Abstract group 800.220: $D_4\times D_{50}$

• Label: 800.220
• Order: \( 2^{5} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 5 \)
• Perm deg.: $31$
• Trans deg.: $200$
• Rank: $4$

Group information

Description:	$D_4\times D_{50}$
Order:	\(800\)\(\medspace = 2^{5} \cdot 5^{2} \)
Exponent:	\(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Automorphism group:	$C_{50}.(C_2^4\times C_{20}).C_2^3$, of order \(128000\)\(\medspace = 2^{10} \cdot 5^{3} \)
Composition factors:	$C_2$ x 5, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	20	25	50	100
Elements	1	311	104	4	44	16	20	220	80	800
Conjugacy classes	1	15	4	2	14	4	10	70	20	140
Divisions	1	15	4	1	7	2	1	7	2	40
Autjugacy classes	1	5	2	1	3	1	1	3	1	18

Dimension	1	2	4	8	20	40
Irr. complex chars.	16	100	24	0	0	0	140
Irr. rational chars.	16	4	8	2	8	2	40

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$200$
Rank:	$4$
Inequivalent generating quadruples:	$976500$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{100}=[a,b]=[a,c]=[a,d]=[b,c]=1, d^{b}=d^{49}, d^{c}=d^{51} \rangle$
Permutation group:	Degree $31$ $\langle(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,18)(12,20)(14,22)(17,23)(19,24)(21,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 74 & 0 \\ 25 & 49 \end{array}\right), \left(\begin{array}{rr} 1 & 50 \\ 50 & 26 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 46 & 18 \\ 30 & 46 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/75\Z)$
Direct product:	$C_2$ $\, \times\, $ $D_4$ $\, \times\, $ $D_{25}$
Semidirect product:	$D_{50}$ $\,\rtimes\,$ $C_2^3$	$C_2^3$ $\,\rtimes\,$ $D_{50}$	$D_{100}$ $\,\rtimes\,$ $C_2^2$	$C_{100}$ $\,\rtimes\,$ $C_2^3$	all 23
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{50}$ . $C_2^3$	$C_{50}$ . $C_2^4$	$(D_4\times C_{10})$ . $D_5$	$(C_5\times D_4)$ . $D_{10}$	all 15

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 3918 subgroups in 354 conjugacy classes, 116 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_{50}$
Commutator:	$G' \simeq$ $C_{50}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_{10}$	$G/\Phi \simeq$ $C_2^2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{50}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4\times D_{50}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times D_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 800: $D_4\times D_{50}$

Order 400: $D_4\times D_{25}$ x 8, $C_2^2\times D_{50}$ x 2, $C_{50}:D_4$ x 2, $D_4\times C_{50}$, $C_2\times D_{100}$, $C_4\times D_{50}$

Order 200: $C_2\times D_{50}$ x 19, $C_{25}:D_4$ x 8, $D_{100}$ x 4, $C_4\times D_{25}$ x 4, $D_4\times C_{25}$ x 4, $C_2^2\times C_{50}$ x 2, $C_2\times C_{100}$, $C_{50}:C_4$

Order 160: $D_4\times D_{10}$

Order 100: $D_{50}$ x 30, $C_2\times C_{50}$ x 9, $C_{100}$ x 2, $C_{25}:C_4$ x 2

Order 80: $D_4\times D_5$ x 8, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 50: $D_{25}$ x 8, $C_{50}$ x 7

Order 40: $C_2\times D_{10}$ x 19, $C_5:D_4$ x 8, $D_{20}$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 32: $C_2^2\times D_4$

Order 25: $C_{25}$

Order 20: $D_{10}$ x 30, $C_2\times C_{10}$ x 9, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 5: $C_5$

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 800: $D_4\times D_{50}$

Order 400: $C_2^2\times D_{50}$, $D_4\times C_{50}$, $C_{50}:D_4$, $D_4\times D_{25}$, $C_2\times D_{100}$, $C_4\times D_{50}$

Order 200: $C_2\times D_{50}$ x 4, $C_2\times C_{100}$, $C_{25}:D_4$, $C_{50}:C_4$, $D_{100}$, $C_4\times D_{25}$, $C_2^2\times C_{50}$, $D_4\times C_{25}$

Order 160: $D_4\times D_{10}$

Order 100: $D_{50}$ x 5, $C_2\times C_{50}$ x 3, $C_{100}$, $C_{25}:C_4$

Order 80: $C_2^2\times D_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $D_4\times D_5$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 50: $C_{50}$ x 3, $D_{25}$ x 2

Order 40: $C_2\times D_{10}$ x 4, $C_2\times C_{20}$, $C_5:D_4$, $C_{10}:C_4$, $D_{20}$, $C_4\times D_5$, $C_2^2\times C_{10}$, $C_5\times D_4$

Order 32: $C_2^2\times D_4$

Order 25: $C_{25}$

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

Order 16: $C_2\times D_4$ x 4, $C_2^4$, $C_2^2\times C_4$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 8: $C_2^3$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 8, $C_4$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 800: $D_4\times D_{50}$ ($C_1$)

