Abstract group 420.25: $C_{15}:D_{14}$

• Label: 420.25
• Order: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 5 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $105$
• Rank: $2$

Group information

Description:	$C_{15}:D_{14}$
Order:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Exponent:	\(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_4\times S_3\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	5	6	7	10	14	15	21	30	35	70	105
Elements	1	31	2	4	14	6	124	18	8	12	56	24	72	48	420
Conjugacy classes	1	3	1	4	1	3	12	3	4	3	4	12	12	12	75
Divisions	1	3	1	1	1	1	3	1	1	1	1	1	1	1	18
Autjugacy classes	1	3	1	1	1	1	3	1	1	1	1	1	1	1	18

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	20	40	15	0	0	0	0	0	75
Irr. rational chars.	4	2	4	2	2	1	2	1	18

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$105$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{105}=[a,b]=1, c^{a}=c^{71}, c^{b}=c^{76} \rangle$
Permutation group:	Degree $15$ $\langle(10,11)(12,13)(14,15), (2,3), (4,5,6,7,8), (1,2,3), (9,10,12,14,15,13,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 29 & 0 \\ 35 & 29 \end{array}\right), \left(\begin{array}{rr} 36 & 45 \\ 35 & 6 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 36 & 35 \\ 35 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/70\Z)$
Direct product:	$C_5$ $\, \times\, $ $S_3$ $\, \times\, $ $D_7$
Semidirect product:	$C_{35}$ $\,\rtimes\,$ $D_6$	$D_{21}$ $\,\rtimes\,$ $C_{10}$	$C_{15}$ $\,\rtimes\,$ $D_{14}$	$C_{105}$ $\,\rtimes\,$ $C_2^2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 184 subgroups in 40 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_5$	$G/Z \simeq$ $S_3\times D_7$
Commutator:	$G' \simeq$ $C_{21}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{15}:D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{105}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{15}:D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{105}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 420: $C_{15}:D_{14}$

Order 210: $C_5\times D_{21}$, $S_3\times C_{35}$, $D_7\times C_{15}$

Order 140: $C_5\times D_{14}$

Order 105: $C_{105}$

Order 84: $S_3\times D_7$

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

Order 60: $S_3\times C_{10}$

Order 42: $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 35: $C_{35}$

Order 30: $C_5\times S_3$ x 2, $C_{30}$

Order 28: $D_{14}$

Order 21: $C_{21}$

Order 20: $C_2\times C_{10}$

Order 15: $C_{15}$

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

Order 12: $D_6$

Order 10: $C_{10}$ x 3

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 420: $C_{15}:D_{14}$

Order 210: $C_5\times D_{21}$, $S_3\times C_{35}$, $D_7\times C_{15}$

Order 140: $C_5\times D_{14}$

Order 105: $C_{105}$

Order 84: $S_3\times D_7$

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

Order 60: $S_3\times C_{10}$

Order 42: $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 35: $C_{35}$

Order 30: $C_5\times S_3$ x 2, $C_{30}$

Order 28: $D_{14}$

Order 21: $C_{21}$

Order 20: $C_2\times C_{10}$

Order 15: $C_{15}$

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

Order 12: $D_6$

Order 10: $C_{10}$ x 3

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 420: $C_{15}:D_{14}$ ($C_1$)

Order 210: $C_5\times D_{21}$ ($C_2$), $S_3\times C_{35}$ ($C_2$), $D_7\times C_{15}$ ($C_2$)

Order 105: $C_{105}$ ($C_2^2$)

Order 84: $S_3\times D_7$ ($C_5$)

Order 70: $C_5\times D_7$ ($S_3$)

Order 42: $D_{21}$ ($C_{10}$), $C_3\times D_7$ ($C_{10}$), $S_3\times C_7$ ($C_{10}$)

Order 35: $C_{35}$ ($D_6$)

Order 30: $C_5\times S_3$ ($D_7$)

Order 21: $C_{21}$ ($C_2\times C_{10}$)

Order 15: $C_{15}$ ($D_{14}$)

Order 14: $D_7$ ($C_5\times S_3$)

Order 7: $C_7$ ($S_3\times C_{10}$)

Order 6: $S_3$ ($C_5\times D_7$)

Order 5: $C_5$ ($S_3\times D_7$)

Order 3: $C_3$ ($C_5\times D_{14}$)

Order 1: $C_1$ ($C_{15}:D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 420: $C_{15}:D_{14}$ ($C_1$)

Order 210: $C_5\times D_{21}$ ($C_2$), $S_3\times C_{35}$ ($C_2$), $D_7\times C_{15}$ ($C_2$)

Order 105: $C_{105}$ ($C_2^2$)

Order 84: $S_3\times D_7$ ($C_5$)

Order 70: $C_5\times D_7$ ($S_3$)

Order 42: $D_{21}$ ($C_{10}$), $C_3\times D_7$ ($C_{10}$), $S_3\times C_7$ ($C_{10}$)

Order 35: $C_{35}$ ($D_6$)

Order 30: $C_5\times S_3$ ($D_7$)

Order 21: $C_{21}$ ($C_2\times C_{10}$)

Order 15: $C_{15}$ ($D_{14}$)

Order 14: $D_7$ ($C_5\times S_3$)

Order 7: $C_7$ ($S_3\times C_{10}$)

Order 6: $S_3$ ($C_5\times D_7$)

Order 5: $C_5$ ($S_3\times D_7$)

Order 3: $C_3$ ($C_5\times D_{14}$)

Order 1: $C_1$ ($C_{15}:D_{14}$)

Series

Derived series

$C_{15}:D_{14}$

$\rhd$

$C_{21}$

$\rhd$

$C_1$

Chief series

$C_{15}:D_{14}$

$\rhd$

$S_3\times C_{35}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{15}:D_{14}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_5$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $75 \times 75$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.