Abstract group 1120.700: $C_{70}:D_8$

• Label: 1120.700
• Order: \( 2^{5} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $560$
• Rank: $3$

Group information

Description:	$C_{70}:D_8$
Order:	\(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \)
Exponent:	\(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Automorphism group:	$C_{14}.(C_2^3\times C_6).C_2^5$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \)
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	40	70	140
Elements	1	67	4	4	6	56	268	66	16	24	24	224	264	96	1120
Conjugacy classes	1	7	2	4	3	4	28	21	8	6	12	16	84	24	220
Divisions	1	7	2	1	1	2	7	5	2	2	1	2	5	2	40
Autjugacy classes	1	4	2	1	1	1	4	3	2	2	1	1	3	2	28

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	40	150	30	0	0	0	0	0	0	220
Irr. rational chars.	8	2	10	4	2	4	2	4	4	40

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$560$
Rank:	$3$
Inequivalent generating triples:	$20832$

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:	$\langle a, b, c \mid a^{2}=b^{8}=c^{70}=[a,c]=1, b^{a}=b^{7}, c^{b}=c^{41} \rangle$
Permutation group:	Degree $22$ $\langle(2,3)(4,5)(6,7)(14,17)(15,19)(18,20), (21,22), (13,14)(15,20)(16,18)(17,19) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_7:D_8)$
Semidirect product:	$C_{70}$ $\,\rtimes\,$ $D_8$ (2)	$(D_4\times C_{10})$ $\,\rtimes\,$ $D_7$	$(C_5\times D_4)$ $\,\rtimes\,$ $D_{14}$ (2)	$D_4$ $\,\rtimes\,$ $(C_5\times D_{14})$ (2)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{140}$ . $D_4$	$C_{140}$ . $C_2^3$	$(C_2\times C_{70})$ . $D_4$	$C_{70}$ . $(C_2\times D_4)$	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:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 700 subgroups in 152 conjugacy classes, 66 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $C_7:D_4$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{10}\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{70}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{70}:D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{70}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_8$
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

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1120: $C_{70}:D_8$

Order 560: $C_{35}:D_8$ x 4, $C_{14}:C_{40}$, $D_4\times C_{70}$, $C_{10}\times D_{28}$

Order 280: $D_4\times C_{35}$ x 3, $C_5\times D_{28}$ x 3, $C_7:C_{40}$ x 2, $C_2^2\times C_{70}$, $C_{10}\times D_{14}$, $C_2\times C_{140}$

Order 224: $C_{14}:D_8$

Order 160: $C_{10}\times D_8$

Order 140: $C_2\times C_{70}$ x 5, $C_5\times D_{14}$ x 4, $C_{140}$ x 2

Order 112: $C_7:D_8$ x 4, $C_{14}:C_8$, $D_4\times C_{14}$, $C_2\times D_{28}$

Order 80: $C_5\times D_8$ x 4, $D_4\times C_{10}$ x 2, $C_2\times C_{40}$

Order 70: $C_{70}$ x 5, $C_5\times D_7$ x 2

Order 56: $C_7\times D_4$ x 3, $D_{28}$ x 3, $C_7:C_8$ x 2, $C_2\times C_{28}$, $C_2^2\times C_{14}$, $C_2\times D_{14}$

Order 40: $C_5\times D_4$ x 6, $C_{40}$ x 2, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$

Order 35: $C_{35}$

Order 32: $C_2\times D_8$

Order 28: $C_2\times C_{14}$ x 5, $D_{14}$ x 4, $C_{28}$ x 2

Order 20: $C_2\times C_{10}$ x 9, $C_{20}$ x 2

Order 16: $D_8$ x 4, $C_2\times D_4$ x 2, $C_2\times C_8$

Order 14: $C_{14}$ x 5, $D_7$ x 2

Order 10: $C_{10}$ x 7

Order 8: $D_4$ x 6, $C_2^3$ x 2, $C_8$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 5: $C_5$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1120: $C_{70}:D_8$

Order 560: $C_{35}:D_8$, $C_{14}:C_{40}$, $D_4\times C_{70}$, $C_{10}\times D_{28}$

Order 280: $D_4\times C_{35}$ x 2, $C_5\times D_{28}$ x 2, $C_2^2\times C_{70}$, $C_{10}\times D_{14}$, $C_2\times C_{140}$, $C_7:C_{40}$

Order 224: $C_{14}:D_8$

Order 160: $C_{10}\times D_8$

Order 140: $C_2\times C_{70}$ x 3, $C_5\times D_{14}$ x 2, $C_{140}$ x 2

Order 112: $C_{14}:C_8$, $D_4\times C_{14}$, $C_2\times D_{28}$, $C_7:D_8$

Order 80: $D_4\times C_{10}$ x 2, $C_5\times D_8$, $C_2\times C_{40}$

Order 70: $C_{70}$ x 3, $C_5\times D_7$

Order 56: $C_7\times D_4$ x 2, $D_{28}$ x 2, $C_2\times C_{28}$, $C_2^2\times C_{14}$, $C_2\times D_{14}$, $C_7:C_8$

Order 40: $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{40}$

Order 35: $C_{35}$

Order 32: $C_2\times D_8$

Order 28: $C_2\times C_{14}$ x 3, $D_{14}$ x 2, $C_{28}$ x 2

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

Order 16: $C_2\times D_4$ x 2, $D_8$, $C_2\times C_8$

Order 14: $C_{14}$ x 3, $D_7$

Order 10: $C_{10}$ x 4

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

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1120: $C_{70}:D_8$ ($C_1$)

