_5:C_{12}$, $C_3\times D_{10}$

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

Order 48: $C_3\times \SD_{16}$

Order 40: $D_{20}$, $C_5:Q_8$, $C_{40}$

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

Order 25: $C_{25}$

Order 24: $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 16: $\SD_{16}$

Order 15: $C_{15}$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

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

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 498000: $C_{1000}.C_{498}$

Order 249000: $C_{249}\times D_{500}$

Order 166000: $C_{1000}.C_{166}$

Order 99600: $C_{200}.C_{498}$

Order 33200: $C_{16600}:C_2$

Order 31125: $C_{31125}$

Order 19920: $C_{120}:C_{166}$

Order 10375: $C_{10375}$

Order 6640: $C_{40}:C_{166}$

Order 6225: $C_{6225}$

Order 6000: $C_{1000}:C_6$

Order 3984: $\SD_{16}\times C_{249}$

Order 2075: $C_{2075}$

Order 2000: $C_{1000}:C_2$

Order 1328: $\SD_{16}\times C_{83}$

Order 1245: $C_{1245}$

Order 1200: $C_{200}:C_6$

Order 415: $C_{415}$

Order 400: $C_{200}:C_2$

Order 375: $C_{375}$

Order 249: $C_{249}$

Order 240: $C_{120}:C_2$

Order 125: $C_{125}$

Order 83: $C_{83}$

Order 80: $C_{40}:C_2$

Order 75: $C_{75}$

Order 48: $C_3\times \SD_{16}$

Order 25: $C_{25}$

Order 16: $\SD_{16}$

Order 15: $C_{15}$

Order 5: $C_5$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 498000: $C_{1000}.C_{498}$ ($C_1$)

Order 249000: $C_{249000}$ ($C_2$), $C_{31125}:Q_8$ ($C_2$), $C_{249}\times D_{500}$ ($C_2$)

Order 166000: $C_{1000}.C_{166}$ ($C_3$)

Order 124500: $C_{124500}$ ($C_2^2$)

Order 83000: $C_{83000}$ ($C_6$), $C_{10375}:Q_8$ ($C_6$), $C_{83}\times D_{500}$ ($C_6$)

Order 62250: $C_{62250}$ ($D_4$)

Order 49800: $C_{49800}$ ($D_5$)

Order 41500: $C_{41500}$ ($C_2\times C_6$)

Order 31125: $C_{31125}$ ($\SD_{16}$)

Order 24900: $C_{24900}$ ($D_{10}$)

Order 20750: $C_{20750}$ ($C_3\times D_4$)

Order 16600: $C_{16600}$ ($C_3\times D_5$)

Order 12450: $C_{12450}$ ($D_{20}$)

Order 10375: $C_{10375}$ ($C_3\times \SD_{16}$)

Order 9960: $C_{9960}$ ($D_{25}$)

Order 8300: $C_{8300}$ ($C_3\times D_{10}$)

Order 6225: $C_{6225}$ ($C_{40}:C_2$)

Order 6000: $C_{1000}:C_6$ ($C_{83}$)

Order 4980: $C_{4980}$ ($D_{50}$)

Order 4150: $C_{4150}$ ($C_3\times D_{20}$)

Order 3320: $C_{3320}$ ($C_3\times D_{25}$)

Order 3000: $C_3\times D_{500}$ ($C_{166}$), $C_{375}:Q_8$ ($C_{166}$), $C_{3000}$ ($C_{166}$)

Order 2490: $C_{2490}$ ($D_{100}$)

Order 2075: $C_{2075}$ ($C_{120}:C_2$)

Order 2000: $C_{1000}:C_2$ ($C_{249}$)

Order 1992: $C_{1992}$ ($D_{125}$)

Order 1660: $C_{1660}$ ($C_3\times D_{50}$)

Order 1500: $C_{1500}$ ($C_2\times C_{166}$)

Order 1245: $C_{1245}$ ($C_{200}:C_2$)

Order 1000: $D_{500}$ ($C_{498}$), $C_{125}:Q_8$ ($C_{498}$), $C_{1000}$ ($C_{498}$)

Order 996: $C_{996}$ ($D_{250}$)

Order 830: $C_{830}$ ($C_3\times D_{100}$)

Order 750: $C_{750}$ ($D_4\times C_{83}$)

Order 664: $C_{664}$ ($C_3\times D_{125}$)

Order 600: $C_{600}$ ($D_5\times C_{83}$)

Order 500: $C_{500}$ ($C_2\times C_{498}$)

Order 498: $C_{498}$ ($D_{500}$)

