Abstract group 504.192: $S_3\times D_{42}$

• Label: 504.192
• Order: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $84$
• Rank: $3$

Group information

Description:	$S_3\times D_{42}$
Order:	\(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$D_6^2\times F_7$, of order \(6048\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	175	8	104	6	42	48	120	504
Conjugacy classes	1	7	3	7	3	9	15	27	72
Divisions	1	7	3	7	1	3	3	5	30
Autjugacy classes	1	4	3	5	1	2	3	4	23

Dimension	1	2	4	6	12	24
Irr. complex chars.	8	44	20	0	0	0	72
Irr. rational chars.	8	8	2	4	6	2	30

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$84$
Rank:	$3$
Inequivalent generating triples:	$5376$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	24
Arbitrary	4	4	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{6}=c^{42}=[a,c]=1, b^{a}=b^{5}, c^{b}=c^{41} \rangle$
Permutation group:	Degree $15$ $\langle(1,2)(3,5)(4,6)(8,9)(10,11)(12,13), (1,2)(3,6)(4,5)(14,15), (14,15), (1,3,4)(2,5,6), (1,4,3)(2,5,6), (7,8,10,12,13,11,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 5 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 21 & 13 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 21 & 22 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $D_{21}$
Semidirect product:	$C_{42}$ $\,\rtimes\,$ $D_6$ (2)	$C_6$ $\,\rtimes\,$ $D_{42}$	$C_{14}$ $\,\rtimes\,$ $S_3^2$	$(S_3\times C_7)$ $\,\rtimes\,$ $D_6$ (2)	all 22
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}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1432 subgroups in 138 conjugacy classes, 42 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times D_{21}$
Commutator:	$G' \simeq$ $C_3\times C_{21}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times D_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $S_3\times D_{42}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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 504: $S_3\times D_{42}$

Order 252: $S_3\times D_{21}$ x 4, $C_3:D_{42}$, $C_3\times D_{42}$, $S_3\times C_{42}$

Order 168: $C_2\times D_{42}$, $S_3\times D_{14}$

Order 126: $C_3:D_{21}$ x 2, $C_3\times D_{21}$ x 2, $S_3\times C_{21}$ x 2, $C_3\times C_{42}$

Order 84: $D_{42}$ x 8, $S_3\times D_7$ x 4, $C_2\times C_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 72: $S_3\times D_6$

Order 63: $C_3\times C_{21}$

Order 56: $C_2\times D_{14}$

Order 42: $D_{21}$ x 8, $C_{42}$ x 5, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2

Order 36: $S_3^2$ x 4, $C_6\times S_3$ x 2, $C_6:S_3$

Order 28: $D_{14}$ x 6, $C_2\times C_{14}$

Order 24: $C_2\times D_6$ x 2

Order 21: $C_{21}$ x 3

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $D_6$ x 13, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 7: $C_7$

Order 6: $S_3$ x 10, $C_6$ x 7

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 504: $S_3\times D_{42}$

Order 252: $C_3:D_{42}$, $C_3\times D_{42}$, $S_3\times C_{42}$, $S_3\times D_{21}$

Order 168: $C_2\times D_{42}$, $S_3\times D_{14}$

Order 126: $C_3\times C_{42}$, $C_3:D_{21}$, $C_3\times D_{21}$, $S_3\times C_{21}$

Order 84: $D_{42}$ x 5, $S_3\times D_7$, $C_2\times C_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 72: $S_3\times D_6$

Order 63: $C_3\times C_{21}$

Order 56: $C_2\times D_{14}$

Order 42: $C_{42}$ x 4, $D_{21}$ x 4, $C_3\times D_7$, $S_3\times C_7$

Order 36: $C_6\times S_3$ x 2, $C_6:S_3$, $S_3^2$

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

Order 24: $C_2\times D_6$ x 2

Order 21: $C_{21}$ x 3

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$, $C_3:S_3$

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

Order 12: $D_6$ x 7, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 5, $S_3$ x 5

