Abstract group 800.310: $D_{20}:C_{20}$

• Label: 800.310
• Order: \( 2^{5} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $18$
• Trans deg.: $80$
• Rank: $2$

Group information

Description:	$D_{20}:C_{20}$
Order:	\(800\)\(\medspace = 2^{5} \cdot 5^{2} \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_5:(C_2^2\times C_4^2\times D_4)$, of order \(2560\)\(\medspace = 2^{9} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	8	10	20	40
Elements	1	23	64	24	8	152	336	192	800
Conjugacy classes	1	3	8	14	2	32	62	48	170
Divisions	1	3	5	4	1	9	11	6	40
Autjugacy classes	1	3	4	3	1	7	8	3	30

Dimension	1	2	4	8	16	32
Irr. complex chars.	40	110	20	0	0	0	170
Irr. rational chars.	4	4	8	15	7	2	40

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$80$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c \mid b^{20}=c^{20}=[a,c]=1, a^{2}=c^{15}, b^{a}=b^{11}c^{15}, c^{b}=c^{9} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,5)(4,7)(6,9)(8,10)(11,12)(13,18)(14,16)(15,17), (11,13,14,12,15,16,18,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 43 & 33 \\ 44 & 32 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 44 & 45 \\ 12 & 11 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right), \left(\begin{array}{rr} 44 & 52 \\ 26 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/55\Z)$
Direct product:	$C_5$ $\, \times\, $ $(D_{20}:C_4)$
Semidirect product:	$D_{20}$ $\,\rtimes\,$ $C_{20}$	$(C_5:Q_8)$ $\,\rtimes\,$ $C_{20}$	$(C_5\times D_{20})$ $\,\rtimes\,$ $C_4$	$C_5$ $\,\rtimes\,$ $(D_4:C_{20})$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}^2$ . $D_4$	$(C_5\times C_{20})$ . $D_4$	$C_{20}$ . $(C_5:D_4)$	$C_{20}$ . $(C_5\times D_4)$	all 22

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

Homology

Abelianization:	$C_{2} \times C_{20} \simeq C_{2} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 364 subgroups in 108 conjugacy classes, 38 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{20}$	$G/Z \simeq$ $C_5:D_4$
Commutator:	$G' \simeq$ $C_{20}$	$G/G' \simeq$ $C_2\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_5\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{16}\times C_5^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{20}:C_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\wr C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 800: $D_{20}:C_{20}$

Order 400: $C_{20}:C_{20}$, $D_{20}:C_{10}$, $\OD_{16}\times C_5^2$

Order 200: $C_{10}\times C_{20}$, $C_{10}\wr C_2$, $C_{10}:C_{20}$, $C_5\times D_{20}$, $D_5\times C_{20}$, $C_5^2:Q_8$, $C_5\times C_{40}$

Order 160: $D_4:C_{20}$, $D_{20}:C_4$

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

Order 80: $C_5\times \OD_{16}$ x 4, $D_4:C_{10}$, $D_{20}:C_2$, $C_4\times C_{20}$, $C_{20}:C_4$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 40: $C_2\times C_{20}$ x 6, $C_{40}$ x 6, $C_5\times D_4$ x 2, $C_5:D_4$, $C_{10}:C_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5\times Q_8$

Order 32: $C_4\wr C_2$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 11, $C_2\times C_{10}$ x 5, $C_5:C_4$ x 3, $D_{10}$

Order 16: $\OD_{16}$, $C_4^2$, $D_4:C_2$

Order 10: $C_{10}$ x 9, $D_5$

Order 8: $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$, $C_8$

Order 5: $C_5$ x 4

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 800: $D_{20}:C_{20}$

Order 400: $C_{20}:C_{20}$, $D_{20}:C_{10}$, $\OD_{16}\times C_5^2$

Order 200: $C_{10}\times C_{20}$, $C_{10}\wr C_2$, $C_{10}:C_{20}$, $C_5\times D_{20}$, $D_5\times C_{20}$, $C_5^2:Q_8$, $C_5\times C_{40}$

Order 160: $D_4:C_{20}$, $D_{20}:C_4$

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

Order 80: $C_5\times \OD_{16}$ x 3, $D_4:C_{10}$, $D_{20}:C_2$, $C_4\times C_{20}$, $C_{20}:C_4$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 40: $C_2\times C_{20}$ x 5, $C_{40}$ x 3, $C_5\times D_4$ x 2, $C_5:D_4$, $C_{10}:C_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5\times Q_8$

Order 32: $C_4\wr C_2$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 8, $C_2\times C_{10}$ x 4, $C_5:C_4$ x 2, $D_{10}$