Order 400: $D_4\times D_{25}$ ($C_2$) x 8, $C_2^2\times D_{50}$ ($C_2$) x 2, $C_{50}:D_4$ ($C_2$) x 2, $D_4\times C_{50}$ ($C_2$), $C_2\times D_{100}$ ($C_2$), $C_4\times D_{50}$ ($C_2$)

Order 200: $C_2\times D_{50}$ ($C_2^2$) x 11, $C_{25}:D_4$ ($C_2^2$) x 8, $D_{100}$ ($C_2^2$) x 4, $C_4\times D_{25}$ ($C_2^2$) x 4, $D_4\times C_{25}$ ($C_2^2$) x 4, $C_2^2\times C_{50}$ ($C_2^2$) x 2, $C_2\times C_{100}$ ($C_2^2$), $C_{50}:C_4$ ($C_2^2$)

Order 100: $D_{50}$ ($C_2^3$) x 6, $C_2\times C_{50}$ ($C_2^3$) x 5, $D_{50}$ ($D_4$) x 4, $C_{100}$ ($C_2^3$) x 2, $C_{25}:C_4$ ($C_2^3$) x 2

Order 80: $D_4\times C_{10}$ ($D_5$)

Order 50: $D_{25}$ ($C_2\times D_4$) x 4, $C_{50}$ ($C_2\times D_4$) x 2, $C_{50}$ ($C_2^4$)

Order 40: $C_5\times D_4$ ($D_{10}$) x 4, $C_2^2\times C_{10}$ ($D_{10}$) x 2, $C_2\times C_{20}$ ($D_{10}$)

Order 25: $C_{25}$ ($C_2^2\times D_4$)

Order 20: $C_2\times C_{10}$ ($C_2\times D_{10}$) x 5, $C_{20}$ ($C_2\times D_{10}$) x 2

Order 16: $C_2\times D_4$ ($D_{25}$)

Order 10: $C_{10}$ ($D_4\times D_5$) x 2, $C_{10}$ ($C_2^2\times D_{10}$)

Order 8: $D_4$ ($D_{50}$) x 4, $C_2^3$ ($D_{50}$) x 2, $C_2\times C_4$ ($D_{50}$)

Order 5: $C_5$ ($D_4\times D_{10}$)

Order 4: $C_2^2$ ($C_2\times D_{50}$) x 5, $C_4$ ($C_2\times D_{50}$) x 2

Order 2: $C_2$ ($D_4\times D_{25}$) x 2, $C_2$ ($C_2^2\times D_{50}$)

Order 1: $C_1$ ($D_4\times D_{50}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 800: $D_4\times D_{50}$ ($C_1$)

Order 400: $C_2^2\times D_{50}$ ($C_2$), $D_4\times C_{50}$ ($C_2$), $C_{50}:D_4$ ($C_2$), $D_4\times D_{25}$ ($C_2$), $C_2\times D_{100}$ ($C_2$), $C_4\times D_{50}$ ($C_2$)

Order 200: $C_2\times D_{50}$ ($C_2^2$) x 3, $C_2\times C_{100}$ ($C_2^2$), $C_{25}:D_4$ ($C_2^2$), $C_{50}:C_4$ ($C_2^2$), $D_{100}$ ($C_2^2$), $C_4\times D_{25}$ ($C_2^2$), $C_2^2\times C_{50}$ ($C_2^2$), $D_4\times C_{25}$ ($C_2^2$)

Order 100: $C_2\times C_{50}$ ($C_2^3$) x 2, $D_{50}$ ($C_2^3$) x 2, $D_{50}$ ($D_4$), $C_{100}$ ($C_2^3$), $C_{25}:C_4$ ($C_2^3$)

Order 80: $D_4\times C_{10}$ ($D_5$)

Order 50: $C_{50}$ ($C_2^4$), $C_{50}$ ($C_2\times D_4$), $D_{25}$ ($C_2\times D_4$)

Order 40: $C_2\times C_{20}$ ($D_{10}$), $C_2^2\times C_{10}$ ($D_{10}$), $C_5\times D_4$ ($D_{10}$)

Order 25: $C_{25}$ ($C_2^2\times D_4$)

Order 20: $C_2\times C_{10}$ ($C_2\times D_{10}$) x 2, $C_{20}$ ($C_2\times D_{10}$)

Order 16: $C_2\times D_4$ ($D_{25}$)

Order 10: $C_{10}$ ($C_2^2\times D_{10}$), $C_{10}$ ($D_4\times D_5$)

Order 8: $C_2^3$ ($D_{50}$), $D_4$ ($D_{50}$), $C_2\times C_4$ ($D_{50}$)

Order 5: $C_5$ ($D_4\times D_{10}$)

Order 4: $C_2^2$ ($C_2\times D_{50}$) x 2, $C_4$ ($C_2\times D_{50}$)

Order 2: $C_2$ ($C_2^2\times D_{50}$), $C_2$ ($D_4\times D_{25}$)

Order 1: $C_1$ ($D_4\times D_{50}$)

Series

Derived series

$D_4\times D_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_1$

Chief series

$D_4\times D_{50}$

$\rhd$

$C_2^2\times D_{50}$

$\rhd$

$C_2\times D_{50}$

$\rhd$

$C_2\times C_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_4\times D_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 50 larger groups in the database.

This group is a maximal quotient of 106 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ 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.