Abstract group 2000.152: $C_{50}:C_{40}$

• Label: 2000.152
• Order: \( 2^{4} \cdot 5^{3} \)
• Exponent: \( 2^{3} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $35$
• Trans deg.: $400$
• Rank: $2$

Group information

Description:	$C_{50}:C_{40}$
Order:	\(2000\)\(\medspace = 2^{4} \cdot 5^{3} \)
Exponent:	\(200\)\(\medspace = 2^{3} \cdot 5^{2} \)
Automorphism group:	$C_{50}.(C_2^3\times C_{20})$, of order \(8000\)\(\medspace = 2^{6} \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_5$ x 3
Derived length:	$2$

This group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).

Group statistics

Order	1	2	4	5	8	10	20	25	40	50	100
Elements	1	3	4	24	200	72	96	100	800	300	400	2000
Conjugacy classes	1	3	4	6	8	18	24	10	32	30	40	176
Divisions	1	3	2	2	2	6	4	3	2	9	6	40
Autjugacy classes	1	2	2	5	1	10	10	5	4	10	10	60

Dimension	1	2	4	8	10	16	20	40
Irr. complex chars.	80	80	0	0	16	0	0	0	176
Irr. rational chars.	4	2	10	12	0	6	4	2	40

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$400$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	11	12	24

Constructions

Presentation:	$\langle a, b \mid a^{40}=b^{50}=1, b^{a}=b^{19} \rangle$
Permutation group:	Degree $35$ $\langle(2,4)(3,8)(5,10)(6,13)(7,17)(9,19)(11,20)(12,15)(14,22)(16,23)(18,25)(21,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 1 & 50 \\ 50 & 26 \end{array}\right), \left(\begin{array}{rr} 46 & 18 \\ 30 & 46 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 70 \\ 55 & 51 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 16 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/75\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_{25}:C_{40})$
Semidirect product:	$C_{50}$ $\,\rtimes\,$ $C_{40}$ (2)	$(C_{50}:C_8)$ $\,\rtimes\,$ $C_5$	$(C_{25}:C_{10})$ $\,\rtimes\,$ $C_8$ (2)	$(C_{25}:C_8)$ $\,\rtimes\,$ $C_{10}$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{100}$ . $C_{20}$	$(C_{10}\times C_{20})$ . $D_5$	$(C_5\times C_{20})$ . $D_{10}$	$C_{20}$ . $(C_5\times D_{10})$	all 29

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

Homology

Abelianization:	$C_{2} \times C_{40} \simeq C_{2} \times C_{8} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 298 subgroups in 82 conjugacy classes, 46 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_{25}:C_{10}$
Commutator:	$G' \simeq$ $C_{25}$	$G/G' \simeq$ $C_2\times C_{40}$
Frattini:	$\Phi \simeq$ $C_{20}$	$G/\Phi \simeq$ $C_5\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{100}:C_{10}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{50}:C_{40}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_5:C_{20}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}:C_5$

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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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 2000: $C_{50}:C_{40}$

Order 1000: $C_{25}:C_{40}$ x 2, $C_{100}:C_{10}$

Order 500: $C_{100}:C_5$ x 2, $C_{50}:C_{10}$

Order 400: $C_{50}:C_8$, $C_{10}:C_{40}$

Order 250: $C_{25}:C_{10}$ x 3

Order 200: $C_2\times C_{100}$ x 3, $C_5:C_{40}$ x 2, $C_{25}:C_8$ x 2, $C_{10}\times C_{20}$

Order 125: $C_{25}:C_5$

Order 100: $C_{100}$ x 6, $C_2\times C_{50}$ x 3, $C_5\times C_{20}$ x 2, $C_{10}^2$

Order 80: $C_{10}:C_8$, $C_2\times C_{40}$

Order 50: $C_{50}$ x 9, $C_5\times C_{10}$ x 3

Order 40: $C_2\times C_{20}$ x 2, $C_{40}$ x 2, $C_5:C_8$ x 2

Order 25: $C_{25}$ x 3, $C_5^2$

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$ x 2

Order 16: $C_2\times C_8$

Order 10: $C_{10}$ x 6

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 5: $C_5$ x 2

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2000: $C_{50}:C_{40}$

Order 1000: $C_{100}:C_{10}$, $C_{25}:C_{40}$

Order 500: $C_{100}:C_5$ x 2, $C_{50}:C_{10}$

Order 400: $C_{50}:C_8$, $C_{10}:C_{40}$

Order 250: $C_{25}:C_{10}$ x 2

Order 200: $C_2\times C_{100}$ x 2, $C_{10}\times C_{20}$, $C_5:C_{40}$, $C_{25}:C_8$

Order 125: $C_{25}:C_5$

Order 100: $C_{100}$ x 4, $C_5\times C_{20}$ x 2, $C_2\times C_{50}$ x 2, $C_{10}^2$

Order 80: $C_{10}:C_8$, $C_2\times C_{40}$

Order 50: $C_{50}$ x 4, $C_5\times C_{10}$ x 2

Order 40: $C_2\times C_{20}$ x 2, $C_{40}$, $C_5:C_8$

Order 25: $C_{25}$ x 2, $C_5^2$

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$ x 2

Order 16: $C_2\times C_8$

Order 10: $C_{10}$ x 4

Order 8: $C_2\times C_4$, $C_8$

Order 5: $C_5$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2000: $C_{50}:C_{40}$ ($C_1$)

