Abstract group 80.30: $C_4\times F_5$

• Label: 80.30
• Order: \( 2^{4} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $9$
• Trans deg.: $20$
• Rank: $2$

Group information

Description:	$C_4\times F_5$
Order:	\(80\)\(\medspace = 2^{4} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_5$
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	20
Elements	1	11	52	4	4	8	80
Conjugacy classes	1	3	12	1	1	2	20
Divisions	1	3	6	1	1	1	13
Autjugacy classes	1	2	4	1	1	1	10

Dimension	1	2	4	8
Irr. complex chars.	16	0	4	0	20
Irr. rational chars.	4	6	2	1	13

Minimal presentations

Permutation degree:	$9$
Transitive degree:	$20$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	6	6

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{20}=1, b^{a}=b^{13} \rangle$
Permutation group:	$\langle(2,3,4,5), (6,7,8,9), (2,4)(3,5), (6,8)(7,9), (1,2,3,5,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{5})$
Transitive group:	20T20	40T53			more information
Direct product:	$C_4$ $\, \times\, $ $F_5$
Semidirect product:	$C_{20}$ $\,\rtimes\,$ $C_4$	$C_5$ $\,\rtimes\,$ $C_4^2$	$(C_5:C_4)$ $\,\rtimes\,$ $C_4$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $C_2^2$	$(C_4\times D_5)$ . $C_2$	$D_5$ . $(C_2\times C_4)$ (2)	$(C_2\times F_5)$ . $C_2$ (2)	all 6
Aut. group:	$\Aut(C_5\times D_5)$	$\Aut(C_5:C_{16})$	$\Aut(C_5\times F_5)$

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

Homology

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

Subgroups

There are 78 subgroups in 30 conjugacy classes, 18 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $F_5$
Commutator:	$G' \simeq$ $C_5$	$G/G' \simeq$ $C_4^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{20}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_4\times F_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 80: $C_4\times F_5$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$

Order 20: $F_5$ x 4, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 16: $C_4^2$

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

Order 8: $C_2\times C_4$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 80: $C_4\times F_5$

Order 40: $C_4\times D_5$, $C_2\times F_5$

Order 20: $D_{10}$, $F_5$, $C_{20}$, $C_5:C_4$

Order 16: $C_4^2$

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

Order 8: $C_2\times C_4$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 3, $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 80: $C_4\times F_5$ ($C_1$)

Order 40: $C_2\times F_5$ ($C_2$) x 2, $C_4\times D_5$ ($C_2$)

Order 20: $F_5$ ($C_4$) x 4, $D_{10}$ ($C_2^2$), $C_{20}$ ($C_4$), $C_5:C_4$ ($C_4$)

Order 10: $D_5$ ($C_2\times C_4$) x 2, $C_{10}$ ($C_2\times C_4$)

Order 5: $C_5$ ($C_4^2$)

Order 4: $C_4$ ($F_5$)

Order 2: $C_2$ ($C_2\times F_5$)

Order 1: $C_1$ ($C_4\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 80: $C_4\times F_5$ ($C_1$)

Order 40: $C_4\times D_5$ ($C_2$), $C_2\times F_5$ ($C_2$)

Order 20: $D_{10}$ ($C_2^2$), $F_5$ ($C_4$), $C_{20}$ ($C_4$), $C_5:C_4$ ($C_4$)

Order 10: $C_{10}$ ($C_2\times C_4$), $D_5$ ($C_2\times C_4$)

Order 5: $C_5$ ($C_4^2$)

Order 4: $C_4$ ($F_5$)

Order 2: $C_2$ ($C_2\times F_5$)

Order 1: $C_1$ ($C_4\times F_5$)

Series

Derived series

$C_4\times F_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

Chief series

$C_4\times F_5$

$\rhd$

$C_2\times F_5$

$\rhd$

$D_{10}$

$\rhd$

$D_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_4\times F_5$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 57 larger groups in the database.

This group is a maximal quotient of 56 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	4C	4D	4E	4F	5A	10A	20A
	Size	1	1	5	5	2	10	10	10	10	10	4	4	8
	2 P	1A	1A	1A	1A	2A	2B	2A	2B	2C	2C	5A	5A	10A
	5 P	1A	2A	2B	2C	4A	4B	4C	4D	4E	4F	1A	2A	4A
80.30.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
80.30.1b		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
80.30.1c		$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
80.30.1d		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
80.30.1e		$2$	$-2$	$-2$	$2$	$0$	$0$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$
80.30.1f		$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$0$	$0$	$0$	$2$	$-2$	$0$
80.30.1g		$2$	$-2$	$2$	$-2$	$0$	$-2$	$0$	$2$	$0$	$0$	$2$	$-2$	$0$
80.30.1h		$2$	$-2$	$2$	$-2$	$0$	$2$	$0$	$-2$	$0$	$0$	$2$	$-2$	$0$
80.30.1i		$2$	$2$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$2$	$2$	$2$	$2$	$-2$
80.30.1j		$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$-2$	$-2$	$2$	$2$	$2$
80.30.4a		$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$
80.30.4b		$4$	$4$	$0$	$0$	$-4$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$1$
80.30.4c		$8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$