Abstract group 1600.1461: $C_2^4.D_{50}$

• Label: 1600.1461
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \cdot 5 \)
• Perm deg.: $39$
• Trans deg.: $800$
• Rank: $4$

Group information

Description:	$C_2^4.D_{50}$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Automorphism group:	$(C_2^6\times C_{50}).C_{10}.C_2.C_2^6$, of order \(4096000\)\(\medspace = 2^{15} \cdot 5^{3} \)
Composition factors:	$C_2$ x 6, $C_5$ 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	15	816	4	60	64	20	300	320	1600
Conjugacy classes	1	11	28	2	30	32	10	150	160	424
Divisions	1	11	16	1	11	8	1	11	8	68
Autjugacy classes	1	4	3	1	4	2	1	4	2	22

Dimension	1	2	4	16	20	80
Irr. complex chars.	32	392	0	0	0	0	424
Irr. rational chars.	16	8	20	4	16	4	68

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$800$
Rank:	$4$
Inequivalent generating quadruples:	$488250$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{4}=d^{50}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, d^{b}=c^{2}d^{49} \rangle$
Permutation group:	Degree $39$ $\langle(1,2)(3,5)(4,6)(7,8)(9,10,11,12)(16,18)(17,21)(19,24)(20,27)(22,30)(23,29) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_2^3.D_{50})$
Semidirect product:	$(C_{50}:C_4^2)$ $\,\rtimes\,$ $C_2$	$C_{50}$ $\,\rtimes\,$ $(C_4^2:C_2)$	$(C_2^2.D_{100})$ $\,\rtimes\,$ $C_2$	$(C_{100}:C_4)$ $\,\rtimes\,$ $C_2^2$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $D_{50}$	$(C_2^3.D_{50})$ . $C_2$	$(C_{50}.D_4)$ . $C_2^2$	$(C_2^3\times C_{20})$ . $D_5$	all 46

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

Homology

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

Subgroups

There are 2927 subgroups in 495 conjugacy classes, 285 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $D_{50}$
Commutator:	$G' \simeq$ $C_{50}$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_{10}$	$G/\Phi \simeq$ $C_2^2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{100}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^4.D_{50}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times D_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 1600: $C_2^4.D_{50}$

Order 800: $C_2^3.D_{50}$ x 8, $C_2^3.D_{50}$ x 2, $C_2^2.D_{100}$ x 2, $C_{50}:C_4^2$ x 2, $C_2^3\times C_{100}$

Order 400: $C_2^2\times C_{100}$ x 14, $C_{50}.D_4$ x 8, $C_{100}:C_4$ x 8, $C_{100}:C_4$ x 8, $C_2^2.D_{50}$ x 4, $C_2^3\times C_{50}$

Order 320: $C_2^4.D_{10}$

Order 200: $C_2\times C_{100}$ x 28, $C_{50}:C_4$ x 16, $C_2^2\times C_{50}$ x 11

Order 160: $C_2^3.D_{10}$ x 8, $C_2^3.D_{10}$ x 2, $C_2^2.D_{20}$ x 2, $C_{10}:C_4^2$ x 2, $C_2^3\times C_{20}$

Order 100: $C_2\times C_{50}$ x 23, $C_{100}$ x 8, $C_{25}:C_4$ x 8

Order 80: $C_2^2\times C_{20}$ x 14, $C_{10}.D_4$ x 8, $C_{20}:C_4$ x 8, $C_{20}:C_4$ x 8, $C_2^2.D_{10}$ x 4, $C_2^3\times C_{10}$

Order 64: $C_4^2:C_2^2$

Order 50: $C_{50}$ x 11

Order 40: $C_2\times C_{20}$ x 28, $C_{10}:C_4$ x 16, $C_2^2\times C_{10}$ x 11

Order 32: $C_4^2:C_2$ x 8, $C_2^2.D_4$ x 2, $C_2^3:C_4$ x 2, $C_2\times C_4^2$ x 2, $C_2^3\times C_4$

Order 25: $C_{25}$

Order 20: $C_2\times C_{10}$ x 23, $C_{20}$ x 8, $C_5:C_4$ x 8

Order 16: $C_2^2\times C_4$ x 18, $C_4:C_4$ x 8, $C_2^2:C_4$ x 8, $C_4^2$ x 8, $C_2^4$

