Abstract group 960.4023: $C_4\times C_{30}.D_4$

• Label: 960.4023
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \)
• Perm deg.: $20$
• Trans deg.: $960$
• Rank: $3$

Group information

Description:	$C_4\times C_{30}.D_4$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5:(C_2^8.C_2^5)$, of order \(40960\)\(\medspace = 2^{13} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
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	7	2	184	4	14	28	368	8	96	56	192	960
Conjugacy classes	1	7	2	32	2	14	14	64	4	48	28	96	312
Divisions	1	7	1	18	1	7	7	18	1	8	7	8	84
Autjugacy classes	1	5	1	5	1	5	5	5	1	2	5	2	38

Dimension	1	2	4	8	16	32
Irr. complex chars.	96	216	0	0	0	0	312
Irr. rational chars.	8	24	22	16	12	2	84

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$960$
Rank:	$3$
Inequivalent generating triples:	$6552$

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:	$\langle a, b, c \mid a^{4}=b^{4}=c^{60}=[a,c]=1, b^{a}=b^{3}, c^{b}=c^{49} \rangle$
Permutation group:	Degree $20$ $\langle(1,2)(3,4)(8,9,10,11)(13,14)(15,16), (1,2)(3,4)(8,10)(9,11)(17,18,19,20) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 39 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 26 & 35 \\ 80 & 31 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 41 & 40 \\ 20 & 1 \end{array}\right), \left(\begin{array}{rr} 69 & 30 \\ 30 & 7 \end{array}\right), \left(\begin{array}{rr} 39 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 77 & 0 \\ 0 & 77 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/95\Z)$
Direct product:	$C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_{10}.D_4)$
Semidirect product:	$C_{60}$ $\,\rtimes\,$ $(C_4:C_4)$	$(C_{20}:C_{12})$ $\,\rtimes\,$ $C_4$	$(C_{20}:C_4)$ $\,\rtimes\,$ $C_{12}$	$C_{20}$ $\,\rtimes\,$ $(C_4:C_{12})$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{60})$ . $Q_8$	$C_{30}$ . $(C_4\times Q_8)$	$C_6$ . $(C_{20}:Q_8)$	$C_2$ . $(C_{60}:Q_8)$	all 70

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

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{12} \simeq C_{2} \times C_{4}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 1020 subgroups in 388 conjugacy classes, 214 normal (50 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{12}$	$G/Z \simeq$ $D_{10}$
Commutator:	$G' \simeq$ $C_{10}$	$G/G' \simeq$ $C_2\times C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_6\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_4\times C_{30}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:C_4^2$
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

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 960: $C_4\times C_{30}.D_4$

Order 480: $C_2\times C_{30}.D_4$ x 2, $C_{30}:C_4^2$ x 2, $C_{30}.C_4^2$ x 2, $C_2\times C_4\times C_{60}$

Order 320: $C_4\times C_{10}.D_4$

Order 240: $C_{30}.D_4$ x 8, $C_{20}:C_{12}$ x 6, $C_{30}.C_2^3$ x 4, $C_2^2\times C_{60}$ x 3, $C_4\times C_{60}$ x 2

Order 192: $C_{12}:C_4^2$

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

Order 120: $C_{10}:C_{12}$ x 18, $C_2\times C_{60}$ x 14, $C_2^2\times C_{30}$

Order 96: $C_2\times C_4\times C_{12}$ x 3, $C_6.C_4^2$ x 2, $C_2\times C_4:C_{12}$ x 2

Order 80: $C_{10}.D_4$ x 8, $C_{20}:C_4$ x 6, $C_2^2.D_{10}$ x 4, $C_2^2\times C_{20}$ x 3, $C_4\times C_{20}$ x 2

Order 64: $C_4:C_4^2$

Order 60: $C_5:C_{12}$ x 10, $C_{60}$ x 8, $C_2\times C_{30}$ x 7

Order 48: $C_4:C_{12}$ x 8, $C_4\times C_{12}$ x 8, $C_2^2\times C_{12}$ x 7

Order 40: $C_{10}:C_4$ x 18, $C_2\times C_{20}$ x 14, $C_2^2\times C_{10}$

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

Order 30: $C_{30}$ x 7

Order 24: $C_2\times C_{12}$ x 32, $C_2^2\times C_6$

Order 20: $C_5:C_4$ x 10, $C_{20}$ x 8, $C_2\times C_{10}$ x 7

Order 16: $C_4:C_4$ x 8, $C_4^2$ x 8, $C_2^2\times C_4$ x 7

Order 15: $C_{15}$

Order 12: $C_{12}$ x 18, $C_2\times C_6$ x 7

Order 10: $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 32, $C_2^3$

Order 6: $C_6$ x 7

Order 5: $C_5$

Order 4: $C_4$ x 18, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_4\times C_{30}.D_4$