Order 16: $\OD_{16}$, $C_4^2$, $D_4:C_2$

Order 10: $C_{10}$ x 7, $D_5$

Order 8: $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$, $C_8$

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 800: $D_{20}:C_{20}$ ($C_1$)

Order 400: $C_{20}:C_{20}$ ($C_2$), $D_{20}:C_{10}$ ($C_2$), $\OD_{16}\times C_5^2$ ($C_2$)

Order 200: $C_{10}\times C_{20}$ ($C_2^2$), $C_5\times D_{20}$ ($C_4$), $C_5^2:Q_8$ ($C_4$)

Order 160: $D_{20}:C_4$ ($C_5$)

Order 100: $C_5\times C_{20}$ ($D_4$), $C_5\times C_{20}$ ($C_2\times C_4$), $C_{10}^2$ ($D_4$)

Order 80: $D_{20}:C_2$ ($C_{10}$), $C_5\times \OD_{16}$ ($C_{10}$), $C_5\times \OD_{16}$ ($D_5$), $C_{20}:C_4$ ($C_{10}$)

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

Order 40: $C_2\times C_{20}$ ($C_2\times C_{10}$), $C_2\times C_{20}$ ($D_{10}$), $D_{20}$ ($C_{20}$), $C_5:Q_8$ ($C_{20}$)

Order 25: $C_5^2$ ($C_4\wr C_2$)

Order 20: $C_2\times C_{10}$ ($D_{20}$), $C_2\times C_{10}$ ($C_5\times D_4$), $C_{20}$ ($C_2\times C_{20}$), $C_{20}$ ($C_5:D_4$), $C_{20}$ ($C_4\times D_5$), $C_{20}$ ($C_5\times D_4$)

Order 16: $\OD_{16}$ ($C_5\times D_5$)

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

Order 8: $C_2\times C_4$ ($C_5\times D_{10}$)

Order 5: $C_5$ ($D_4:C_{20}$), $C_5$ ($D_{20}:C_4$)

Order 4: $C_2^2$ ($C_5\times D_{20}$), $C_4$ ($C_{10}\wr C_2$), $C_4$ ($D_5\times C_{20}$)

Order 2: $C_2$ ($D_{10}:C_{20}$)

Order 1: $C_1$ ($D_{20}:C_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 800: $D_{20}:C_{20}$ ($C_1$)

Order 400: $C_{20}:C_{20}$ ($C_2$), $D_{20}:C_{10}$ ($C_2$), $\OD_{16}\times C_5^2$ ($C_2$)

Order 200: $C_{10}\times C_{20}$ ($C_2^2$), $C_5\times D_{20}$ ($C_4$), $C_5^2:Q_8$ ($C_4$)

Order 160: $D_{20}:C_4$ ($C_5$)

Order 100: $C_5\times C_{20}$ ($D_4$), $C_5\times C_{20}$ ($C_2\times C_4$), $C_{10}^2$ ($D_4$)

Order 80: $D_{20}:C_2$ ($C_{10}$), $C_5\times \OD_{16}$ ($C_{10}$), $C_5\times \OD_{16}$ ($D_5$), $C_{20}:C_4$ ($C_{10}$)

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

Order 40: $C_2\times C_{20}$ ($C_2\times C_{10}$), $C_2\times C_{20}$ ($D_{10}$), $D_{20}$ ($C_{20}$), $C_5:Q_8$ ($C_{20}$)

Order 25: $C_5^2$ ($C_4\wr C_2$)

Order 20: $C_2\times C_{10}$ ($D_{20}$), $C_2\times C_{10}$ ($C_5\times D_4$), $C_{20}$ ($C_2\times C_{20}$), $C_{20}$ ($C_5:D_4$), $C_{20}$ ($C_4\times D_5$), $C_{20}$ ($C_5\times D_4$)

Order 16: $\OD_{16}$ ($C_5\times D_5$)

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

Order 8: $C_2\times C_4$ ($C_5\times D_{10}$)

Order 5: $C_5$ ($D_4:C_{20}$), $C_5$ ($D_{20}:C_4$)

Order 4: $C_2^2$ ($C_5\times D_{20}$), $C_4$ ($C_{10}\wr C_2$), $C_4$ ($D_5\times C_{20}$)

Order 2: $C_2$ ($D_{10}:C_{20}$)

Order 1: $C_1$ ($D_{20}:C_{20}$)

Series

Derived series

$D_{20}:C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_1$

Chief series

$D_{20}:C_{20}$

$\rhd$

$\OD_{16}\times C_5^2$

$\rhd$

$C_{10}\times C_{20}$

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_{20}:C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_{20}$

$\lhd$

$C_2\times C_{20}$

$\lhd$

$C_5\times \OD_{16}$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

See the $170 \times 170$ 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.