Abstract group 240.26: $C_6.D_{20}$

• Label: 240.26
• Order: \( 2^{4} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $120$
• Rank: $2$

Group information

Description:	$C_6.D_{20}$
Order:	\(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^3\times D_6\times F_5$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	15	20	30
Elements	1	23	2	72	4	46	12	8	48	24	240
Conjugacy classes	1	5	1	4	2	7	6	2	8	6	42
Divisions	1	5	1	2	1	5	3	1	1	3	23
Autjugacy classes	1	4	1	2	1	4	3	1	1	3	21

Dimension	1	2	4	8
Irr. complex chars.	8	26	8	0	42
Irr. rational chars.	4	8	4	7	23

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$120$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{6}=c^{10}=[a,c]=1, b^{a}=b^{5}c^{5}, c^{b}=c^{9} \rangle$
Permutation group:	Degree $16$ $\langle(2,3)(4,5)(6,7)(8,9)(13,14)(15,16), (7,9)(11,12)(13,15,14,16), (13,14)(15,16), (6,8)(7,9), (10,11,12), (1,2,4,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 10 \\ 15 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 2 \\ 10 & 11 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6:C_4)$ $\,\rtimes\,$ $D_5$	$C_5$ $\,\rtimes\,$ $(C_6.D_4)$	$(C_3\times D_{10})$ $\,\rtimes\,$ $C_4$ (2)	$D_{10}$ $\,\rtimes\,$ $(C_3:C_4)$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{30}$ . $D_4$ (2)	$C_6$ . $D_{20}$	$C_6$ . $(C_5:D_4)$	$C_6$ . $(C_4\times D_5)$	all 16

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

Homology

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

Subgroups

There are 272 subgroups in 68 conjugacy classes, 30 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3\times D_5$
Commutator:	$G' \simeq$ $C_{30}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $S_3\times D_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{30}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6.D_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 240: $C_6.D_{20}$

Order 120: $C_6\times D_{10}$, $C_{30}:C_4$, $C_6:C_{20}$

Order 80: $D_{10}:C_4$

Order 60: $C_3\times D_{10}$ x 4, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_6.D_4$

Order 40: $C_2\times C_{20}$, $C_{10}:C_4$, $C_2\times D_{10}$

Order 30: $C_{30}$ x 3, $C_3\times D_5$ x 2

Order 24: $C_6:C_4$ x 2, $C_2^2\times C_6$

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

Order 16: $C_2^2:C_4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 5, $C_3:C_4$ x 2

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

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

Order 6: $C_6$ x 5

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 240: $C_6.D_{20}$

Order 120: $C_6\times D_{10}$, $C_{30}:C_4$, $C_6:C_{20}$

Order 80: $D_{10}:C_4$

Order 60: $C_3\times D_{10}$ x 3, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_6.D_4$

Order 40: $C_2\times C_{20}$, $C_{10}:C_4$, $C_2\times D_{10}$

Order 30: $C_{30}$ x 3, $C_3\times D_5$

Order 24: $C_6:C_4$ x 2, $C_2^2\times C_6$

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

Order 16: $C_2^2:C_4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 4, $C_3:C_4$ x 2

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

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

Order 6: $C_6$ x 4

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 240: $C_6.D_{20}$ ($C_1$)

Order 120: $C_6\times D_{10}$ ($C_2$), $C_{30}:C_4$ ($C_2$), $C_6:C_{20}$ ($C_2$)

Order 60: $C_3\times D_{10}$ ($C_4$) x 2, $C_2\times C_{30}$ ($C_2^2$)

Order 40: $C_2\times D_{10}$ ($S_3$)

Order 30: $C_{30}$ ($D_4$) x 2, $C_{30}$ ($C_2\times C_4$)

Order 24: $C_6:C_4$ ($D_5$)

Order 20: $D_{10}$ ($C_3:C_4$) x 2, $C_2\times C_{10}$ ($D_6$)

Order 15: $C_{15}$ ($C_2^2:C_4$)

Order 12: $C_2\times C_6$ ($D_{10}$)

Order 10: $C_{10}$ ($C_3:D_4$) x 2, $C_{10}$ ($C_6:C_4$)

Order 6: $C_6$ ($C_5:D_4$), $C_6$ ($D_{20}$), $C_6$ ($C_4\times D_5$)

Order 5: $C_5$ ($C_6.D_4$)

Order 4: $C_2^2$ ($S_3\times D_5$)

Order 3: $C_3$ ($D_{10}:C_4$)

Order 2: $C_2$ ($C_6.D_{10}$), $C_2$ ($C_3:D_{20}$), $C_2$ ($C_{15}:D_4$)

Order 1: $C_1$ ($C_6.D_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 240: $C_6.D_{20}$ ($C_1$)

Order 120: $C_6\times D_{10}$ ($C_2$), $C_{30}:C_4$ ($C_2$), $C_6:C_{20}$ ($C_2$)

Order 60: $C_2\times C_{30}$ ($C_2^2$), $C_3\times D_{10}$ ($C_4$)

Order 40: $C_2\times D_{10}$ ($S_3$)

Order 30: $C_{30}$ ($D_4$) x 2, $C_{30}$ ($C_2\times C_4$)

Order 24: $C_6:C_4$ ($D_5$)

Order 20: $C_2\times C_{10}$ ($D_6$), $D_{10}$ ($C_3:C_4$)

Order 15: $C_{15}$ ($C_2^2:C_4$)

Order 12: $C_2\times C_6$ ($D_{10}$)

Order 10: $C_{10}$ ($C_3:D_4$) x 2, $C_{10}$ ($C_6:C_4$)

Order 6: $C_6$ ($C_5:D_4$), $C_6$ ($D_{20}$), $C_6$ ($C_4\times D_5$)

Order 5: $C_5$ ($C_6.D_4$)

Order 4: $C_2^2$ ($S_3\times D_5$)

Order 3: $C_3$ ($D_{10}:C_4$)

Order 2: $C_2$ ($C_6.D_{10}$), $C_2$ ($C_3:D_{20}$), $C_2$ ($C_{15}:D_4$)

Order 1: $C_1$ ($C_6.D_{20}$)

Series

Derived series

$C_6.D_{20}$

$\rhd$

$C_{30}$

$\rhd$

$C_1$

Chief series

$C_6.D_{20}$

$\rhd$

$C_6:C_{20}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_6.D_{20}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 75 larger groups in the database.

This group is a maximal quotient of 28 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $23 \times 23$ rational character table.