Abstract group 1008.787: $D_6.D_{42}$

• Label: 1008.787
• Order: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $21$
• Trans deg.: $168$
• Rank: $3$

Group information

Description:	$D_6.D_{42}$
Order:	\(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$D_6\times C_2^2\times S_3\times F_7$, of order \(12096\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	51	8	300	120	6	96	54	48	36	216	72	1008
Conjugacy classes	1	4	3	5	9	3	2	9	15	3	45	6	105
Divisions	1	4	3	4	9	1	2	3	3	1	8	1	40
Autjugacy classes	1	4	3	4	8	1	2	3	3	1	7	1	38

Dimension	1	2	4	6	12	24
Irr. complex chars.	8	46	51	0	0	0	105
Irr. rational chars.	8	8	7	4	7	6	40

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$168$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	8	14

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{42}=c^{4}=d^{3}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{41}c^{2}, c^{b}=c^{3}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $21$ $\langle(1,2)(3,5)(4,7)(6,8)(13,14)(15,16)(17,18)(20,21), (1,3,4,8)(2,5,7,6)(10,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_7$ $\,\rtimes\,$ $(D_6.D_6)$	$(D_6:C_{14})$ $\,\rtimes\,$ $S_3$	$(C_{21}:D_4)$ $\,\rtimes\,$ $S_3$	$C_{21}$ $\,\rtimes\,$ $(D_4:S_3)$ (2)	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{42}$ . $D_6$	$D_6$ . $D_{42}$	$C_6^2$ . $D_{14}$	$(S_3\times C_{14})$ . $D_6$	all 27

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 1628 subgroups in 176 conjugacy classes, 50 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times D_{42}$
Commutator:	$G' \simeq$ $C_3\times C_{42}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3\times D_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_6.D_{42}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
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 1008: $D_6.D_{42}$

Order 504: $C_6.D_{42}$, $D_{42}:C_6$, $D_6:C_{42}$, $C_6.D_{42}$, $D_6:D_{21}$, $C_6.D_{42}$, $C_6.D_{42}$

Order 336: $D_4:D_{21}$, $D_6.D_{14}$

Order 252: $C_6.D_{21}$ x 2, $C_6\times C_{42}$, $C_3\times D_{42}$, $S_3\times C_{42}$, $C_{21}:C_{12}$, $C_3:C_{84}$

Order 168: $C_{42}:C_4$ x 4, $C_{21}:D_4$ x 2, $D_4\times C_{21}$, $C_4\times D_{21}$, $C_{21}:Q_8$, $D_6:C_{14}$, $C_{21}:D_4$, $C_{21}:Q_8$, $C_{21}:D_4$, $C_6.D_{14}$, $C_6.D_{14}$

Order 144: $D_6.D_6$

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

Order 112: $D_4:D_7$

Order 84: $C_{21}:C_4$ x 7, $C_2\times C_{42}$ x 4, $C_{84}$, $C_7:C_{12}$, $C_3:C_{28}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 72: $C_6\wr C_2$ x 2, $C_6.D_6$ x 2, $C_6.D_6$, $C_3^2:Q_8$, $D_6:S_3$

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

Order 56: $C_7:D_4$ x 2, $C_{14}:C_4$ x 2, $C_7\times D_4$, $C_4\times D_7$, $C_7:Q_8$

Order 48: $D_4:S_3$ x 2

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

Order 36: $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 28: $C_7:C_4$ x 3, $C_2\times C_{14}$ x 2, $D_{14}$, $C_{28}$

Order 24: $C_6:C_4$ x 5, $C_3:D_4$ x 4, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2

Order 21: $C_{21}$ x 3

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

Order 16: $D_4:C_2$

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

Order 12: $C_3:C_4$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

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

Order 7: $C_7$

Order 6: $C_6$ x 9, $S_3$ x 2

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1008: $D_6.D_{42}$

Order 504: $C_6.D_{42}$, $D_{42}:C_6$, $D_6:C_{42}$, $C_6.D_{42}$, $D_6:D_{21}$, $C_6.D_{42}$, $C_6.D_{42}$

