Abstract group 960.6493: $(C_4\times D_{10}):C_{12}$

• Label: 960.6493
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $16$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$(C_4\times D_{10}):C_{12}$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5:(C_2^8.C_2^3)$, of order \(10240\)\(\medspace = 2^{11} \cdot 5 \)
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	47	2	208	4	94	28	416	8	32	56	64	960
Conjugacy classes	1	11	2	16	1	22	7	32	2	8	14	16	132
Divisions	1	11	1	10	1	11	5	10	1	3	5	3	62
Autjugacy classes	1	7	1	6	1	7	4	6	1	2	4	2	42

Dimension	1	2	4	8	16	32
Irr. complex chars.	48	36	48	0	0	0	132
Irr. rational chars.	8	20	18	10	5	1	62

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$26208$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	8	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{6}=d^{20}=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{a}=bc^{3}d^{10}, d^{b}=d^{7} \rangle$
Permutation group:	Degree $16$ $\langle(2,3,4,5)(9,10)(13,14,15,16), (9,11,10,12)(13,15)(14,16), (9,10)(11,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 29 & 91 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 30 \\ 15 & 61 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 19 & 47 \\ 5 & 2 \end{array}\right), \left(\begin{array}{rr} 76 & 0 \\ 0 & 76 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 64 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	$C_3$ $\, \times\, $ $((C_4\times D_{10}):C_4)$
Semidirect product:	$(D_{10}.D_4)$ $\,\rtimes\,$ $C_6$ (2)	$(C_{12}\times D_{10})$ $\,\rtimes\,$ $C_4$ (2)	$(C_4\times D_{10})$ $\,\rtimes\,$ $C_{12}$ (2)	$D_{10}$ $\,\rtimes\,$ $(C_4:C_{12})$ (2)	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times D_{10})$ . $Q_8$ (2)	$D_{10}$ . $(C_6\times Q_8)$	$(D_5\times C_{12})$ . $D_4$ (4)	$(C_6\times D_{10})$ . $D_4$ (2)	all 56

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

Homology

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

Subgroups

There are 2004 subgroups in 468 conjugacy classes, 140 normal (64 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_2^2\times F_5$
Commutator:	$G' \simeq$ $C_2\times C_{10}$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_{10}:C_{12}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $(C_4\times D_{10}):C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.D_4$
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

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 960: $(C_4\times D_{10}):C_{12}$

Order 480: $C_{30}.C_4^2$ x 2, $C_2\times D_{10}:C_{12}$ x 2, $C_2\times C_{20}:C_{12}$ x 2, $C_{60}:C_2^3$

Order 320: $(C_4\times D_{10}):C_4$

Order 240: $C_{12}\times D_{10}$ x 8, $D_{10}:C_{12}$ x 4, $D_{10}:C_{12}$ x 4, $C_{20}:C_{12}$ x 4, $C_{15}:C_2^4$, $C_2^2\times C_{60}$, $C_{30}.C_2^3$

Order 192: $(C_2^2\times C_{12}):C_4$

Order 160: $C_{10}.C_4^2$ x 2, $C_2^3:F_5$ x 2, $D_{10}.D_4$ x 2, $C_{20}:C_2^3$

Order 120: $C_6\times F_5$ x 12, $C_6\times D_{10}$ x 10, $D_5\times C_{12}$ x 10, $C_2\times C_{60}$ x 4, $C_{10}:C_{12}$ x 4, $C_2^2\times C_{30}$

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

Order 80: $C_4\times D_{10}$ x 8, $C_2^2\times F_5$ x 4, $D_{10}:C_4$ x 4, $C_{20}:C_4$ x 4, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_2^3.D_4$

Order 60: $C_3\times D_{10}$ x 18, $C_2\times C_{30}$ x 5, $C_3\times F_5$ x 4, $C_{60}$ x 3, $C_5:C_{12}$ x 3

Order 48: $C_2^2\times C_{12}$ x 14, $C_4:C_{12}$ x 4, $C_2^2:C_{12}$ x 4, $C_2^3\times C_6$

Order 40: $C_2\times F_5$ x 12, $C_4\times D_5$ x 10, $C_2\times D_{10}$ x 10, $C_2\times C_{20}$ x 4, $C_{10}:C_4$ x 4, $C_2^2\times C_{10}$