Order 560: $C_{35}:D_8$ ($C_2$) x 4, $C_{14}:C_{40}$ ($C_2$), $D_4\times C_{70}$ ($C_2$), $C_{10}\times D_{28}$ ($C_2$)

Order 280: $D_4\times C_{35}$ ($C_2^2$) x 2, $C_7:C_{40}$ ($C_2^2$) x 2, $C_5\times D_{28}$ ($C_2^2$) x 2, $C_2\times C_{140}$ ($C_2^2$)

Order 224: $C_{14}:D_8$ ($C_5$)

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

Order 112: $C_7:D_8$ ($C_{10}$) x 4, $C_{14}:C_8$ ($C_{10}$), $D_4\times C_{14}$ ($C_{10}$), $C_2\times D_{28}$ ($C_{10}$)

Order 80: $D_4\times C_{10}$ ($D_7$)

Order 70: $C_{70}$ ($D_8$) x 2, $C_{70}$ ($C_2\times D_4$)

Order 56: $C_7\times D_4$ ($C_2\times C_{10}$) x 2, $D_{28}$ ($C_2\times C_{10}$) x 2, $C_7:C_8$ ($C_2\times C_{10}$) x 2, $C_2\times C_{28}$ ($C_2\times C_{10}$)

Order 40: $C_5\times D_4$ ($D_{14}$) x 2, $C_2\times C_{20}$ ($D_{14}$)

Order 35: $C_{35}$ ($C_2\times D_8$)

Order 28: $C_2\times C_{14}$ ($C_5\times D_4$), $C_{28}$ ($C_2^2\times C_{10}$), $C_{28}$ ($C_5\times D_4$)

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: $C_2\times D_4$ ($C_5\times D_7$)

Order 14: $C_{14}$ ($C_5\times D_8$) x 2, $C_{14}$ ($D_4\times C_{10}$)

Order 10: $C_{10}$ ($C_7:D_8$) x 2, $C_{10}$ ($C_{14}:D_4$)

Order 8: $D_4$ ($C_5\times D_{14}$) x 2, $C_2\times C_4$ ($C_5\times D_{14}$)

Order 7: $C_7$ ($C_{10}\times D_8$)

Order 5: $C_5$ ($C_{14}:D_8$)

Order 4: $C_2^2$ ($C_{35}:D_4$), $C_4$ ($C_{10}\times D_{14}$), $C_4$ ($C_{35}:D_4$)

Order 2: $C_2$ ($C_{35}:D_8$) x 2, $C_2$ ($C_{70}:D_4$)

Order 1: $C_1$ ($C_{70}:D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1120: $C_{70}:D_8$ ($C_1$)

Order 560: $C_{35}:D_8$ ($C_2$), $C_{14}:C_{40}$ ($C_2$), $D_4\times C_{70}$ ($C_2$), $C_{10}\times D_{28}$ ($C_2$)

Order 280: $D_4\times C_{35}$ ($C_2^2$), $C_2\times C_{140}$ ($C_2^2$), $C_7:C_{40}$ ($C_2^2$), $C_5\times D_{28}$ ($C_2^2$)

Order 224: $C_{14}:D_8$ ($C_5$)

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

Order 112: $C_{14}:C_8$ ($C_{10}$), $D_4\times C_{14}$ ($C_{10}$), $C_2\times D_{28}$ ($C_{10}$), $C_7:D_8$ ($C_{10}$)

Order 80: $D_4\times C_{10}$ ($D_7$)

Order 70: $C_{70}$ ($D_8$), $C_{70}$ ($C_2\times D_4$)

Order 56: $C_7\times D_4$ ($C_2\times C_{10}$), $C_2\times C_{28}$ ($C_2\times C_{10}$), $D_{28}$ ($C_2\times C_{10}$), $C_7:C_8$ ($C_2\times C_{10}$)

Order 40: $C_2\times C_{20}$ ($D_{14}$), $C_5\times D_4$ ($D_{14}$)

Order 35: $C_{35}$ ($C_2\times D_8$)

Order 28: $C_2\times C_{14}$ ($C_5\times D_4$), $C_{28}$ ($C_2^2\times C_{10}$), $C_{28}$ ($C_5\times D_4$)

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: $C_2\times D_4$ ($C_5\times D_7$)

Order 14: $C_{14}$ ($D_4\times C_{10}$), $C_{14}$ ($C_5\times D_8$)

Order 10: $C_{10}$ ($C_{14}:D_4$), $C_{10}$ ($C_7:D_8$)

Order 8: $D_4$ ($C_5\times D_{14}$), $C_2\times C_4$ ($C_5\times D_{14}$)

Order 7: $C_7$ ($C_{10}\times D_8$)

Order 5: $C_5$ ($C_{14}:D_8$)

Order 4: $C_2^2$ ($C_{35}:D_4$), $C_4$ ($C_{10}\times D_{14}$), $C_4$ ($C_{35}:D_4$)

Order 2: $C_2$ ($C_{35}:D_8$), $C_2$ ($C_{70}:D_4$)

Order 1: $C_1$ ($C_{70}:D_8$)

Series

Derived series

$C_{70}:D_8$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_{70}:D_8$

$\rhd$

$C_{35}:D_8$

$\rhd$

$D_4\times C_{35}$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{70}:D_8$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

$\lhd$

$C_2\times C_{20}$

$\lhd$

$D_4\times C_{10}$

Character theory

Complex character table

See the $220 \times 220$ 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 $40 \times 40$ rational character table.