Abstract group 1008.598: $D_{42}:D_6$

• Label: 1008.598
• 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.: $17$
• Trans deg.: $84$
• Rank: $3$

Group information

Description:	$D_{42}:D_6$
Order:	\(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^3\times S_3^2\times F_7$, 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	207	8	48	216	6	96	150	48	36	120	72	1008
Conjugacy classes	1	7	3	2	9	3	2	9	9	3	15	6	69
Divisions	1	7	3	2	9	1	2	3	3	1	4	1	37
Autjugacy classes	1	6	3	2	8	1	2	3	3	1	4	1	35

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	8	22	33	0	6	0	0	69
Irr. rational chars.	8	10	6	4	0	5	4	37

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$84$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	24
Arbitrary	6	6	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{84}=d^{3}=[a,d]=[b,d]=1, b^{a}=bc^{42}, c^{a}=c^{71}, c^{b}=c^{55}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $17$ $\langle(2,3)(4,5)(6,7)(12,13), (9,10)(11,12)(13,14), (11,13,14,12)(16,17), (11,14)(12,13), (8,9,10), (15,16,17), (1,2,4,6,7,5,3)\rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{42}$ $\,\rtimes\,$ $D_6$	$D_{14}$ $\,\rtimes\,$ $S_3^2$	$D_6$ $\,\rtimes\,$ $(S_3\times D_7)$	$C_7$ $\,\rtimes\,$ $(D_6:D_6)$	all 34
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{14}$ . $(S_3\times D_6)$	$C_6$ . $(S_3\times D_{14})$ (2)	$C_{42}$ . $(C_2\times D_6)$ (2)	$C_2$ . $(D_7\times S_3^2)$	all 7

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 3212 subgroups in 248 conjugacy classes, 48 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_7\times S_3^2$
Commutator:	$G' \simeq$ $C_3\times C_{42}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_7\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $D_{42}:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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: $D_{42}:D_6$

Order 504: $D_{42}:C_6$, $C_{42}.D_6$, $S_3\times D_{42}$, $C_{42}:D_6$, $C_{21}:D_{12}$, $C_6.D_{42}$, $C_{21}:D_{12}$

Order 336: $D_6:D_{14}$, $C_{12}:D_{14}$

Order 252: $S_3\times D_{21}$ x 2, $C_{21}:D_6$ x 2, $C_3:D_{42}$, $C_{42}:S_3$, $C_3\times D_{42}$, $S_3\times C_{42}$, $C_3^2\times D_{14}$, $C_{21}:C_{12}$, $C_3:C_{84}$

Order 168: $S_3\times D_{14}$ x 4, $C_{21}:D_4$ x 2, $C_2\times D_{42}$, $C_4\times D_{21}$, $D_6:C_{14}$, $C_7\times D_{12}$, $C_{21}:D_4$, $C_3\times D_{28}$, $C_7:D_{12}$, $C_3:D_{28}$, $D_{21}:C_4$

Order 144: $D_6:D_6$

Order 126: $C_3:D_{21}$ x 2, $C_3\times C_{42}$, $C_{21}:S_3$, $C_3\times D_{21}$, $S_3\times C_{21}$, $C_3^2\times D_7$

Order 112: $D_4\times D_7$

Order 84: $S_3\times D_7$ x 10, $D_{42}$ x 6, $S_3\times C_{14}$ x 4, $C_3\times D_{14}$ x 4, $C_{84}$, $C_{21}:C_4$, $C_7:C_{12}$, $C_3:C_{28}$, $C_2\times C_{42}$

Order 72: $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6:D_6$, $S_3\times D_6$, $C_6.D_6$

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

Order 56: $C_7:D_4$ x 2, $C_2\times D_{14}$ x 2, $C_7\times D_4$, $D_{28}$, $C_4\times D_7$

Order 48: $S_3\times D_4$ x 2

Order 42: $D_{21}$ x 7, $C_3\times D_7$ x 5, $S_3\times C_7$ x 5, $C_{42}$ x 4

Order 36: $C_6:S_3$ x 4, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $S_3^2$ x 2, $C_6^2$

Order 28: $D_{14}$ x 7, $C_2\times C_{14}$ x 2, $C_{28}$, $C_7:C_4$

Order 24: $C_2\times D_6$ x 5, $C_3:D_4$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2

Order 21: $C_{21}$ x 3

Order 18: $C_3:S_3$ x 3, $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$

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

Order 12: $D_6$ x 20, $C_2\times C_6$ x 5, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

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

Order 7: $C_7$

Order 6: $S_3$ x 12, $C_6$ x 9

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1008: $D_{42}:D_6$

Order 504: $D_{42}:C_6$, $C_{42}.D_6$, $S_3\times D_{42}$, $C_{42}:D_6$, $C_{21}:D_{12}$, $C_6.D_{42}$, $C_{21}:D_{12}$

Order 336: $D_6:D_{14}$, $C_{12}:D_{14}$

Order 252: $C_3:D_{42}$, $C_{42}:S_3$, $C_3\times D_{42}$, $S_3\times C_{42}$, $C_3^2\times D_{14}$, $S_3\times D_{21}$, $C_{21}:D_6$, $C_{21}:C_{12}$, $C_3:C_{84}$