Order 4: $C_2^2$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 504: $S_3\times D_{42}$ ($C_1$)

Order 252: $S_3\times D_{21}$ ($C_2$) x 4, $C_3:D_{42}$ ($C_2$), $C_3\times D_{42}$ ($C_2$), $S_3\times C_{42}$ ($C_2$)

Order 126: $C_3:D_{21}$ ($C_2^2$) x 2, $C_3\times D_{21}$ ($C_2^2$) x 2, $S_3\times C_{21}$ ($C_2^2$) x 2, $C_3\times C_{42}$ ($C_2^2$)

Order 84: $D_{42}$ ($S_3$), $S_3\times C_{14}$ ($S_3$)

Order 63: $C_3\times C_{21}$ ($C_2^3$)

Order 42: $C_{42}$ ($D_6$) x 2, $D_{21}$ ($D_6$) x 2, $S_3\times C_7$ ($D_6$) x 2

Order 36: $C_6\times S_3$ ($D_7$)

Order 21: $C_{21}$ ($C_2\times D_6$) x 2

Order 18: $C_3\times S_3$ ($D_{14}$) x 2, $C_3\times C_6$ ($D_{14}$)

Order 14: $C_{14}$ ($S_3^2$)

Order 12: $D_6$ ($D_{21}$)

Order 9: $C_3^2$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($S_3\times D_6$)

Order 6: $S_3$ ($D_{42}$) x 2, $C_6$ ($S_3\times D_7$), $C_6$ ($D_{42}$)

Order 3: $C_3$ ($C_2\times D_{42}$), $C_3$ ($S_3\times D_{14}$)

Order 2: $C_2$ ($S_3\times D_{21}$)

Order 1: $C_1$ ($S_3\times D_{42}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 504: $S_3\times D_{42}$ ($C_1$)

Order 252: $C_3:D_{42}$ ($C_2$), $C_3\times D_{42}$ ($C_2$), $S_3\times C_{42}$ ($C_2$), $S_3\times D_{21}$ ($C_2$)

Order 126: $C_3\times C_{42}$ ($C_2^2$), $C_3:D_{21}$ ($C_2^2$), $C_3\times D_{21}$ ($C_2^2$), $S_3\times C_{21}$ ($C_2^2$)

Order 84: $D_{42}$ ($S_3$), $S_3\times C_{14}$ ($S_3$)

Order 63: $C_3\times C_{21}$ ($C_2^3$)

Order 42: $C_{42}$ ($D_6$) x 2, $D_{21}$ ($D_6$), $S_3\times C_7$ ($D_6$)

Order 36: $C_6\times S_3$ ($D_7$)

Order 21: $C_{21}$ ($C_2\times D_6$) x 2

Order 18: $C_3\times C_6$ ($D_{14}$), $C_3\times S_3$ ($D_{14}$)

Order 14: $C_{14}$ ($S_3^2$)

Order 12: $D_6$ ($D_{21}$)

Order 9: $C_3^2$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($S_3\times D_6$)

Order 6: $C_6$ ($S_3\times D_7$), $C_6$ ($D_{42}$), $S_3$ ($D_{42}$)

Order 3: $C_3$ ($C_2\times D_{42}$), $C_3$ ($S_3\times D_{14}$)

Order 2: $C_2$ ($S_3\times D_{21}$)

Order 1: $C_1$ ($S_3\times D_{42}$)

Series

Derived series

$S_3\times D_{42}$

$\rhd$

$C_3\times C_{21}$

$\rhd$

$C_1$

Chief series

$S_3\times D_{42}$

$\rhd$

$S_3\times C_{42}$

$\rhd$

$C_3\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$S_3\times D_{42}$

$\rhd$

$C_3\times C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 27 larger groups in the database.

This group is a maximal quotient of 36 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.