Order 480: $C_{30}:C_4^2$ x 2, $C_2\times C_4\times C_{60}$, $C_2\times C_{30}.D_4$, $C_{30}.C_4^2$

Order 320: $C_4\times C_{10}.D_4$

Order 240: $C_{30}.C_2^3$ x 3, $C_{20}:C_{12}$ x 2, $C_2^2\times C_{60}$ x 2, $C_4\times C_{60}$, $C_{30}.D_4$

Order 192: $C_{12}:C_4^2$

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

Order 120: $C_{10}:C_{12}$ x 7, $C_2\times C_{60}$ x 6, $C_2^2\times C_{30}$

Order 96: $C_2\times C_4\times C_{12}$ x 3, $C_6.C_4^2$, $C_2\times C_4:C_{12}$

Order 80: $C_2^2.D_{10}$ x 3, $C_2^2\times C_{20}$ x 2, $C_{20}:C_4$ x 2, $C_4\times C_{20}$, $C_{10}.D_4$

Order 64: $C_4:C_4^2$

Order 60: $C_2\times C_{30}$ x 5, $C_5:C_{12}$ x 3, $C_{60}$ x 2

Order 48: $C_2^2\times C_{12}$ x 5, $C_4\times C_{12}$ x 3, $C_4:C_{12}$

Order 40: $C_{10}:C_4$ x 7, $C_2\times C_{20}$ x 6, $C_2^2\times C_{10}$

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

Order 30: $C_{30}$ x 5

Order 24: $C_2\times C_{12}$ x 13, $C_2^2\times C_6$

Order 20: $C_2\times C_{10}$ x 5, $C_5:C_4$ x 3, $C_{20}$ x 2

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

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 5, $C_{12}$ x 5

Order 10: $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 13, $C_2^3$

Order 6: $C_6$ x 5

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_4\times C_{30}.D_4$ ($C_1$)

Order 480: $C_2\times C_{30}.D_4$ ($C_2$) x 2, $C_{30}:C_4^2$ ($C_2$) x 2, $C_{30}.C_4^2$ ($C_2$) x 2, $C_2\times C_4\times C_{60}$ ($C_2$)

Order 320: $C_4\times C_{10}.D_4$ ($C_3$)

Order 240: $C_{30}.D_4$ ($C_4$) x 8, $C_{20}:C_{12}$ ($C_4$) x 4, $C_{30}.C_2^3$ ($C_2^2$) x 4, $C_2^2\times C_{60}$ ($C_2^2$) x 3

Order 160: $C_{10}.C_4^2$ ($C_6$) x 2, $C_2^3.D_{10}$ ($C_6$) x 2, $C_{10}:C_4^2$ ($C_6$) x 2, $C_2\times C_4\times C_{20}$ ($C_6$)

Order 120: $C_{10}:C_{12}$ ($C_2\times C_4$) x 12, $C_2\times C_{60}$ ($C_2\times C_4$) x 6, $C_2\times C_{60}$ ($Q_8$) x 2, $C_2\times C_{60}$ ($D_4$) x 2, $C_2^2\times C_{30}$ ($C_2^3$)

Order 96: $C_2\times C_4\times C_{12}$ ($D_5$)

Order 80: $C_{10}.D_4$ ($C_{12}$) x 8, $C_2^2.D_{10}$ ($C_2\times C_6$) x 4, $C_{20}:C_4$ ($C_{12}$) x 4, $C_2^2\times C_{20}$ ($C_2\times C_6$) x 3

Order 60: $C_{60}$ ($C_4:C_4$) x 4, $C_5:C_{12}$ ($C_4^2$) x 4, $C_2\times C_{30}$ ($C_2^2\times C_4$) x 3, $C_2\times C_{30}$ ($D_4:C_2$) x 2, $C_2\times C_{30}$ ($C_2\times Q_8$), $C_2\times C_{30}$ ($C_2\times D_4$)

Order 48: $C_2^2\times C_{12}$ ($D_{10}$) x 3

Order 40: $C_{10}:C_4$ ($C_2\times C_{12}$) x 12, $C_2\times C_{20}$ ($C_2\times C_{12}$) x 6, $C_2\times C_{20}$ ($C_3\times Q_8$) x 2, $C_2\times C_{20}$ ($C_3\times D_4$) x 2, $C_2^2\times C_{10}$ ($C_2^2\times C_6$)

