Abstract group 240.60: $C_{15}:D_8$

• Label: 240.60
• Order: \( 2^{4} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $120$
• Rank: $2$

Group information

Description:	$C_{15}:D_8$
Order:	\(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_4\times D_4\times D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
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	8	10	12	15	20	30	40	60
Elements	1	17	2	2	4	10	12	68	4	8	8	40	48	16	240
Conjugacy classes	1	3	1	1	4	3	2	12	1	4	4	12	8	4	60
Divisions	1	3	1	1	1	2	1	3	1	1	1	2	1	1	20
Autjugacy classes	1	3	1	1	1	2	1	3	1	1	1	2	1	1	20

Dimension	1	2	4	8	16
Irr. complex chars.	20	35	5	0	0	60
Irr. rational chars.	4	3	7	3	3	20

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$120$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{8}=c^{15}=[a,c]=1, b^{a}=b^{7}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $16$ $\langle(2,3)(9,10)(11,13)(12,15)(14,16), (10,13)(11,16)(14,15), (4,5,6,7,8), (9,11,12,16)(10,14,15,13), (9,12)(10,15)(11,16)(13,14), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 61 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 47 & 0 \\ 0 & 47 \end{array}\right), \left(\begin{array}{rr} 32 & 31 \\ 31 & 1 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 31 & 20 \\ 48 & 31 \end{array}\right), \left(\begin{array}{rr} 58 & 49 \\ 25 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/62\Z)$
Direct product:	$C_5$ $\, \times\, $ $(C_3:D_8)$
Semidirect product:	$C_{15}$ $\,\rtimes\,$ $D_8$	$D_{12}$ $\,\rtimes\,$ $C_{10}$	$(C_5\times D_4)$ $\,\rtimes\,$ $S_3$	$D_4$ $\,\rtimes\,$ $(C_5\times S_3)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{30}$ . $D_4$	$C_{20}$ . $D_6$	$C_{60}$ . $C_2^2$	$C_6$ . $(C_5\times D_4)$	all 8

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

Homology

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

Subgroups

There are 112 subgroups in 44 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_3:D_4$
Commutator:	$G' \simeq$ $C_{12}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_3\times C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{15}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{15}:D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{30}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8$
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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Classes of subgroups up to conjugation

Order 240: $C_{15}:D_8$

Order 120: $D_4\times C_{15}$, $C_5\times D_{12}$, $C_3:C_{40}$

Order 80: $C_5\times D_8$

Order 60: $C_{60}$, $C_2\times C_{30}$, $S_3\times C_{10}$

Order 48: $C_3:D_8$

Order 40: $C_5\times D_4$ x 2, $C_{40}$

Order 30: $C_{30}$ x 2, $C_5\times S_3$

Order 24: $D_{12}$, $C_3\times D_4$, $C_3:C_8$

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

Order 16: $D_8$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $D_6$, $C_{12}$

Order 10: $C_{10}$ x 3

Order 8: $D_4$ x 2, $C_8$

Order 6: $C_6$ x 2, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 240: $C_{15}:D_8$

Order 120: $D_4\times C_{15}$, $C_5\times D_{12}$, $C_3:C_{40}$

Order 80: $C_5\times D_8$

Order 60: $C_{60}$, $C_2\times C_{30}$, $S_3\times C_{10}$

Order 48: $C_3:D_8$

Order 40: $C_5\times D_4$ x 2, $C_{40}$

Order 30: $C_{30}$ x 2, $C_5\times S_3$

Order 24: $D_{12}$, $C_3\times D_4$, $C_3:C_8$

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

Order 16: $D_8$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $D_6$, $C_{12}$

Order 10: $C_{10}$ x 3

Order 8: $D_4$ x 2, $C_8$

Order 6: $C_6$ x 2, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 240: $C_{15}:D_8$ ($C_1$)

Order 120: $D_4\times C_{15}$ ($C_2$), $C_5\times D_{12}$ ($C_2$), $C_3:C_{40}$ ($C_2$)

Order 60: $C_{60}$ ($C_2^2$)

Order 48: $C_3:D_8$ ($C_5$)

Order 40: $C_5\times D_4$ ($S_3$)

Order 30: $C_{30}$ ($D_4$)

Order 24: $D_{12}$ ($C_{10}$), $C_3\times D_4$ ($C_{10}$), $C_3:C_8$ ($C_{10}$)

Order 20: $C_{20}$ ($D_6$)

Order 15: $C_{15}$ ($D_8$)

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

Order 10: $C_{10}$ ($C_3:D_4$)

Order 8: $D_4$ ($C_5\times S_3$)

Order 6: $C_6$ ($C_5\times D_4$)

Order 5: $C_5$ ($C_3:D_8$)

Order 4: $C_4$ ($S_3\times C_{10}$)

Order 3: $C_3$ ($C_5\times D_8$)

Order 2: $C_2$ ($C_{15}:D_4$)

Order 1: $C_1$ ($C_{15}:D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 240: $C_{15}:D_8$ ($C_1$)

Order 120: $D_4\times C_{15}$ ($C_2$), $C_5\times D_{12}$ ($C_2$), $C_3:C_{40}$ ($C_2$)

Order 60: $C_{60}$ ($C_2^2$)

Order 48: $C_3:D_8$ ($C_5$)

Order 40: $C_5\times D_4$ ($S_3$)

Order 30: $C_{30}$ ($D_4$)

Order 24: $D_{12}$ ($C_{10}$), $C_3\times D_4$ ($C_{10}$), $C_3:C_8$ ($C_{10}$)

Order 20: $C_{20}$ ($D_6$)

Order 15: $C_{15}$ ($D_8$)

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

Order 10: $C_{10}$ ($C_3:D_4$)

Order 8: $D_4$ ($C_5\times S_3$)

Order 6: $C_6$ ($C_5\times D_4$)

Order 5: $C_5$ ($C_3:D_8$)

Order 4: $C_4$ ($S_3\times C_{10}$)

Order 3: $C_3$ ($C_5\times D_8$)

Order 2: $C_2$ ($C_{15}:D_4$)

Order 1: $C_1$ ($C_{15}:D_8$)

Series

Derived series

$C_{15}:D_8$

$\rhd$

$C_{12}$

$\rhd$

$C_1$

Chief series

$C_{15}:D_8$

$\rhd$

$D_4\times C_{15}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{15}:D_8$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{20}$

$\lhd$

$C_5\times D_4$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.