Abstract group 80.43: $C_2^2.D_{10}$

• Label: 80.43
• Order: \( 2^{4} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $80$
• Rank: $3$

Group information

Description:	$C_2^2.D_{10}$
Order:	\(80\)\(\medspace = 2^{4} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$F_5\times C_2^3:S_4$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	4	5	10
Elements	1	7	40	4	28	80
Conjugacy classes	1	7	8	2	14	32
Divisions	1	7	4	1	7	20
Autjugacy classes	1	2	1	1	2	7

Dimension	1	2	4
Irr. complex chars.	16	16	0	32
Irr. rational chars.	8	4	8	20

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$80$
Rank:	$3$
Inequivalent generating triples:	$42$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	5	7

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{10}=[a,b]=[a,c]=1, c^{b}=c^{9} \rangle$
Permutation group:	Degree $13$ $\langle(5,6,7,8)(10,11)(12,13), (1,2)(3,4), (1,3)(2,4)(5,7)(6,8), (5,7)(6,8), (9,10,12,13,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 3 & 2 & 4 \\ 3 & 2 & 0 & 2 \\ 3 & 0 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 3 & 2 & 2 & 2 \\ 4 & 4 & 0 & 1 \\ 4 & 4 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 4 & 2 & 4 & 4 \\ 2 & 3 & 3 & 2 \\ 0 & 3 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 3 & 1 & 2 \\ 0 & 1 & 4 & 1 \\ 1 & 4 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_5:C_4)$
Semidirect product:	$(C_2\times C_{10})$ $\,\rtimes\,$ $C_4$ (4)	$C_{10}$ $\,\rtimes\,$ $(C_2\times C_4)$ (6)	$C_5$ $\,\rtimes\,$ $(C_2^2\times C_4)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $D_5$	$C_2^2$ . $D_{10}$ (3)	$C_{10}$ . $C_2^3$	$C_2$ . $(C_2\times D_{10})$	all 6

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 98 subgroups in 54 conjugacy classes, 43 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $D_5$
Commutator:	$G' \simeq$ $C_5$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^2.D_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
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

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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 80: $C_2^2.D_{10}$

Order 40: $C_{10}:C_4$ x 6, $C_2^2\times C_{10}$

Order 20: $C_2\times C_{10}$ x 7, $C_5:C_4$ x 4

Order 16: $C_2^2\times C_4$

Order 10: $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 5: $C_5$

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 80: $C_2^2.D_{10}$

Order 40: $C_{10}:C_4$, $C_2^2\times C_{10}$

Order 20: $C_2\times C_{10}$ x 2, $C_5:C_4$

Order 16: $C_2^2\times C_4$

Order 10: $C_{10}$ x 2

Order 8: $C_2^3$, $C_2\times C_4$

Order 5: $C_5$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 80: $C_2^2.D_{10}$ ($C_1$)

Order 40: $C_{10}:C_4$ ($C_2$) x 6, $C_2^2\times C_{10}$ ($C_2$)

Order 20: $C_2\times C_{10}$ ($C_4$) x 4, $C_5:C_4$ ($C_2^2$) x 4, $C_2\times C_{10}$ ($C_2^2$) x 3

Order 10: $C_{10}$ ($C_2\times C_4$) x 6, $C_{10}$ ($C_2^3$)

Order 8: $C_2^3$ ($D_5$)

Order 5: $C_5$ ($C_2^2\times C_4$)

Order 4: $C_2^2$ ($C_5:C_4$) x 4, $C_2^2$ ($D_{10}$) x 3

Order 2: $C_2$ ($C_{10}:C_4$) x 6, $C_2$ ($C_2\times D_{10}$)

Order 1: $C_1$ ($C_2^2.D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 80: $C_2^2.D_{10}$ ($C_1$)

Order 40: $C_{10}:C_4$ ($C_2$), $C_2^2\times C_{10}$ ($C_2$)

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

Order 10: $C_{10}$ ($C_2^3$), $C_{10}$ ($C_2\times C_4$)

Order 8: $C_2^3$ ($D_5$)

Order 5: $C_5$ ($C_2^2\times C_4$)

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

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

Order 1: $C_1$ ($C_2^2.D_{10}$)

Series

Derived series

$C_2^2.D_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Chief series

$C_2^2.D_{10}$

$\rhd$

$C_{10}:C_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2^2.D_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 52 larger groups in the database.

This group is a maximal quotient of 39 larger groups in the database.

Character theory

Complex character table

See the $32 \times 32$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.