Order 32: $C_2\times C_4^2$ ($C_3\times D_5$)

Order 30: $C_{30}$ ($C_4\times Q_8$) x 2, $C_{30}$ ($C_4\times D_4$) x 2, $C_{30}$ ($C_4^2:C_2$), $C_{30}$ ($C_2^2.D_4$), $C_{30}$ ($C_2\times C_4^2$)

Order 24: $C_2\times C_{12}$ ($C_4\times D_5$) x 6, $C_2\times C_{12}$ ($C_5:D_4$) x 2, $C_2\times C_{12}$ ($C_5:Q_8$) x 2, $C_2^2\times C_6$ ($C_2\times D_{10}$)

Order 20: $C_{20}$ ($C_4:C_{12}$) x 4, $C_5:C_4$ ($C_4\times C_{12}$) x 4, $C_2\times C_{10}$ ($C_2^2\times C_{12}$) x 3, $C_2\times C_{10}$ ($D_4:C_6$) x 2, $C_2\times C_{10}$ ($C_6\times Q_8$), $C_2\times C_{10}$ ($C_6\times D_4$)

Order 16: $C_2^2\times C_4$ ($C_3\times D_{10}$) x 3

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

Order 12: $C_{12}$ ($C_{10}.D_4$) x 4, $C_2\times C_6$ ($C_4\times D_{10}$) x 3, $C_2\times C_6$ ($D_{20}:C_2$) x 2, $C_2\times C_6$ ($C_{10}:D_4$), $C_2\times C_6$ ($C_{10}:Q_8$)

Order 10: $C_{10}$ ($Q_8\times C_{12}$) x 2, $C_{10}$ ($D_4\times C_{12}$) x 2, $C_{10}$ ($C_4^2:C_6$), $C_{10}$ ($C_2\times C_4:C_{12}$), $C_{10}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2\times C_4$ ($D_5\times C_{12}$) x 6, $C_2\times C_4$ ($C_{15}:D_4$) x 2, $C_2\times C_4$ ($C_{15}:Q_8$) x 2, $C_2^3$ ($C_6\times D_{10}$)

Order 6: $C_6$ ($C_{20}:Q_8$) x 2, $C_6$ ($C_{20}:D_4$) x 2, $C_6$ ($C_4^2:D_5$), $C_6$ ($D_5\times C_4^2$), $C_6$ ($C_2^3.D_{10}$)

Order 5: $C_5$ ($C_{12}:C_4^2$)

Order 4: $C_4$ ($C_{30}.D_4$) x 4, $C_2^2$ ($C_{12}\times D_{10}$) x 3, $C_2^2$ ($D_{20}:C_6$) x 2, $C_2^2$ ($C_{30}:D_4$), $C_2^2$ ($C_{30}:Q_8$)

Order 3: $C_3$ ($C_4\times C_{10}.D_4$)

Order 2: $C_2$ ($C_{60}:D_4$) x 2, $C_2$ ($C_{60}:Q_8$) x 2, $C_2$ ($C_2\times C_{30}.D_4$), $C_2$ ($C_3\times C_4^2:D_5$), $C_2$ ($C_4\times D_5\times C_{12}$)

Order 1: $C_1$ ($C_4\times C_{30}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_4\times C_{30}.D_4$ ($C_1$)

Order 480: $C_{30}:C_4^2$ ($C_2$) x 2, $C_2\times C_4\times C_{60}$ ($C_2$), $C_2\times C_{30}.D_4$ ($C_2$), $C_{30}.C_4^2$ ($C_2$)

Order 320: $C_4\times C_{10}.D_4$ ($C_3$)

Order 240: $C_{30}.C_2^3$ ($C_2^2$) x 3, $C_2^2\times C_{60}$ ($C_2^2$) x 2, $C_{30}.D_4$ ($C_4$), $C_{20}:C_{12}$ ($C_4$)

Order 160: $C_{10}:C_4^2$ ($C_6$) x 2, $C_{10}.C_4^2$ ($C_6$), $C_2\times C_4\times C_{20}$ ($C_6$), $C_2^3.D_{10}$ ($C_6$)

Order 120: $C_{10}:C_{12}$ ($C_2\times C_4$) x 4, $C_2\times C_{60}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{30}$ ($C_2^3$), $C_2\times C_{60}$ ($Q_8$), $C_2\times C_{60}$ ($D_4$)

Order 96: $C_2\times C_4\times C_{12}$ ($D_5$)