Order 336: $D_4:D_{21}$, $D_6.D_{14}$

Order 252: $C_6.D_{21}$ x 2, $C_6\times C_{42}$, $C_3\times D_{42}$, $S_3\times C_{42}$, $C_{21}:C_{12}$, $C_3:C_{84}$

Order 168: $C_{42}:C_4$ x 4, $C_{21}:D_4$ x 2, $D_4\times C_{21}$, $C_4\times D_{21}$, $C_{21}:Q_8$, $D_6:C_{14}$, $C_{21}:D_4$, $C_{21}:Q_8$, $C_{21}:D_4$, $C_6.D_{14}$, $C_6.D_{14}$

Order 144: $D_6.D_6$

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

Order 112: $D_4:D_7$

Order 84: $C_{21}:C_4$ x 7, $C_2\times C_{42}$ x 4, $C_{84}$, $C_7:C_{12}$, $C_3:C_{28}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 72: $C_6\wr C_2$ x 2, $C_6.D_6$ x 2, $C_6.D_6$, $C_3^2:Q_8$, $D_6:S_3$

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

Order 56: $C_7:D_4$ x 2, $C_{14}:C_4$ x 2, $C_7\times D_4$, $C_4\times D_7$, $C_7:Q_8$

Order 48: $D_4:S_3$ x 2

Order 42: $C_{42}$ x 7, $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 36: $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 28: $C_7:C_4$ x 3, $C_2\times C_{14}$ x 2, $D_{14}$, $C_{28}$

Order 24: $C_6:C_4$ x 5, $C_3:D_4$ x 4, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2

Order 21: $C_{21}$ x 3

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

Order 16: $D_4:C_2$

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

Order 12: $C_3:C_4$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

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

Order 7: $C_7$

Order 6: $C_6$ x 8, $S_3$ x 2

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1008: $D_6.D_{42}$ ($C_1$)

Order 504: $C_6.D_{42}$ ($C_2$), $D_{42}:C_6$ ($C_2$), $D_6:C_{42}$ ($C_2$), $C_6.D_{42}$ ($C_2$), $D_6:D_{21}$ ($C_2$), $C_6.D_{42}$ ($C_2$), $C_6.D_{42}$ ($C_2$)

Order 252: $C_6.D_{21}$ ($C_2^2$) x 2, $C_6\times C_{42}$ ($C_2^2$), $C_3\times D_{42}$ ($C_2^2$), $S_3\times C_{42}$ ($C_2^2$), $C_{21}:C_{12}$ ($C_2^2$), $C_3:C_{84}$ ($C_2^2$)

Order 168: $C_{21}:D_4$ ($S_3$), $D_6:C_{14}$ ($S_3$)

Order 126: $C_3\times C_{42}$ ($C_2^3$)

Order 84: $C_2\times C_{42}$ ($D_6$) x 2, $C_{21}:C_4$ ($D_6$), $C_3:C_{28}$ ($D_6$), $D_{42}$ ($D_6$), $S_3\times C_{14}$ ($D_6$)

Order 72: $C_6\wr C_2$ ($D_7$)

Order 63: $C_3\times C_{21}$ ($D_4:C_2$)

Order 42: $C_{42}$ ($C_2\times D_6$) x 2

Order 36: $C_3:C_{12}$ ($D_{14}$), $C_6^2$ ($D_{14}$), $C_6\times S_3$ ($D_{14}$)

Order 28: $C_2\times C_{14}$ ($S_3^2$)

Order 24: $C_3:D_4$ ($D_{21}$)

Order 21: $C_{21}$ ($D_4:S_3$) x 2

Order 18: $C_3\times C_6$ ($C_2\times D_{14}$)

