Abstract group 800.936: $C_{20}^2:C_2$

• Label: 800.936
• Order: \( 2^{5} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5^{2} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $400$
• Rank: $3$

Group information

Description:	$C_{20}^2:C_2$
Order:	\(800\)\(\medspace = 2^{5} \cdot 5^{2} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_2^7.C_2^3.S_5$, of order \(122880\)\(\medspace = 2^{13} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_5$ 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	24	24	168	576	800
Conjugacy classes	1	5	14	24	120	336	500
Divisions	1	5	8	6	30	48	98
Autjugacy classes	1	3	3	1	3	3	14

Dimension	1	2	4	8	16
Irr. complex chars.	400	100	0	0	0	500
Irr. rational chars.	8	4	50	24	12	98

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$400$
Rank:	$3$
Inequivalent generating triples:	$1302$

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:	$\langle a, b, c \mid a^{2}=b^{20}=c^{20}=[a,c]=[b,c]=1, b^{a}=bc^{10} \rangle$
Permutation group:	Degree $22$ $\langle(2,5)(6,8)(9,10,11,12), (1,2)(3,6)(4,5)(7,8)(9,11)(10,12), (1,3,4,7)(2,6,5,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 22 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 45 & 0 \\ 0 & 45 \end{array}\right), \left(\begin{array}{rr} 1 & 44 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 43 & 11 \\ 44 & 65 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 44 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/88\Z)$
Direct product:	$C_5$ ${}^2$ $\, \times\, $ $(C_4^2:C_2)$
Semidirect product:	$C_{20}^2$ $\,\rtimes\,$ $C_2$	$(C_{20}:C_{20})$ $\,\rtimes\,$ $C_2$	$(C_{10}\times C_{20})$ $\,\rtimes\,$ $C_4$	$(C_4\times C_{20})$ $\,\rtimes\,$ $C_{10}$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}^2$ . $C_2^3$	$C_2^3$ . $C_{10}^2$	$C_{10}$ . $(D_4:C_{10})$	$C_{20}$ . $(C_2\times C_{20})$ (2)	all 27

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:	$C_{2} \times C_{20}$
Commutator length:	$1$

Subgroups

There are 368 subgroups in 304 conjugacy classes, 240 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}\times C_{20}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2\times C_{10}\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times C_{10}^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{20}^2:C_2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{20}^2:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2$
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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 800: $C_{20}^2:C_2$

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

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

Order 160: $C_4^2:C_{10}$ x 6

Order 100: $C_5\times C_{20}$ x 8, $C_{10}^2$ x 5

Order 80: $C_4:C_{20}$ x 12, $C_2^2:C_{20}$ x 12, $C_4\times C_{20}$ x 12, $C_2^2\times C_{20}$ x 6

Order 50: $C_5\times C_{10}$ x 5

Order 40: $C_2\times C_{20}$ x 60, $C_2^2\times C_{10}$ x 6

Order 32: $C_4^2:C_2$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 48, $C_2\times C_{10}$ x 30

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

Order 10: $C_{10}$ x 30

Order 8: $C_2\times C_4$ x 10, $C_2^3$

Order 5: $C_5$ x 6

Order 4: $C_4$ x 8, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 800: $C_{20}^2:C_2$

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

Order 200: $C_{10}\times C_{20}$ x 4, $C_2\times C_{10}^2$

Order 160: $C_4^2:C_{10}$

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

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

Order 50: $C_5\times C_{10}$ x 3

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

Order 32: $C_4^2:C_2$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 3, $C_{20}$ x 3

Order 16: $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2^2\times C_4$

Order 10: $C_{10}$ x 3

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

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 800: $C_{20}^2:C_2$ ($C_1$)

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

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

Order 160: $C_4^2:C_{10}$ ($C_5$) x 6

Order 100: $C_5\times C_{20}$ ($C_2\times C_4$) x 4, $C_{10}^2$ ($C_2\times C_4$) x 2, $C_{10}^2$ ($C_2^3$)

Order 80: $C_4:C_{20}$ ($C_{10}$) x 12, $C_2^2:C_{20}$ ($C_{10}$) x 12, $C_4\times C_{20}$ ($C_{10}$) x 12, $C_2^2\times C_{20}$ ($C_{10}$) x 6

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

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

Order 32: $C_4^2:C_2$ ($C_5^2$)

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

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

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

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

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

Order 5: $C_5$ ($C_4^2:C_{10}$) x 6

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

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

Order 1: $C_1$ ($C_{20}^2:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 800: $C_{20}^2:C_2$ ($C_1$)

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

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

Order 160: $C_4^2:C_{10}$ ($C_5$)

Order 100: $C_5\times C_{20}$ ($C_2\times C_4$) x 2, $C_{10}^2$ ($C_2^3$), $C_{10}^2$ ($C_2\times C_4$)

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

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

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

Order 32: $C_4^2:C_2$ ($C_5^2$)

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

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

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

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

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

Order 5: $C_5$ ($C_4^2:C_{10}$)

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

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

Order 1: $C_1$ ($C_{20}^2:C_2$)

Series

Derived series

$C_{20}^2:C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{20}^2:C_2$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}^2:C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

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

$\lhd$

$C_{20}^2:C_2$

Supergroups

This group is a maximal subgroup of 104 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $500 \times 500$ 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 $98 \times 98$ rational character table.