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

Order 30: $C_3\times D_5$ x 6, $C_{30}$ x 5

Order 24: $C_2\times C_{12}$ x 30, $C_2^2\times C_6$ x 11

Order 20: $D_{10}$ x 18, $C_2\times C_{10}$ x 5, $F_5$ x 4, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 16: $C_2^2\times C_4$ x 14, $C_4:C_4$ x 4, $C_2^2:C_4$ x 4, $C_2^4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 23, $C_{12}$ x 10

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 30, $C_2^3$ x 11

Order 6: $C_6$ x 11

Order 5: $C_5$

Order 4: $C_2^2$ x 23, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $(C_4\times D_{10}):C_{12}$

Order 480: $C_{30}.C_4^2$, $C_{60}:C_2^3$, $C_2\times D_{10}:C_{12}$, $C_2\times C_{20}:C_{12}$

Order 320: $(C_4\times D_{10}):C_4$

Order 240: $C_{12}\times D_{10}$ x 5, $C_{15}:C_2^4$, $D_{10}:C_{12}$, $C_2^2\times C_{60}$, $C_{30}.C_2^3$, $D_{10}:C_{12}$, $C_{20}:C_{12}$

Order 192: $(C_2^2\times C_{12}):C_4$

Order 160: $C_{10}.C_4^2$, $C_{20}:C_2^3$, $C_2^3:F_5$, $D_{10}.D_4$

Order 120: $C_6\times D_{10}$ x 6, $D_5\times C_{12}$ x 4, $C_6\times F_5$ x 3, $C_2\times C_{60}$ x 3, $C_{10}:C_{12}$ x 3, $C_2^2\times C_{30}$

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

Order 80: $C_4\times D_{10}$ x 5, $C_2^2\times D_{10}$, $C_2^2\times F_5$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$, $D_{10}:C_4$, $C_{20}:C_4$

Order 64: $C_2^3.D_4$

Order 60: $C_3\times D_{10}$ x 9, $C_2\times C_{30}$ x 4, $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_3\times F_5$

Order 48: $C_2^2\times C_{12}$ x 8, $C_2^3\times C_6$, $C_4:C_{12}$, $C_2^2:C_{12}$

Order 40: $C_2\times D_{10}$ x 6, $C_4\times D_5$ x 4, $C_2\times C_{20}$ x 3, $C_{10}:C_4$ x 3, $C_2\times F_5$ x 3, $C_2^2\times C_{10}$

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

Order 30: $C_{30}$ x 4, $C_3\times D_5$ x 3

Order 24: $C_2\times C_{12}$ x 13, $C_2^2\times C_6$ x 7

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 4, $C_{20}$ x 2, $C_5:C_4$ x 2, $F_5$

Order 16: $C_2^2\times C_4$ x 8, $C_4:C_4$, $C_2^2:C_4$, $C_2^4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 13, $C_{12}$ x 5

Order 10: $C_{10}$ x 4, $D_5$ x 3

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 7

Order 6: $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 13, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $(C_4\times D_{10}):C_{12}$ ($C_1$)

Order 480: $C_{30}.C_4^2$ ($C_2$) x 2, $C_2\times D_{10}:C_{12}$ ($C_2$) x 2, $C_2\times C_{20}:C_{12}$ ($C_2$) x 2, $C_{60}:C_2^3$ ($C_2$)

Order 320: $(C_4\times D_{10}):C_4$ ($C_3$)

Order 240: $D_{10}:C_{12}$ ($C_2^2$) x 4, $C_{12}\times D_{10}$ ($C_2^2$) x 2, $C_{12}\times D_{10}$ ($C_4$) x 2, $C_{15}:C_2^4$ ($C_2^2$), $C_2^2\times C_{60}$ ($C_4$), $C_{30}.C_2^3$ ($C_4$)

Order 160: $C_{10}.C_4^2$ ($C_6$) x 2, $C_2^3:F_5$ ($C_6$) x 2, $D_{10}.D_4$ ($C_6$) x 2, $C_{20}:C_2^3$ ($C_6$)

Order 120: $D_5\times C_{12}$ ($D_4$) x 4, $C_6\times D_{10}$ ($Q_8$) x 2, $C_6\times D_{10}$ ($D_4$) x 2, $C_2\times C_{60}$ ($C_2\times C_4$) x 2, $C_{10}:C_{12}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{30}$ ($C_2\times C_4$), $C_6\times D_{10}$ ($C_2^3$), $C_6\times D_{10}$ ($C_2\times C_4$)