Order 80: $C_2^2.D_{10}$ ($C_2\times C_6$) x 3, $C_2^2\times C_{20}$ ($C_2\times C_6$) x 2, $C_{10}.D_4$ ($C_{12}$), $C_{20}:C_4$ ($C_{12}$)

Order 60: $C_2\times C_{30}$ ($C_2^2\times C_4$) x 2, $C_{60}$ ($C_4:C_4$), $C_5:C_{12}$ ($C_4^2$), $C_2\times C_{30}$ ($D_4:C_2$), $C_2\times C_{30}$ ($C_2\times Q_8$), $C_2\times C_{30}$ ($C_2\times D_4$)

Order 48: $C_2^2\times C_{12}$ ($D_{10}$) x 2

Order 40: $C_{10}:C_4$ ($C_2\times C_{12}$) x 4, $C_2\times C_{20}$ ($C_2\times C_{12}$) x 2, $C_2\times C_{20}$ ($C_3\times Q_8$), $C_2\times C_{20}$ ($C_3\times D_4$), $C_2^2\times C_{10}$ ($C_2^2\times C_6$)

Order 32: $C_2\times C_4^2$ ($C_3\times D_5$)

Order 30: $C_{30}$ ($C_4\times Q_8$), $C_{30}$ ($C_4\times D_4$), $C_{30}$ ($C_4^2:C_2$), $C_{30}$ ($C_2^2.D_4$), $C_{30}$ ($C_2\times C_4^2$)

Order 24: $C_2\times C_{12}$ ($C_4\times D_5$) x 2, $C_2\times C_{12}$ ($C_5:D_4$), $C_2\times C_{12}$ ($C_5:Q_8$), $C_2^2\times C_6$ ($C_2\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_2^2\times C_{12}$) x 2, $C_2\times C_{10}$ ($D_4:C_6$), $C_2\times C_{10}$ ($C_6\times Q_8$), $C_2\times C_{10}$ ($C_6\times D_4$), $C_{20}$ ($C_4:C_{12}$), $C_5:C_4$ ($C_4\times C_{12}$)

Order 16: $C_2^2\times C_4$ ($C_3\times D_{10}$) x 2

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

Order 12: $C_2\times C_6$ ($C_4\times D_{10}$) x 2, $C_2\times C_6$ ($C_{10}:D_4$), $C_2\times C_6$ ($D_{20}:C_2$), $C_2\times C_6$ ($C_{10}:Q_8$), $C_{12}$ ($C_{10}.D_4$)

Order 10: $C_{10}$ ($Q_8\times C_{12}$), $C_{10}$ ($D_4\times C_{12}$), $C_{10}$ ($C_4^2:C_6$), $C_{10}$ ($C_2\times C_4:C_{12}$), $C_{10}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2\times C_4$ ($D_5\times C_{12}$) x 2, $C_2^3$ ($C_6\times D_{10}$), $C_2\times C_4$ ($C_{15}:D_4$), $C_2\times C_4$ ($C_{15}:Q_8$)

Order 6: $C_6$ ($C_4^2:D_5$), $C_6$ ($D_5\times C_4^2$), $C_6$ ($C_{20}:Q_8$), $C_6$ ($C_{20}:D_4$), $C_6$ ($C_2^3.D_{10}$)

Order 5: $C_5$ ($C_{12}:C_4^2$)

Order 4: $C_2^2$ ($C_{12}\times D_{10}$) x 2, $C_2^2$ ($C_{30}:D_4$), $C_2^2$ ($D_{20}:C_6$), $C_2^2$ ($C_{30}:Q_8$), $C_4$ ($C_{30}.D_4$)

Order 3: $C_3$ ($C_4\times C_{10}.D_4$)

Order 2: $C_2$ ($C_{60}:D_4$), $C_2$ ($C_2\times C_{30}.D_4$), $C_2$ ($C_3\times C_4^2:D_5$), $C_2$ ($C_4\times D_5\times C_{12}$), $C_2$ ($C_{60}:Q_8$)

Order 1: $C_1$ ($C_4\times C_{30}.D_4$)

Series

Derived series

$C_4\times C_{30}.D_4$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$C_4\times C_{30}.D_4$

$\rhd$

$C_2\times C_{30}.D_4$

$\rhd$

$C_2^2\times C_{60}$

$\rhd$

$C_2^2\times C_{30}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_4\times C_{30}.D_4$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{12}$

$\lhd$

$C_2\times C_4\times C_{12}$

Character theory

Complex character table

See the $312 \times 312$ 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 $84 \times 84$ rational character table.