Order 10: $C_{10}$ x 11

Order 8: $C_2\times C_4$ x 44, $C_2^3$ x 11

Order 5: $C_5$

Order 4: $C_2^2$ x 23, $C_4$ x 16

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $C_2^4.D_{50}$

Order 800: $C_2^3\times C_{100}$, $C_2^3.D_{50}$, $C_2^3.D_{50}$, $C_2^2.D_{100}$, $C_{50}:C_4^2$

Order 400: $C_2^2\times C_{100}$ x 4, $C_2^3\times C_{50}$, $C_2^2.D_{50}$, $C_{50}.D_4$, $C_{100}:C_4$, $C_{100}:C_4$

Order 320: $C_2^4.D_{10}$

Order 200: $C_2\times C_{100}$ x 5, $C_2^2\times C_{50}$ x 4, $C_{50}:C_4$ x 2

Order 160: $C_2^3\times C_{20}$, $C_2^3.D_{10}$, $C_2^3.D_{10}$, $C_2^2.D_{20}$, $C_{10}:C_4^2$

Order 100: $C_2\times C_{50}$ x 6, $C_{100}$ x 2, $C_{25}:C_4$

Order 80: $C_2^2\times C_{20}$ x 4, $C_2^3\times C_{10}$, $C_2^2.D_{10}$, $C_{10}.D_4$, $C_{20}:C_4$, $C_{20}:C_4$

Order 64: $C_4^2:C_2^2$

Order 50: $C_{50}$ x 4

Order 40: $C_2\times C_{20}$ x 5, $C_2^2\times C_{10}$ x 4, $C_{10}:C_4$ x 2

Order 32: $C_2^3\times C_4$, $C_4^2:C_2$, $C_2^2.D_4$, $C_2^3:C_4$, $C_2\times C_4^2$

Order 25: $C_{25}$

Order 20: $C_2\times C_{10}$ x 6, $C_{20}$ x 2, $C_5:C_4$

Order 16: $C_2^2\times C_4$ x 5, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2^4$

Order 10: $C_{10}$ x 4

Order 8: $C_2\times C_4$ x 7, $C_2^3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 6, $C_4$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $C_2^4.D_{50}$ ($C_1$)

Order 800: $C_2^3.D_{50}$ ($C_2$) x 8, $C_2^3.D_{50}$ ($C_2$) x 2, $C_2^2.D_{100}$ ($C_2$) x 2, $C_{50}:C_4^2$ ($C_2$) x 2, $C_2^3\times C_{100}$ ($C_2$)

Order 400: $C_2^2\times C_{100}$ ($C_4$) x 8, $C_{50}.D_4$ ($C_2^2$) x 8, $C_{100}:C_4$ ($C_2^2$) x 8, $C_{100}:C_4$ ($C_2^2$) x 8, $C_2^2\times C_{100}$ ($C_2^2$) x 6, $C_2^2.D_{50}$ ($C_2^2$) x 4, $C_2^3\times C_{50}$ ($C_2^2$)

Order 200: $C_2\times C_{100}$ ($C_2\times C_4$) x 24, $C_{50}:C_4$ ($C_2^3$) x 8, $C_2\times C_{100}$ ($C_2^3$) x 4, $C_2^2\times C_{50}$ ($C_2\times C_4$) x 4, $C_2^2\times C_{50}$ ($C_2^3$) x 3

Order 160: $C_2^3\times C_{20}$ ($D_5$)

Order 100: $C_{100}$ ($C_2^2\times C_4$) x 8, $C_2\times C_{50}$ ($C_2^2\times C_4$) x 6, $C_2\times C_{50}$ ($D_4:C_2$) x 4, $C_2\times C_{50}$ ($C_2^4$)

Order 80: $C_2^2\times C_{20}$ ($C_5:C_4$) x 8, $C_2^2\times C_{20}$ ($D_{10}$) x 6, $C_2^3\times C_{10}$ ($D_{10}$)

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

Order 40: $C_2\times C_{20}$ ($C_{10}:C_4$) x 24, $C_2\times C_{20}$ ($C_2\times D_{10}$) x 4, $C_2^2\times C_{10}$ ($C_{10}:C_4$) x 4, $C_2^2\times C_{10}$ ($C_2\times D_{10}$) x 3

Order 32: $C_2^3\times C_4$ ($D_{25}$)

