Abstract group 240.165: $C_{30}:Q_8$

• Label: 240.165
• Order: \( 2^{4} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $18$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$C_{30}:Q_8$
Order:	\(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$(D_6\times C_4^2):D_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
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	10	12	15	20	30	60
Elements	1	3	2	28	4	6	12	8	8	112	24	32	240
Conjugacy classes	1	3	1	6	4	3	12	4	4	24	12	16	90
Divisions	1	3	1	6	1	3	3	2	1	6	3	2	32
Autjugacy classes	1	2	1	2	1	2	2	1	1	2	2	1	18

Dimension	1	2	4	8	16
Irr. complex chars.	40	50	0	0	0	90
Irr. rational chars.	8	6	10	6	2	32

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$2604$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	5	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{60}=[a,b]=[a,c]=1, b^{2}=c^{30}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $18$ $\langle(1,2,3,5)(4,8,7,6)(10,11), (1,3)(2,5)(4,7)(6,8)(17,18), (1,4,3,7)(2,6,5,8), (12,13,14,15,16), (1,3)(2,5)(4,7)(6,8), (9,10,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 2 & 1 & 6 \\ 2 & 0 & 8 \\ 4 & 2 & 7 \end{array}\right), \left(\begin{array}{rrr} 5 & 10 & 5 \\ 10 & 9 & 3 \\ 5 & 5 & 0 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 10 & 10 & 0 \\ 2 & 4 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 3 & 8 & 0 \\ 5 & 10 & 2 \end{array}\right), \left(\begin{array}{rrr} 8 & 7 & 9 \\ 1 & 1 & 1 \\ 3 & 4 & 1 \end{array}\right), \left(\begin{array}{rrr} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_3:Q_8)$
Semidirect product:	$C_{30}$ $\,\rtimes\,$ $Q_8$ (2)	$C_6$ $\,\rtimes\,$ $(C_5\times Q_8)$ (2)	$C_{15}$ $\,\rtimes\,$ $(C_2\times Q_8)$	$C_3$ $\,\rtimes\,$ $(Q_8\times C_{10})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{20}$ . $D_6$ (2)	$C_{60}$ . $C_2^2$ (2)	$C_{30}$ . $C_2^3$	$(C_2\times C_{20})$ . $S_3$	all 20

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 120 subgroups in 76 conjugacy classes, 54 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{10}\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{30}:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
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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Subgroup information

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

Classes of subgroups up to conjugation

Order 240: $C_{30}:Q_8$

Order 120: $C_{15}:Q_8$ x 4, $C_6:C_{20}$ x 2, $C_2\times C_{60}$

Order 80: $Q_8\times C_{10}$

Order 60: $C_3:C_{20}$ x 4, $C_{60}$ x 2, $C_2\times C_{30}$

Order 48: $C_6:Q_8$

Order 40: $C_5\times Q_8$ x 4, $C_2\times C_{20}$ x 3

Order 30: $C_{30}$ x 3

Order 24: $C_3:Q_8$ x 4, $C_6:C_4$ x 2, $C_2\times C_{12}$

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

Order 16: $C_2\times Q_8$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 240: $C_{30}:Q_8$

Order 120: $C_2\times C_{60}$, $C_6:C_{20}$, $C_{15}:Q_8$

Order 80: $Q_8\times C_{10}$

Order 60: $C_{60}$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_6:Q_8$

Order 40: $C_2\times C_{20}$ x 2, $C_5\times Q_8$

Order 30: $C_{30}$ x 2

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_3:Q_8$

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

Order 16: $C_2\times Q_8$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 2

Order 8: $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 2

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 240: $C_{30}:Q_8$ ($C_1$)

Order 120: $C_{15}:Q_8$ ($C_2$) x 4, $C_6:C_{20}$ ($C_2$) x 2, $C_2\times C_{60}$ ($C_2$)

Order 60: $C_3:C_{20}$ ($C_2^2$) x 4, $C_{60}$ ($C_2^2$) x 2, $C_2\times C_{30}$ ($C_2^2$)

Order 48: $C_6:Q_8$ ($C_5$)

Order 40: $C_2\times C_{20}$ ($S_3$)

Order 30: $C_{30}$ ($Q_8$) x 2, $C_{30}$ ($C_2^3$)

Order 24: $C_3:Q_8$ ($C_{10}$) x 4, $C_6:C_4$ ($C_{10}$) x 2, $C_2\times C_{12}$ ($C_{10}$)

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

Order 15: $C_{15}$ ($C_2\times Q_8$)

Order 12: $C_3:C_4$ ($C_2\times C_{10}$) x 4, $C_{12}$ ($C_2\times C_{10}$) x 2, $C_2\times C_6$ ($C_2\times C_{10}$)

Order 10: $C_{10}$ ($C_3:Q_8$) x 2, $C_{10}$ ($C_2\times D_6$)

Order 8: $C_2\times C_4$ ($C_5\times S_3$)

Order 6: $C_6$ ($C_5\times Q_8$) x 2, $C_6$ ($C_2^2\times C_{10}$)

Order 5: $C_5$ ($C_6:Q_8$)

Order 4: $C_4$ ($S_3\times C_{10}$) x 2, $C_2^2$ ($S_3\times C_{10}$)

Order 3: $C_3$ ($Q_8\times C_{10}$)

Order 2: $C_2$ ($C_{15}:Q_8$) x 2, $C_2$ ($C_{10}\times D_6$)

Order 1: $C_1$ ($C_{30}:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 240: $C_{30}:Q_8$ ($C_1$)

Order 120: $C_2\times C_{60}$ ($C_2$), $C_6:C_{20}$ ($C_2$), $C_{15}:Q_8$ ($C_2$)

Order 60: $C_{60}$ ($C_2^2$), $C_2\times C_{30}$ ($C_2^2$), $C_3:C_{20}$ ($C_2^2$)

Order 48: $C_6:Q_8$ ($C_5$)

Order 40: $C_2\times C_{20}$ ($S_3$)

Order 30: $C_{30}$ ($C_2^3$), $C_{30}$ ($Q_8$)

Order 24: $C_2\times C_{12}$ ($C_{10}$), $C_6:C_4$ ($C_{10}$), $C_3:Q_8$ ($C_{10}$)

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

Order 15: $C_{15}$ ($C_2\times Q_8$)

Order 12: $C_2\times C_6$ ($C_2\times C_{10}$), $C_{12}$ ($C_2\times C_{10}$), $C_3:C_4$ ($C_2\times C_{10}$)

Order 10: $C_{10}$ ($C_3:Q_8$), $C_{10}$ ($C_2\times D_6$)

Order 8: $C_2\times C_4$ ($C_5\times S_3$)

Order 6: $C_6$ ($C_2^2\times C_{10}$), $C_6$ ($C_5\times Q_8$)

Order 5: $C_5$ ($C_6:Q_8$)

Order 4: $C_2^2$ ($S_3\times C_{10}$), $C_4$ ($S_3\times C_{10}$)

Order 3: $C_3$ ($Q_8\times C_{10}$)

Order 2: $C_2$ ($C_{10}\times D_6$), $C_2$ ($C_{15}:Q_8$)

Order 1: $C_1$ ($C_{30}:Q_8$)

Series

Derived series

$C_{30}:Q_8$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_{30}:Q_8$

$\rhd$

$C_6:C_{20}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{30}:Q_8$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

$\lhd$

$C_2\times C_{20}$

Supergroups

This group is a maximal subgroup of 76 larger groups in the database.

This group is a maximal quotient of 31 larger groups in the database.

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.