Abstract group 1600.5478: $C_{10}.(D_4\times C_{20})$

• Label: 1600.5478
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $1600$
• Rank: $3$

Group information

Description:	$C_{10}.(D_4\times C_{20})$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	Group of order \(491520\)\(\medspace = 2^{15} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	20
Elements	1	7	56	24	168	1344	1600
Conjugacy classes	1	7	20	24	168	480	700
Divisions	1	7	14	6	42	84	154
Autjugacy classes	1	7	5	1	7	5	26

Dimension	1	2	4	8	16
Irr. complex chars.	400	300	0	0	0	700
Irr. rational chars.	8	12	50	72	12	154

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$1600$
Rank:	$3$
Inequivalent generating triples:	$2604$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{20}=c^{20}=[a,c]=1, b^{a}=b^{11}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $22$ $\langle(2,4)(5,6,8,7)(9,10)(11,12), (1,2)(3,4)(5,7,8,6)(9,11)(10,12), (1,3)(2,4) \!\cdots\! \rangle$
Direct product:	$C_5$ ${}^2$ $\, \times\, $ $(C_2^3.C_2^3)$
Semidirect product:	$(C_{20}:C_{20})$ $\,\rtimes\,$ $C_4$	$(C_4:C_{20})$ $\,\rtimes\,$ $C_{20}$	$C_{20}$ $\,\rtimes\,$ $(C_4:C_{20})$	$C_4$ $\,\rtimes\,$ $(C_{20}:C_{20})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{10}\times C_{20})$ . $Q_8$ (2)	$C_{10}$ . $(Q_8\times C_{20})$	$C_{10}$ . $(C_{20}:Q_8)$	$C_{10}$ . $(C_{20}.Q_8)$	all 58

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

Homology

Abelianization:	$C_{2} \times C_{10} \times C_{20} \simeq C_{2}^{2} \times C_{4} \times C_{5}^{2}$
Schur multiplier:	$C_{2}^{2} \times C_{10}$
Commutator length:	$1$

Subgroups

There are 968 subgroups in 680 conjugacy classes, 424 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}^2$	$G/Z \simeq$ $C_2^3$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2\times C_{10}\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2\times C_{10}^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}.(D_4\times C_{20})$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{10}.(D_4\times C_{20})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1600: $C_{10}.(D_4\times C_{20})$

Order 800: $C_{10}^2.D_4$ x 4, $C_2.C_{20}^2$ x 2, $C_2\times C_{20}^2$

Order 400: $C_{20}:C_{20}$ x 10, $C_2\times C_{10}\times C_{20}$ x 7, $C_{20}^2$ x 2

Order 320: $C_2.(D_4\times C_{20})$ x 6

Order 200: $C_{10}\times C_{20}$ x 28, $C_2\times C_{10}^2$

Order 160: $C_2\times C_4:C_{20}$ x 24, $C_{10}.C_4^2$ x 12, $C_2\times C_4\times C_{20}$ x 6

Order 100: $C_5\times C_{20}$ x 14, $C_{10}^2$ x 7

Order 80: $C_4:C_{20}$ x 60, $C_2^2\times C_{20}$ x 42, $C_4\times C_{20}$ x 12

Order 64: $C_2^3.C_2^3$

Order 50: $C_5\times C_{10}$ x 7

Order 40: $C_2\times C_{20}$ x 168, $C_2^2\times C_{10}$ x 6

Order 32: $C_2^2.D_4$ x 4, $C_2.C_4^2$ x 2, $C_2\times C_4^2$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 84, $C_2\times C_{10}$ x 42

Order 16: $C_4:C_4$ x 10, $C_2^2\times C_4$ x 7, $C_4^2$ x 2

Order 10: $C_{10}$ x 42

Order 8: $C_2\times C_4$ x 28, $C_2^3$

Order 5: $C_5$ x 6

Order 4: $C_4$ x 14, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $C_{10}.(D_4\times C_{20})$

Order 800: $C_{10}^2.D_4$ x 3, $C_2\times C_{20}^2$, $C_2.C_{20}^2$

Order 400: $C_2\times C_{10}\times C_{20}$ x 5, $C_{20}:C_{20}$ x 3, $C_{20}^2$

Order 320: $C_2.(D_4\times C_{20})$

Order 200: $C_{10}\times C_{20}$ x 15, $C_2\times C_{10}^2$

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

Order 100: $C_{10}^2$ x 7, $C_5\times C_{20}$ x 5

Order 80: $C_2^2\times C_{20}$ x 5, $C_4:C_{20}$ x 3, $C_4\times C_{20}$

Order 64: $C_2^3.C_2^3$

Order 50: $C_5\times C_{10}$ x 7

Order 40: $C_2\times C_{20}$ x 15, $C_2^2\times C_{10}$

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

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 7, $C_{20}$ x 5

Order 16: $C_2^2\times C_4$ x 5, $C_4:C_4$ x 3, $C_4^2$

Order 10: $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 15, $C_2^3$

Order 5: $C_5$

Order 4: $C_2^2$ x 7, $C_4$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $C_{10}.(D_4\times C_{20})$ ($C_1$)