Order 14: $C_{14}$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($S_3\times D_7$), $C_2\times C_6$ ($D_{42}$), $D_6$ ($D_{42}$), $C_3:C_4$ ($D_{42}$)

Order 9: $C_3^2$ ($D_4:D_7$)

Order 7: $C_7$ ($D_6.D_6$)

Order 6: $C_6$ ($C_2\times D_{42}$), $C_6$ ($S_3\times D_{14}$)

Order 4: $C_2^2$ ($S_3\times D_{21}$)

Order 3: $C_3$ ($D_4:D_{21}$), $C_3$ ($D_6.D_{14}$)

Order 2: $C_2$ ($S_3\times D_{42}$)

Order 1: $C_1$ ($D_6.D_{42}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1008: $D_6.D_{42}$ ($C_1$)

Order 504: $C_6.D_{42}$ ($C_2$), $D_{42}:C_6$ ($C_2$), $D_6:C_{42}$ ($C_2$), $C_6.D_{42}$ ($C_2$), $D_6:D_{21}$ ($C_2$), $C_6.D_{42}$ ($C_2$), $C_6.D_{42}$ ($C_2$)

Order 252: $C_6.D_{21}$ ($C_2^2$) x 2, $C_6\times C_{42}$ ($C_2^2$), $C_3\times D_{42}$ ($C_2^2$), $S_3\times C_{42}$ ($C_2^2$), $C_{21}:C_{12}$ ($C_2^2$), $C_3:C_{84}$ ($C_2^2$)

Order 168: $C_{21}:D_4$ ($S_3$), $D_6:C_{14}$ ($S_3$)

Order 126: $C_3\times C_{42}$ ($C_2^3$)

Order 84: $C_2\times C_{42}$ ($D_6$) x 2, $C_{21}:C_4$ ($D_6$), $C_3:C_{28}$ ($D_6$), $D_{42}$ ($D_6$), $S_3\times C_{14}$ ($D_6$)

Order 72: $C_6\wr C_2$ ($D_7$)

Order 63: $C_3\times C_{21}$ ($D_4:C_2$)

Order 42: $C_{42}$ ($C_2\times D_6$) x 2

Order 36: $C_3:C_{12}$ ($D_{14}$), $C_6^2$ ($D_{14}$), $C_6\times S_3$ ($D_{14}$)

Order 28: $C_2\times C_{14}$ ($S_3^2$)

Order 24: $C_3:D_4$ ($D_{21}$)

Order 21: $C_{21}$ ($D_4:S_3$) x 2

Order 18: $C_3\times C_6$ ($C_2\times D_{14}$)

Order 14: $C_{14}$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($S_3\times D_7$), $C_2\times C_6$ ($D_{42}$), $D_6$ ($D_{42}$), $C_3:C_4$ ($D_{42}$)

Order 9: $C_3^2$ ($D_4:D_7$)

Order 7: $C_7$ ($D_6.D_6$)

Order 6: $C_6$ ($C_2\times D_{42}$), $C_6$ ($S_3\times D_{14}$)

Order 4: $C_2^2$ ($S_3\times D_{21}$)

Order 3: $C_3$ ($D_4:D_{21}$), $C_3$ ($D_6.D_{14}$)

Order 2: $C_2$ ($S_3\times D_{42}$)

Order 1: $C_1$ ($D_6.D_{42}$)

Series

Derived series

$D_6.D_{42}$

$\rhd$

$C_3\times C_{42}$

$\rhd$

$C_1$

Chief series

$D_6.D_{42}$

$\rhd$

$C_6.D_{42}$

$\rhd$

$C_3:C_{84}$

$\rhd$

$C_3:C_{28}$

$\rhd$

$C_3:C_4$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_6.D_{42}$

$\rhd$

$C_3\times C_{42}$

$\rhd$

$C_3\times C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Character theory

Complex character table

See the $105 \times 105$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

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