Abstract group 1120.880: $C_{140}.D_4$

• Label: 1120.880
• Order: \( 2^{5} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $28$
• Trans deg.: $560$
• Rank: $3$

Group information

Description:	$C_{140}.D_4$
Order:	\(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \)
Exponent:	\(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Automorphism group:	$(C_2\times C_{70}).C_6.C_2^6$, of order \(53760\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_5$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	7	8	10	14	20	28	35	70	140
Elements	1	143	152	4	6	280	12	18	48	72	24	72	288	1120
Conjugacy classes	1	3	5	2	3	2	6	9	12	18	12	36	72	181
Divisions	1	3	5	1	1	2	2	2	4	4	1	2	4	32
Autjugacy classes	1	3	4	1	1	1	2	2	3	3	1	2	3	27

Dimension	1	2	4	6	8	12	16	24	48	96
Irr. complex chars.	8	138	35	0	0	0	0	0	0	0	181
Irr. rational chars.	8	2	5	4	2	2	1	5	2	1	32

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$560$
Rank:	$3$
Inequivalent generating triples:	$8064$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	96
Arbitrary	4	8	18

Constructions

Presentation:	$\langle a, b, c \mid c^{140}=1, a^{2}=c^{70}, b^{4}=c^{70}, b^{a}=b^{3}, c^{a}=c^{71}, c^{b}=c^{139} \rangle$
Permutation group:	Degree $28$ $\langle(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,22)(24,28) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8:D_{35})$ $\,\rtimes\,$ $C_2$ (2)	$(Q_8\times C_{14})$ $\,\rtimes\,$ $D_5$	$(Q_8\times C_{10})$ $\,\rtimes\,$ $D_7$	$(C_2\times Q_8)$ $\,\rtimes\,$ $D_{35}$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $D_{70}$ (2)	$C_{140}$ . $D_4$	$D_{140}$ . $C_2^2$	$C_{140}$ . $C_2^3$	all 28

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 1180 subgroups in 120 conjugacy classes, 47 normal (33 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{70}:D_4$
Commutator:	$G' \simeq$ $C_{140}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times D_{70}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{70}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{140}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{70}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1120: $C_{140}.D_4$

Order 560: $C_{35}:Q_{16}$ x 2, $Q_8:D_{35}$ x 2, $C_{140}.C_4$, $Q_8\times C_{70}$, $D_{140}:C_2$

Order 280: $Q_8\times C_{35}$ x 3, $C_{35}:C_8$ x 2, $C_2\times C_{140}$ x 2, $C_{35}:D_4$, $D_{140}$, $C_4\times D_{35}$, $C_{35}:Q_8$

Order 224: $C_{28}.D_4$

Order 160: $C_{20}.D_4$

Order 140: $C_{140}$ x 4, $C_{35}:C_4$, $C_2\times C_{70}$, $D_{70}$

Order 112: $C_7:Q_{16}$ x 2, $Q_8:D_7$ x 2, $C_7:\OD_{16}$, $Q_8\times C_{14}$, $D_{28}:C_2$

Order 80: $C_5:Q_{16}$ x 2, $Q_8:D_5$ x 2, $Q_8\times C_{10}$, $D_{20}:C_2$, $C_5:\OD_{16}$

Order 70: $C_{70}$ x 2, $D_{35}$

Order 56: $C_7\times Q_8$ x 3, $C_2\times C_{28}$ x 2, $C_7:C_8$ x 2, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$

Order 40: $C_5\times Q_8$ x 3, $C_2\times C_{20}$ x 2, $C_5:C_8$ x 2, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$

Order 35: $C_{35}$

Order 32: $Q_{16}:C_2$

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

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

Order 16: $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 14: $C_{14}$ x 2, $D_7$

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

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$ x 2

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1120: $C_{140}.D_4$

Order 560: $C_{35}:Q_{16}$, $Q_8:D_{35}$, $C_{140}.C_4$, $Q_8\times C_{70}$, $D_{140}:C_2$

Order 280: $Q_8\times C_{35}$ x 2, $C_2\times C_{140}$ x 2, $C_{35}:C_8$, $C_{35}:D_4$, $D_{140}$, $C_4\times D_{35}$, $C_{35}:Q_8$

Order 224: $C_{28}.D_4$

Order 160: $C_{20}.D_4$

Order 140: $C_{140}$ x 3, $C_{35}:C_4$, $C_2\times C_{70}$, $D_{70}$

Order 112: $C_7:\OD_{16}$, $Q_8\times C_{14}$, $D_{28}:C_2$, $C_7:Q_{16}$, $Q_8:D_7$

Order 80: $Q_8\times C_{10}$, $D_{20}:C_2$, $C_5:Q_{16}$, $Q_8:D_5$, $C_5:\OD_{16}$

Order 70: $C_{70}$ x 2, $D_{35}$

Order 56: $C_2\times C_{28}$ x 2, $C_7\times Q_8$ x 2, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$, $C_7:C_8$

Order 40: $C_2\times C_{20}$ x 2, $C_5\times Q_8$ x 2, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5:C_8$