Order 800: $C_{10}^2.D_4$ ($C_2$) x 4, $C_2.C_{20}^2$ ($C_2$) x 2, $C_2\times C_{20}^2$ ($C_2$)

Order 400: $C_2\times C_{10}\times C_{20}$ ($C_2^2$) x 7, $C_{20}:C_{20}$ ($C_4$) x 4

Order 320: $C_2.(D_4\times C_{20})$ ($C_5$) x 6

Order 200: $C_{10}\times C_{20}$ ($C_2\times C_4$) x 6, $C_{10}\times C_{20}$ ($Q_8$) x 4, $C_{10}\times C_{20}$ ($D_4$) x 4, $C_2\times C_{10}^2$ ($C_2^3$)

Order 160: $C_2\times C_4:C_{20}$ ($C_{10}$) x 24, $C_{10}.C_4^2$ ($C_{10}$) x 12, $C_2\times C_4\times C_{20}$ ($C_{10}$) x 6

Order 100: $C_5\times C_{20}$ ($C_4:C_4$) x 4, $C_{10}^2$ ($D_4:C_2$) x 2, $C_{10}^2$ ($C_2\times Q_8$) x 2, $C_{10}^2$ ($C_2\times D_4$) x 2, $C_{10}^2$ ($C_2^2\times C_4$)

Order 80: $C_2^2\times C_{20}$ ($C_2\times C_{10}$) x 42, $C_4:C_{20}$ ($C_{20}$) x 24

Order 64: $C_2^3.C_2^3$ ($C_5^2$)

Order 50: $C_5\times C_{10}$ ($C_4:Q_8$), $C_5\times C_{10}$ ($C_4.Q_8$), $C_5\times C_{10}$ ($C_2^2:Q_8$), $C_5\times C_{10}$ ($C_4:D_4$), $C_5\times C_{10}$ ($C_4\times Q_8$), $C_5\times C_{10}$ ($C_4\times D_4$), $C_5\times C_{10}$ ($C_2^2.D_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_{20}$) x 36, $C_2\times C_{20}$ ($C_5\times Q_8$) x 24, $C_2\times C_{20}$ ($C_5\times D_4$) x 24, $C_2^2\times C_{10}$ ($C_2^2\times C_{10}$) x 6

Order 32: $C_2^2.D_4$ ($C_5\times C_{10}$) x 4, $C_2.C_4^2$ ($C_5\times C_{10}$) x 2, $C_2\times C_4^2$ ($C_5\times C_{10}$)

Order 25: $C_5^2$ ($C_2^3.C_2^3$)

Order 20: $C_{20}$ ($C_4:C_{20}$) x 24, $C_2\times C_{10}$ ($D_4:C_{10}$) x 12, $C_2\times C_{10}$ ($Q_8\times C_{10}$) x 12, $C_2\times C_{10}$ ($D_4\times C_{10}$) x 12, $C_2\times C_{10}$ ($C_2^2\times C_{20}$) x 6

Order 16: $C_2^2\times C_4$ ($C_{10}^2$) x 7, $C_4:C_4$ ($C_5\times C_{20}$) x 4

Order 10: $C_{10}$ ($C_{20}:Q_8$) x 6, $C_{10}$ ($C_{20}.Q_8$) x 6, $C_{10}$ ($C_{20}.D_4$) x 6, $C_{10}$ ($C_{20}:D_4$) x 6, $C_{10}$ ($Q_8\times C_{20}$) x 6, $C_{10}$ ($D_4\times C_{20}$) x 6, $C_{10}$ ($C_2\times C_4:C_{20}$) x 6

Order 8: $C_2\times C_4$ ($C_{10}\times C_{20}$) x 6, $C_2\times C_4$ ($Q_8\times C_5^2$) x 4, $C_2\times C_4$ ($D_4\times C_5^2$) x 4, $C_2^3$ ($C_2\times C_{10}^2$)

Order 5: $C_5$ ($C_2.(D_4\times C_{20})$) x 6

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

Order 2: $C_2$ ($C_{20}^2.C_2$), $C_2$ ($C_{20}^2.C_2$), $C_2$ ($C_{10}^2:Q_8$), $C_2$ ($C_{10}^2:D_4$), $C_2$ ($C_{20}^2.C_2$), $C_2$ ($C_{20}^2:C_2$), $C_2$ ($C_{10}^2.D_4$)