Order 415: $C_{415}$ ($C_{200}:C_6$)

Order 375: $C_{375}$ ($\SD_{16}\times C_{83}$)

Order 332: $C_{332}$ ($C_3\times D_{250}$)

Order 300: $C_{300}$ ($D_5\times C_{166}$)

Order 250: $C_{250}$ ($D_4\times C_{249}$)

Order 249: $C_{249}$ ($C_{1000}:C_2$)

Order 200: $C_{200}$ ($D_5\times C_{249}$)

Order 166: $C_{166}$ ($C_3\times D_{500}$)

Order 150: $C_{150}$ ($D_{20}\times C_{83}$)

Order 125: $C_{125}$ ($\SD_{16}\times C_{249}$)

Order 120: $C_{120}$ ($D_{25}\times C_{83}$)

Order 100: $C_{100}$ ($D_5\times C_{498}$)

Order 83: $C_{83}$ ($C_{1000}:C_6$)

Order 75: $C_{75}$ ($C_{40}:C_{166}$)

Order 60: $C_{60}$ ($C_{83}\times D_{50}$)

Order 50: $C_{50}$ ($D_{20}\times C_{249}$)

Order 40: $C_{40}$ ($D_{25}\times C_{249}$)

Order 30: $C_{30}$ ($C_{83}\times D_{100}$)

Order 25: $C_{25}$ ($C_{120}:C_{166}$)

Order 24: $C_{24}$ ($C_{83}\times D_{125}$)

Order 20: $C_{20}$ ($D_{50}\times C_{249}$)

Order 15: $C_{15}$ ($C_{16600}:C_2$)

Order 12: $C_{12}$ ($C_{83}\times D_{250}$)

Order 10: $C_{10}$ ($D_{100}\times C_{249}$)

Order 8: $C_8$ ($C_{249}\times D_{125}$)

Order 6: $C_6$ ($C_{83}\times D_{500}$)

Order 5: $C_5$ ($C_{200}.C_{498}$)

Order 4: $C_4$ ($C_{249}\times D_{250}$)

Order 3: $C_3$ ($C_{1000}.C_{166}$)

Order 2: $C_2$ ($C_{249}\times D_{500}$)

Order 1: $C_1$ ($C_{1000}.C_{498}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 498000: $C_{1000}.C_{498}$ ($C_1$)

Order 249000: $C_{249}\times D_{500}$ ($C_2$)

Order 166000: $C_{1000}.C_{166}$ ($C_3$)

Order 31125: $C_{31125}$ ($\SD_{16}$)

Order 10375: $C_{10375}$ ($C_3\times \SD_{16}$)

Order 6225: $C_{6225}$ ($C_{40}:C_2$)

Order 6000: $C_{1000}:C_6$ ($C_{83}$)

Order 2075: $C_{2075}$ ($C_{120}:C_2$)

Order 2000: $C_{1000}:C_2$ ($C_{249}$)

Order 1245: $C_{1245}$ ($C_{200}:C_2$)

Order 415: $C_{415}$ ($C_{200}:C_6$)

Order 375: $C_{375}$ ($\SD_{16}\times C_{83}$)

Order 249: $C_{249}$ ($C_{1000}:C_2$)

Order 125: $C_{125}$ ($\SD_{16}\times C_{249}$)

Order 83: $C_{83}$ ($C_{1000}:C_6$)

Order 75: $C_{75}$ ($C_{40}:C_{166}$)

Order 25: $C_{25}$ ($C_{120}:C_{166}$)

Order 15: $C_{15}$ ($C_{16600}:C_2$)

Order 5: $C_5$ ($C_{200}.C_{498}$)

Order 3: $C_3$ ($C_{1000}.C_{166}$)

Order 1: $C_1$ ($C_{1000}.C_{498}$)

Series

Derived series

$C_{1000}.C_{498}$

$\rhd$

$C_{500}$

$\rhd$

$C_1$

Chief series

$C_{1000}.C_{498}$

$\rhd$

$C_{249000}$

$\rhd$

$C_{83000}$

$\rhd$

$C_{1000}$

$\rhd$

$C_{500}$

$\rhd$

$C_{250}$

$\rhd$

$C_{125}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{1000}.C_{498}$

$\rhd$

$C_{500}$

$\rhd$

$C_{250}$

$\rhd$

$C_{125}$

Upper central series

$C_1$

$\lhd$

$C_{498}$

$\lhd$

$C_{996}$

$\lhd$

$C_{1992}$

Character theory

The character tables for this group have not been computed.