Order 1000: $C_{25}:C_{40}$ ($C_2$) x 2, $C_{100}:C_{10}$ ($C_2$)

Order 500: $C_{50}:C_{10}$ ($C_4$), $C_{100}:C_5$ ($C_2^2$), $C_{100}:C_5$ ($C_4$)

Order 400: $C_{50}:C_8$ ($C_5$)

Order 250: $C_{25}:C_{10}$ ($C_8$) x 2, $C_{25}:C_{10}$ ($C_2\times C_4$)

Order 200: $C_{25}:C_8$ ($C_{10}$) x 2, $C_2\times C_{100}$ ($C_{10}$), $C_{10}\times C_{20}$ ($D_5$)

Order 125: $C_{25}:C_5$ ($C_2\times C_8$)

Order 100: $C_5\times C_{20}$ ($D_{10}$), $C_5\times C_{20}$ ($C_5:C_4$), $C_2\times C_{50}$ ($C_{20}$), $C_{100}$ ($C_2\times C_{10}$), $C_{100}$ ($C_{20}$), $C_{10}^2$ ($C_5:C_4$)

Order 50: $C_5\times C_{10}$ ($C_5:C_8$) x 2, $C_{50}$ ($C_{40}$) x 2, $C_5\times C_{10}$ ($C_{10}:C_4$), $C_{50}$ ($C_2\times C_{20}$)

Order 40: $C_2\times C_{20}$ ($C_5\times D_5$)

Order 25: $C_5^2$ ($C_{10}:C_8$), $C_{25}$ ($C_2\times C_{40}$)

Order 20: $C_2\times C_{10}$ ($C_5:C_{20}$), $C_{20}$ ($C_5:C_{20}$), $C_{20}$ ($C_5\times D_{10}$)

Order 10: $C_{10}$ ($C_5:C_{40}$) x 2, $C_{10}$ ($C_{10}:C_{20}$)

Order 8: $C_2\times C_4$ ($C_{25}:C_{10}$)

Order 5: $C_5$ ($C_{10}:C_{40}$)

Order 4: $C_2^2$ ($C_{25}:C_{20}$), $C_4$ ($C_{25}:C_{20}$), $C_4$ ($C_{50}:C_{10}$)

Order 2: $C_2$ ($C_{25}:C_{40}$) x 2, $C_2$ ($C_{50}:C_{20}$)

Order 1: $C_1$ ($C_{50}:C_{40}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2000: $C_{50}:C_{40}$ ($C_1$)

Order 1000: $C_{100}:C_{10}$ ($C_2$), $C_{25}:C_{40}$ ($C_2$)

Order 500: $C_{50}:C_{10}$ ($C_4$), $C_{100}:C_5$ ($C_2^2$), $C_{100}:C_5$ ($C_4$)

Order 400: $C_{50}:C_8$ ($C_5$)

Order 250: $C_{25}:C_{10}$ ($C_2\times C_4$), $C_{25}:C_{10}$ ($C_8$)

Order 200: $C_2\times C_{100}$ ($C_{10}$), $C_{10}\times C_{20}$ ($D_5$), $C_{25}:C_8$ ($C_{10}$)

Order 125: $C_{25}:C_5$ ($C_2\times C_8$)

Order 100: $C_5\times C_{20}$ ($D_{10}$), $C_5\times C_{20}$ ($C_5:C_4$), $C_2\times C_{50}$ ($C_{20}$), $C_{100}$ ($C_2\times C_{10}$), $C_{100}$ ($C_{20}$), $C_{10}^2$ ($C_5:C_4$)

Order 50: $C_5\times C_{10}$ ($C_{10}:C_4$), $C_5\times C_{10}$ ($C_5:C_8$), $C_{50}$ ($C_2\times C_{20}$), $C_{50}$ ($C_{40}$)

Order 40: $C_2\times C_{20}$ ($C_5\times D_5$)

Order 25: $C_5^2$ ($C_{10}:C_8$), $C_{25}$ ($C_2\times C_{40}$)

Order 20: $C_2\times C_{10}$ ($C_5:C_{20}$), $C_{20}$ ($C_5:C_{20}$), $C_{20}$ ($C_5\times D_{10}$)

Order 10: $C_{10}$ ($C_{10}:C_{20}$), $C_{10}$ ($C_5:C_{40}$)

Order 8: $C_2\times C_4$ ($C_{25}:C_{10}$)

Order 5: $C_5$ ($C_{10}:C_{40}$)

Order 4: $C_2^2$ ($C_{25}:C_{20}$), $C_4$ ($C_{25}:C_{20}$), $C_4$ ($C_{50}:C_{10}$)

Order 2: $C_2$ ($C_{50}:C_{20}$), $C_2$ ($C_{25}:C_{40}$)

Order 1: $C_1$ ($C_{50}:C_{40}$)

Series

Derived series

$C_{50}:C_{40}$

$\rhd$

$C_{25}$

$\rhd$

$C_1$

Chief series

$C_{50}:C_{40}$

$\rhd$

$C_{100}:C_{10}$

$\rhd$

$C_{50}:C_{10}$

$\rhd$

$C_{25}:C_{10}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{50}:C_{40}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Character theory

Complex character table

See the $176 \times 176$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.