Order 1: $C_1$ ($C_{10}.(D_4\times C_{20})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $C_{10}.(D_4\times C_{20})$ ($C_1$)

Order 800: $C_{10}^2.D_4$ ($C_2$) x 3, $C_2\times C_{20}^2$ ($C_2$), $C_2.C_{20}^2$ ($C_2$)

Order 400: $C_2\times C_{10}\times C_{20}$ ($C_2^2$) x 5, $C_{20}:C_{20}$ ($C_4$)

Order 320: $C_2.(D_4\times C_{20})$ ($C_5$)

Order 200: $C_{10}\times C_{20}$ ($Q_8$) x 2, $C_{10}\times C_{20}$ ($D_4$) x 2, $C_{10}\times C_{20}$ ($C_2\times C_4$) x 2, $C_2\times C_{10}^2$ ($C_2^3$)

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

Order 100: $C_{10}^2$ ($D_4:C_2$) x 2, $C_{10}^2$ ($C_2\times Q_8$) x 2, $C_{10}^2$ ($C_2\times D_4$) x 2, $C_5\times C_{20}$ ($C_4:C_4$), $C_{10}^2$ ($C_2^2\times C_4$)

Order 80: $C_2^2\times C_{20}$ ($C_2\times C_{10}$) x 5, $C_4:C_{20}$ ($C_{20}$)

Order 64: $C_2^3.C_2^3$ ($C_5^2$)

Order 50: $C_5\times C_{10}$ ($C_4:Q_8$), $C_5\times C_{10}$ ($C_4.Q_8$), $C_5\times C_{10}$ ($C_2^2:Q_8$), $C_5\times C_{10}$ ($C_4:D_4$), $C_5\times C_{10}$ ($C_4\times Q_8$), $C_5\times C_{10}$ ($C_4\times D_4$), $C_5\times C_{10}$ ($C_2^2.D_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_{20}$) x 2, $C_2\times C_{20}$ ($C_5\times Q_8$) x 2, $C_2\times C_{20}$ ($C_5\times D_4$) x 2, $C_2^2\times C_{10}$ ($C_2^2\times C_{10}$)

Order 32: $C_2^2.D_4$ ($C_5\times C_{10}$) x 3, $C_2\times C_4^2$ ($C_5\times C_{10}$), $C_2.C_4^2$ ($C_5\times C_{10}$)

Order 25: $C_5^2$ ($C_2^3.C_2^3$)

Order 20: $C_2\times C_{10}$ ($D_4:C_{10}$) x 2, $C_2\times C_{10}$ ($Q_8\times C_{10}$) x 2, $C_2\times C_{10}$ ($D_4\times C_{10}$) x 2, $C_2\times C_{10}$ ($C_2^2\times C_{20}$), $C_{20}$ ($C_4:C_{20}$)

Order 16: $C_2^2\times C_4$ ($C_{10}^2$) x 5, $C_4:C_4$ ($C_5\times C_{20}$)

Order 10: $C_{10}$ ($C_{20}:Q_8$), $C_{10}$ ($C_{20}.Q_8$), $C_{10}$ ($C_{20}.D_4$), $C_{10}$ ($C_{20}:D_4$), $C_{10}$ ($Q_8\times C_{20}$), $C_{10}$ ($D_4\times C_{20}$), $C_{10}$ ($C_2\times C_4:C_{20}$)

Order 8: $C_2\times C_4$ ($Q_8\times C_5^2$) x 2, $C_2\times C_4$ ($D_4\times C_5^2$) x 2, $C_2\times C_4$ ($C_{10}\times C_{20}$) x 2, $C_2^3$ ($C_2\times C_{10}^2$)

Order 5: $C_5$ ($C_2.(D_4\times C_{20})$)

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

Order 2: $C_2$ ($C_{20}^2.C_2$), $C_2$ ($C_{20}^2.C_2$), $C_2$ ($C_{10}^2:Q_8$), $C_2$ ($C_{10}^2:D_4$), $C_2$ ($C_{20}^2.C_2$), $C_2$ ($C_{20}^2:C_2$), $C_2$ ($C_{10}^2.D_4$)

Order 1: $C_1$ ($C_{10}.(D_4\times C_{20})$)

Series

Derived series

$C_{10}.(D_4\times C_{20})$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{10}.(D_4\times C_{20})$

$\rhd$

$C_2\times C_{20}^2$

$\rhd$

$C_2\times C_{10}\times C_{20}$

$\rhd$

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

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{10}.(D_4\times C_{20})$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}^2$

$\lhd$

$C_{10}.(D_4\times C_{20})$

Character theory

Complex character table

The $700 \times 700$ character table is not available for this group.

Rational character table

See the $154 \times 154$ rational character table (warning: may be slow to load).