Order 80: $C_2^2\times F_5$ ($C_2\times C_6$) x 4, $C_4\times D_{10}$ ($C_2\times C_6$) x 2, $C_4\times D_{10}$ ($C_{12}$) x 2, $C_2^2\times D_{10}$ ($C_2\times C_6$), $C_2^2\times C_{20}$ ($C_{12}$), $C_2^2.D_{10}$ ($C_{12}$)

Order 60: $C_3\times D_{10}$ ($C_2\times D_4$) x 3, $C_{60}$ ($C_2^2:C_4$) x 2, $C_5:C_{12}$ ($C_2^2:C_4$) x 2, $C_2\times C_{30}$ ($C_4:C_4$) x 2, $C_3\times D_{10}$ ($C_4:C_4$) x 2, $C_3\times D_{10}$ ($D_4:C_2$) x 2, $C_2\times C_{30}$ ($C_2^2\times C_4$), $C_3\times D_{10}$ ($C_2\times Q_8$)

Order 48: $C_2^2\times C_{12}$ ($F_5$)

Order 40: $C_4\times D_5$ ($C_3\times D_4$) x 4, $C_2\times C_{20}$ ($C_2\times C_{12}$) x 2, $C_{10}:C_4$ ($C_2\times C_{12}$) x 2, $C_2\times D_{10}$ ($C_3\times Q_8$) x 2, $C_2\times D_{10}$ ($C_3\times D_4$) x 2, $C_2^2\times C_{10}$ ($C_2\times C_{12}$), $C_2\times D_{10}$ ($C_2\times C_{12}$), $C_2\times D_{10}$ ($C_2^2\times C_6$)

Order 30: $C_3\times D_5$ ($C_2^2:Q_8$) x 2, $C_3\times D_5$ ($C_4:D_4$) x 2, $C_{30}$ ($C_4^2:C_2$), $C_{30}$ ($C_2^2.D_4$), $C_{30}$ ($C_2^3:C_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times F_5$) x 2, $C_2^2\times C_6$ ($C_2\times F_5$)

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

Order 16: $C_2^2\times C_4$ ($C_3\times F_5$)

Order 15: $C_{15}$ ($C_2^3.D_4$)

Order 12: $C_2\times C_6$ ($C_{20}:C_4$) x 2, $C_{12}$ ($D_{10}:C_4$) x 2, $C_2\times C_6$ ($C_2^2\times F_5$)

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

Order 8: $C_2\times C_4$ ($C_6\times F_5$) x 2, $C_2^3$ ($C_6\times F_5$)

Order 6: $C_6$ ($C_2^3:F_5$), $C_6$ ($D_{10}.C_2^3$), $C_6$ ($D_{10}.D_4$)

Order 5: $C_5$ ($(C_2^2\times C_{12}):C_4$)

Order 4: $C_2^2$ ($C_{20}:C_{12}$) x 2, $C_4$ ($D_{10}:C_{12}$) x 2, $C_2^2$ ($D_{10}:C_{12}$)

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

Order 2: $C_2$ ($C_2\times D_{10}:C_{12}$), $C_2$ ($(C_2\times C_{20}):C_{12}$), $C_2$ ($C_2\times C_{20}:C_{12}$)

Order 1: $C_1$ ($(C_4\times D_{10}):C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $(C_4\times D_{10}):C_{12}$ ($C_1$)

Order 480: $C_{30}.C_4^2$ ($C_2$), $C_{60}:C_2^3$ ($C_2$), $C_2\times D_{10}:C_{12}$ ($C_2$), $C_2\times C_{20}:C_{12}$ ($C_2$)

Order 320: $(C_4\times D_{10}):C_4$ ($C_3$)

Order 240: $C_{12}\times D_{10}$ ($C_2^2$) x 2, $C_{12}\times D_{10}$ ($C_4$) x 2, $C_{15}:C_2^4$ ($C_2^2$), $D_{10}:C_{12}$ ($C_2^2$), $C_2^2\times C_{60}$ ($C_4$), $C_{30}.C_2^3$ ($C_4$)

Order 160: $C_{10}.C_4^2$ ($C_6$), $C_{20}:C_2^3$ ($C_6$), $C_2^3:F_5$ ($C_6$), $D_{10}.D_4$ ($C_6$)