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

Order 20: $C_{20}$ ($C_2^2.D_{10}$) x 8, $C_2\times C_{10}$ ($C_2^2.D_{10}$) x 6, $C_2\times C_{10}$ ($D_{20}:C_2$) x 4, $C_2\times C_{10}$ ($C_2^2\times D_{10}$)

Order 16: $C_2^2\times C_4$ ($C_{25}:C_4$) x 8, $C_2^2\times C_4$ ($D_{50}$) x 6, $C_2^4$ ($D_{50}$)

Order 10: $C_{10}$ ($C_2^3.D_{10}$) x 4, $C_{10}$ ($D_{20}:C_2^2$) x 2, $C_{10}$ ($C_{10}.C_2^4$)

Order 8: $C_2\times C_4$ ($C_{50}:C_4$) x 24, $C_2^3$ ($C_{50}:C_4$) x 4, $C_2\times C_4$ ($C_2\times D_{50}$) x 4, $C_2^3$ ($C_2\times D_{50}$) x 3

Order 5: $C_5$ ($C_2^4.D_{10}$)

Order 4: $C_4$ ($C_2^2.D_{50}$) x 8, $C_2^2$ ($C_2^2.D_{50}$) x 6, $C_2^2$ ($D_{100}:C_2$) x 4, $C_2^2$ ($C_2^2\times D_{50}$)

Order 2: $C_2$ ($C_2^3.D_{50}$) x 4, $C_2$ ($D_{100}:C_2^2$) x 2, $C_2$ ($C_2^3.D_{50}$)

Order 1: $C_1$ ($C_2^4.D_{50}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $C_2^4.D_{50}$ ($C_1$)

Order 800: $C_2^3\times C_{100}$ ($C_2$), $C_2^3.D_{50}$ ($C_2$), $C_2^3.D_{50}$ ($C_2$), $C_2^2.D_{100}$ ($C_2$), $C_{50}:C_4^2$ ($C_2$)

Order 400: $C_2^2\times C_{100}$ ($C_2^2$) x 3, $C_2^3\times C_{50}$ ($C_2^2$), $C_2^2\times C_{100}$ ($C_4$), $C_2^2.D_{50}$ ($C_2^2$), $C_{50}.D_4$ ($C_2^2$), $C_{100}:C_4$ ($C_2^2$), $C_{100}:C_4$ ($C_2^2$)

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

Order 160: $C_2^3\times C_{20}$ ($D_5$)

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

Order 80: $C_2^2\times C_{20}$ ($D_{10}$) x 3, $C_2^3\times C_{10}$ ($D_{10}$), $C_2^2\times C_{20}$ ($C_5:C_4$)

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

Order 40: $C_2\times C_{20}$ ($C_{10}:C_4$) x 3, $C_2\times C_{20}$ ($C_2\times D_{10}$) x 2, $C_2^2\times C_{10}$ ($C_2\times D_{10}$) x 2, $C_2^2\times C_{10}$ ($C_{10}:C_4$)

Order 32: $C_2^3\times C_4$ ($D_{25}$)

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

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

Order 16: $C_2^2\times C_4$ ($D_{50}$) x 3, $C_2^4$ ($D_{50}$), $C_2^2\times C_4$ ($C_{25}:C_4$)

Order 10: $C_{10}$ ($C_{10}.C_2^4$), $C_{10}$ ($D_{20}:C_2^2$), $C_{10}$ ($C_2^3.D_{10}$)

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

Order 5: $C_5$ ($C_2^4.D_{10}$)

Order 4: $C_2^2$ ($C_2^2.D_{50}$) x 2, $C_4$ ($C_2^2.D_{50}$) x 2, $C_2^2$ ($C_2^2\times D_{50}$), $C_2^2$ ($D_{100}:C_2$)

Order 2: $C_2$ ($C_2^3.D_{50}$), $C_2$ ($D_{100}:C_2^2$), $C_2$ ($C_2^3.D_{50}$)

Order 1: $C_1$ ($C_2^4.D_{50}$)

Series

Derived series

$C_2^4.D_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_1$

Chief series

$C_2^4.D_{50}$

$\rhd$

$C_{50}:C_4^2$

$\rhd$

$C_2^2.D_{50}$

$\rhd$

$C_2^2\times C_{50}$

$\rhd$

$C_2\times C_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2^4.D_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^3\times C_4$

Character theory

Complex character table

See the $424 \times 424$ 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 $68 \times 68$ rational character table.