Order 168: $S_3\times D_{14}$ x 4, $C_{21}:D_4$ x 2, $C_2\times D_{42}$, $C_4\times D_{21}$, $D_6:C_{14}$, $C_7\times D_{12}$, $C_{21}:D_4$, $C_3\times D_{28}$, $C_7:D_{12}$, $C_3:D_{28}$, $D_{21}:C_4$

Order 144: $D_6:D_6$

Order 126: $C_3\times C_{42}$, $C_3:D_{21}$, $C_{21}:S_3$, $C_3\times D_{21}$, $S_3\times C_{21}$, $C_3^2\times D_7$

Order 112: $D_4\times D_7$

Order 84: $D_{42}$ x 5, $S_3\times D_7$ x 4, $S_3\times C_{14}$ x 4, $C_3\times D_{14}$ x 4, $C_{84}$, $C_{21}:C_4$, $C_7:C_{12}$, $C_3:C_{28}$, $C_2\times C_{42}$

Order 72: $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6:D_6$, $S_3\times D_6$, $C_6.D_6$

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

Order 56: $C_7:D_4$ x 2, $C_2\times D_{14}$ x 2, $C_7\times D_4$, $D_{28}$, $C_4\times D_7$

Order 48: $S_3\times D_4$ x 2

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

Order 36: $C_6:S_3$ x 3, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$, $S_3^2$

Order 28: $D_{14}$ x 5, $C_2\times C_{14}$ x 2, $C_{28}$, $C_7:C_4$

Order 24: $C_2\times D_6$ x 5, $C_3:D_4$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2

Order 21: $C_{21}$ x 3

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

Order 16: $C_2\times D_4$

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

Order 12: $D_6$ x 13, $C_2\times C_6$ x 5, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

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

Order 7: $C_7$

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

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 504: $D_{42}:C_6$ ($C_2$), $C_{42}.D_6$ ($C_2$), $S_3\times D_{42}$ ($C_2$), $C_{42}:D_6$ ($C_2$), $C_{21}:D_{12}$ ($C_2$), $C_6.D_{42}$ ($C_2$), $C_{21}:D_{12}$ ($C_2$)

Order 252: $C_3:D_{42}$ ($C_2^2$), $C_{42}:S_3$ ($C_2^2$), $C_3\times D_{42}$ ($C_2^2$), $S_3\times C_{42}$ ($C_2^2$), $C_3^2\times D_{14}$ ($C_2^2$), $C_{21}:C_{12}$ ($C_2^2$), $C_3:C_{84}$ ($C_2^2$)

Order 168: $C_3:D_{28}$ ($S_3$), $C_{21}:D_4$ ($S_3$)

Order 126: $C_3:D_{21}$ ($D_4$) x 2, $C_3\times C_{42}$ ($C_2^3$)

Order 84: $C_3\times D_{14}$ ($D_6$) 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: $C_3:D_{12}$ ($D_7$)

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

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

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

Order 28: $D_{14}$ ($S_3^2$)

Order 21: $C_{21}$ ($S_3\times D_4$) 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: $D_6$ ($S_3\times D_7$), $C_3:C_4$ ($S_3\times D_7$)

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

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

Order 6: $C_6$ ($S_3\times D_{14}$) x 2

Order 3: $C_3$ ($D_6:D_{14}$), $C_3$ ($C_{12}:D_{14}$)

Order 2: $C_2$ ($D_7\times S_3^2$)

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 504: $D_{42}:C_6$ ($C_2$), $C_{42}.D_6$ ($C_2$), $S_3\times D_{42}$ ($C_2$), $C_{42}:D_6$ ($C_2$), $C_{21}:D_{12}$ ($C_2$), $C_6.D_{42}$ ($C_2$), $C_{21}:D_{12}$ ($C_2$)

Order 252: $C_3:D_{42}$ ($C_2^2$), $C_{42}:S_3$ ($C_2^2$), $C_3\times D_{42}$ ($C_2^2$), $S_3\times C_{42}$ ($C_2^2$), $C_3^2\times D_{14}$ ($C_2^2$), $C_{21}:C_{12}$ ($C_2^2$), $C_3:C_{84}$ ($C_2^2$)

Order 168: $C_3:D_{28}$ ($S_3$), $C_{21}:D_4$ ($S_3$)

Order 126: $C_3\times C_{42}$ ($C_2^3$), $C_3:D_{21}$ ($D_4$)

Order 84: $C_3\times D_{14}$ ($D_6$) 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: $C_3:D_{12}$ ($D_7$)

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

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

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

Order 28: $D_{14}$ ($S_3^2$)

Order 21: $C_{21}$ ($S_3\times D_4$) 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: $D_6$ ($S_3\times D_7$), $C_3:C_4$ ($S_3\times D_7$)

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

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

Order 6: $C_6$ ($S_3\times D_{14}$) x 2

Order 3: $C_3$ ($D_6:D_{14}$), $C_3$ ($C_{12}:D_{14}$)

Order 2: $C_2$ ($D_7\times S_3^2$)

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

Series

Derived series

$D_{42}:D_6$

$\rhd$

$C_3\times C_{42}$

$\rhd$

$C_1$

Chief series

$D_{42}:D_6$

$\rhd$

$C_{42}.D_6$

$\rhd$

$S_3\times C_{42}$

$\rhd$

$C_3\times C_{42}$

$\rhd$

$C_3\times C_{21}$

$\rhd$

$C_{21}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{42}:D_6$

$\rhd$

$C_3\times C_{42}$

$\rhd$

$C_3\times C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $69 \times 69$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $37 \times 37$ rational character table.