Order 120: $C_2\times C_{60}$ ($C_2\times C_4$) x 2, $C_{10}:C_{12}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{30}$ ($C_2\times C_4$), $C_6\times D_{10}$ ($C_2^3$), $C_6\times D_{10}$ ($Q_8$), $C_6\times D_{10}$ ($D_4$), $C_6\times D_{10}$ ($C_2\times C_4$), $D_5\times C_{12}$ ($D_4$)

Order 80: $C_4\times D_{10}$ ($C_2\times C_6$) x 2, $C_4\times D_{10}$ ($C_{12}$) x 2, $C_2^2\times D_{10}$ ($C_2\times C_6$), $C_2^2\times F_5$ ($C_2\times C_6$), $C_2^2\times C_{20}$ ($C_{12}$), $C_2^2.D_{10}$ ($C_{12}$)

Order 60: $C_3\times D_{10}$ ($C_2\times D_4$) x 2, $C_{60}$ ($C_2^2:C_4$), $C_5:C_{12}$ ($C_2^2:C_4$), $C_2\times C_{30}$ ($C_4:C_4$), $C_2\times C_{30}$ ($C_2^2\times C_4$), $C_3\times D_{10}$ ($C_4:C_4$), $C_3\times D_{10}$ ($D_4:C_2$), $C_3\times D_{10}$ ($C_2\times Q_8$)

Order 48: $C_2^2\times C_{12}$ ($F_5$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_{12}$) x 2, $C_{10}:C_4$ ($C_2\times C_{12}$) x 2, $C_4\times D_5$ ($C_3\times D_4$), $C_2^2\times C_{10}$ ($C_2\times C_{12}$), $C_2\times D_{10}$ ($C_2\times C_{12}$), $C_2\times D_{10}$ ($C_2^2\times C_6$), $C_2\times D_{10}$ ($C_3\times Q_8$), $C_2\times D_{10}$ ($C_3\times D_4$)

Order 30: $C_{30}$ ($C_4^2:C_2$), $C_{30}$ ($C_2^2.D_4$), $C_{30}$ ($C_2^3:C_4$), $C_3\times D_5$ ($C_2^2:Q_8$), $C_3\times D_5$ ($C_4:D_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times F_5$) x 2, $C_2^2\times C_6$ ($C_2\times F_5$)

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

Order 16: $C_2^2\times C_4$ ($C_3\times F_5$)

Order 15: $C_{15}$ ($C_2^3.D_4$)

Order 12: $C_2\times C_6$ ($C_2^2\times F_5$), $C_2\times C_6$ ($C_{20}:C_4$), $C_{12}$ ($D_{10}:C_4$)

Order 10: $C_{10}$ ($C_4^2:C_6$), $C_{10}$ ($C_2\times C_4:C_{12}$), $C_{10}$ ($C_2^3:C_{12}$), $D_5$ ($C_{12}.D_4$), $D_5$ ($C_{12}:D_4$)

Order 8: $C_2\times C_4$ ($C_6\times F_5$) x 2, $C_2^3$ ($C_6\times F_5$)

Order 6: $C_6$ ($C_2^3:F_5$), $C_6$ ($D_{10}.C_2^3$), $C_6$ ($D_{10}.D_4$)

Order 5: $C_5$ ($(C_2^2\times C_{12}):C_4$)

Order 4: $C_2^2$ ($D_{10}:C_{12}$), $C_2^2$ ($C_{20}:C_{12}$), $C_4$ ($D_{10}:C_{12}$)

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

Order 2: $C_2$ ($C_2\times D_{10}:C_{12}$), $C_2$ ($(C_2\times C_{20}):C_{12}$), $C_2$ ($C_2\times C_{20}:C_{12}$)

Order 1: $C_1$ ($(C_4\times D_{10}):C_{12}$)

Series

Derived series

$(C_4\times D_{10}):C_{12}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_1$

Chief series

$(C_4\times D_{10}):C_{12}$

$\rhd$

$C_{60}:C_2^3$

$\rhd$

$C_{12}\times D_{10}$

$\rhd$

$D_5\times C_{12}$

$\rhd$

$C_{60}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$(C_4\times D_{10}):C_{12}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2^2\times C_{12}$

Character theory

Complex character table

See the $132 \times 132$ 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 $62 \times 62$ rational character table.