Abstract group 200.12: $C_{50}:C_4$

• Label: 200.12
• Order: \( 2^{3} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 5 \)
• Perm deg.: $27$
• Trans deg.: $50$
• Rank: $2$

Group information

Description:	$C_{50}:C_4$
Order:	\(200\)\(\medspace = 2^{3} \cdot 5^{2} \)
Exponent:	\(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Automorphism group:	$C_{50}:C_{20}$, of order \(1000\)\(\medspace = 2^{3} \cdot 5^{3} \)
Composition factors:	$C_2$ x 3, $C_5$ x 2
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	4	5	10	25	50
Elements	1	51	100	4	4	20	20	200
Conjugacy classes	1	3	4	1	1	5	5	20
Divisions	1	3	2	1	1	1	1	10
Autjugacy classes	1	3	2	1	1	1	1	10

Dimension	1	2	4	20
Irr. complex chars.	8	0	12	0	20
Irr. rational chars.	4	2	2	2	10

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$50$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	20
Arbitrary	4	4	20

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{50}=1, b^{a}=b^{43} \rangle$
Permutation group:	Degree $27$ $\langle(2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25)(13,16,21,24)(26,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 24 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 24 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 24 & 0 \\ 0 & 24 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/25\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 25 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/50\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_{25}:C_4)$
Semidirect product:	$D_{25}$ $\,\rtimes\,$ $C_4$	$C_{50}$ $\,\rtimes\,$ $C_4$	$C_{25}$ $\,\rtimes\,$ $(C_2\times C_4)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{50}$ . $C_2$	$C_{10}$ . $F_5$	$D_{25}$ . $C_2^2$	$C_5$ . $(C_2\times F_5)$	more information

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 192 subgroups in 24 conjugacy classes, 12 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{25}:C_4$
Commutator:	$G' \simeq$ $C_{25}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $C_2\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{50}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_{50}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $F_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$

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 200: $C_{50}:C_4$

Order 100: $C_{25}:C_4$ x 2, $D_{50}$

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

Order 40: $C_2\times F_5$

Order 25: $C_{25}$

Order 20: $F_5$ x 2, $D_{10}$

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

Order 8: $C_2\times C_4$

Order 5: $C_5$

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 200: $C_{50}:C_4$

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

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

Order 40: $C_2\times F_5$

Order 25: $C_{25}$

Order 20: $D_{10}$, $F_5$

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

Order 8: $C_2\times C_4$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 200: $C_{50}:C_4$ ($C_1$)

Order 100: $C_{25}:C_4$ ($C_2$) x 2, $D_{50}$ ($C_2$)

Order 50: $C_{50}$ ($C_4$), $D_{25}$ ($C_2^2$), $D_{25}$ ($C_4$)

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

Order 10: $C_{10}$ ($F_5$)

Order 5: $C_5$ ($C_2\times F_5$)

Order 2: $C_2$ ($C_{25}:C_4$)

Order 1: $C_1$ ($C_{50}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 200: $C_{50}:C_4$ ($C_1$)

Order 100: $D_{50}$ ($C_2$), $C_{25}:C_4$ ($C_2$)

Order 50: $C_{50}$ ($C_4$), $D_{25}$ ($C_2^2$), $D_{25}$ ($C_4$)

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

Order 10: $C_{10}$ ($F_5$)

Order 5: $C_5$ ($C_2\times F_5$)

Order 2: $C_2$ ($C_{25}:C_4$)

Order 1: $C_1$ ($C_{50}:C_4$)

Series

Derived series

$C_{50}:C_4$

$\rhd$

$C_{25}$

$\rhd$

$C_1$

Chief series

$C_{50}:C_4$

$\rhd$

$C_{25}:C_4$

$\rhd$

$D_{25}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{50}:C_4$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	5A	10A	25A	50A
	Size	1	1	25	25	50	50	4	4	20	20
	2 P	1A	1A	1A	1A	2B	2B	5A	5A	25A	25A
	5 P	1A	2A	2B	2C	4A	4B	1A	2A	5A	10A
200.12.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
200.12.1b		$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
200.12.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
200.12.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
200.12.1e		$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$-2$	$2$	$-2$
200.12.1f		$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$2$	$2$	$2$
200.12.4a		$4$	$4$	$0$	$0$	$0$	$0$	$4$	$4$	$-1$	$-1$
200.12.4b		$4$	$-4$	$0$	$0$	$0$	$0$	$4$	$-4$	$-1$	$1$
200.12.4c		$20$	$20$	$0$	$0$	$0$	$0$	$-5$	$-5$	$0$	$0$
200.12.4d		$20$	$-20$	$0$	$0$	$0$	$0$	$-5$	$5$	$0$	$0$