Order 35: $C_{35}$

Order 32: $Q_{16}:C_2$

Order 28: $C_{28}$ x 3, $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

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

Order 16: $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 14: $C_{14}$ x 2, $D_7$

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

Order 8: $Q_8$ x 3, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1120: $C_{140}.D_4$ ($C_1$)

Order 560: $C_{35}:Q_{16}$ ($C_2$) x 2, $Q_8:D_{35}$ ($C_2$) x 2, $C_{140}.C_4$ ($C_2$), $Q_8\times C_{70}$ ($C_2$), $D_{140}:C_2$ ($C_2$)

Order 280: $Q_8\times C_{35}$ ($C_2^2$) x 2, $C_{35}:C_8$ ($C_2^2$) x 2, $C_2\times C_{140}$ ($C_2^2$), $D_{140}$ ($C_2^2$), $C_{35}:Q_8$ ($C_2^2$)

Order 140: $C_{140}$ ($C_2^3$), $C_{140}$ ($D_4$), $C_2\times C_{70}$ ($D_4$)

Order 112: $Q_8\times C_{14}$ ($D_5$)

Order 80: $Q_8\times C_{10}$ ($D_7$)

Order 70: $C_{70}$ ($C_2\times D_4$)

Order 56: $C_7\times Q_8$ ($D_{10}$) x 2, $C_2\times C_{28}$ ($D_{10}$)

Order 40: $C_5\times Q_8$ ($D_{14}$) x 2, $C_2\times C_{20}$ ($D_{14}$)

Order 35: $C_{35}$ ($Q_{16}:C_2$)

Order 28: $C_2\times C_{14}$ ($C_5:D_4$), $C_{28}$ ($C_5:D_4$), $C_{28}$ ($C_2\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_7:D_4$), $C_{20}$ ($C_7:D_4$), $C_{20}$ ($C_2\times D_{14}$)

Order 16: $C_2\times Q_8$ ($D_{35}$)

Order 14: $C_{14}$ ($C_{10}:D_4$)

Order 10: $C_{10}$ ($C_{14}:D_4$)

Order 8: $Q_8$ ($D_{70}$) x 2, $C_2\times C_4$ ($D_{70}$)

Order 7: $C_7$ ($C_{20}.D_4$)

Order 5: $C_5$ ($C_{28}.D_4$)

Order 4: $C_2^2$ ($C_{35}:D_4$), $C_4$ ($C_2\times D_{70}$), $C_4$ ($C_{35}:D_4$)

Order 2: $C_2$ ($C_{70}:D_4$)

Order 1: $C_1$ ($C_{140}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1120: $C_{140}.D_4$ ($C_1$)

Order 560: $C_{35}:Q_{16}$ ($C_2$), $Q_8:D_{35}$ ($C_2$), $C_{140}.C_4$ ($C_2$), $Q_8\times C_{70}$ ($C_2$), $D_{140}:C_2$ ($C_2$)

Order 280: $Q_8\times C_{35}$ ($C_2^2$), $C_{35}:C_8$ ($C_2^2$), $C_2\times C_{140}$ ($C_2^2$), $D_{140}$ ($C_2^2$), $C_{35}:Q_8$ ($C_2^2$)

Order 140: $C_{140}$ ($C_2^3$), $C_{140}$ ($D_4$), $C_2\times C_{70}$ ($D_4$)

Order 112: $Q_8\times C_{14}$ ($D_5$)

Order 80: $Q_8\times C_{10}$ ($D_7$)

Order 70: $C_{70}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($D_{10}$), $C_7\times Q_8$ ($D_{10}$)

Order 40: $C_2\times C_{20}$ ($D_{14}$), $C_5\times Q_8$ ($D_{14}$)

Order 35: $C_{35}$ ($Q_{16}:C_2$)

Order 28: $C_2\times C_{14}$ ($C_5:D_4$), $C_{28}$ ($C_5:D_4$), $C_{28}$ ($C_2\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_7:D_4$), $C_{20}$ ($C_7:D_4$), $C_{20}$ ($C_2\times D_{14}$)

Order 16: $C_2\times Q_8$ ($D_{35}$)

Order 14: $C_{14}$ ($C_{10}:D_4$)

Order 10: $C_{10}$ ($C_{14}:D_4$)

Order 8: $Q_8$ ($D_{70}$), $C_2\times C_4$ ($D_{70}$)

Order 7: $C_7$ ($C_{20}.D_4$)

Order 5: $C_5$ ($C_{28}.D_4$)

Order 4: $C_2^2$ ($C_{35}:D_4$), $C_4$ ($C_2\times D_{70}$), $C_4$ ($C_{35}:D_4$)

Order 2: $C_2$ ($C_{70}:D_4$)

Order 1: $C_1$ ($C_{140}.D_4$)

Series

Derived series

$C_{140}.D_4$

$\rhd$

$C_{140}$

$\rhd$

$C_1$

Chief series

$C_{140}.D_4$

$\rhd$

$Q_8:D_{35}$

$\rhd$

$Q_8\times C_{35}$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{140}.D_4$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times Q_8$

Character theory

Complex character table

See the $181 \times 181$ 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 $32 \times 32$ rational character table.