Abstract group 84.14: $D_{42}$

• Label: 84.14
• Order: \( 2^{2} \cdot 3 \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$D_{42}$
Order:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$D_6\times F_7$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	43	2	2	6	6	12	12	84
Conjugacy classes	1	3	1	1	3	3	6	6	24
Divisions	1	3	1	1	1	1	1	1	10
Autjugacy classes	1	2	1	1	1	1	1	1	9

Dimension	1	2	6	12
Irr. complex chars.	4	20	0	0	24
Irr. rational chars.	4	2	2	2	10

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	12
Arbitrary	2	2	8

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{42}=1, b^{a}=b^{41} \rangle$
Permutation group:	Degree $12$ $\langle(2,3)(4,5)(6,7)(9,10), (11,12), (8,9,10), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 33 & 38 \\ 20 & 33 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 40 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{41})$
Transitive group:	42T11				more information
Direct product:	$C_2$ $\, \times\, $ $D_{21}$
Semidirect product:	$C_7$ $\,\rtimes\,$ $D_6$	$C_6$ $\,\rtimes\,$ $D_7$	$C_{14}$ $\,\rtimes\,$ $S_3$	$C_3$ $\,\rtimes\,$ $D_{14}$	all 6
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}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 104 subgroups in 20 conjugacy classes, 11 normal (9 characteristic).

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

Special subgroups

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

Order 42: $D_{21}$ x 2, $C_{42}$

Order 28: $D_{14}$

Order 21: $C_{21}$

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

Order 12: $D_6$

Order 7: $C_7$

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

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 84: $D_{42}$

Order 42: $C_{42}$, $D_{21}$

Order 28: $D_{14}$

Order 21: $C_{21}$

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

Order 12: $D_6$

Order 7: $C_7$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 84: $D_{42}$ ($C_1$)

Order 42: $D_{21}$ ($C_2$) x 2, $C_{42}$ ($C_2$)

Order 21: $C_{21}$ ($C_2^2$)

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

Order 7: $C_7$ ($D_6$)

Order 6: $C_6$ ($D_7$)

Order 3: $C_3$ ($D_{14}$)

Order 2: $C_2$ ($D_{21}$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 84: $D_{42}$ ($C_1$)

Order 42: $C_{42}$ ($C_2$), $D_{21}$ ($C_2$)

Order 21: $C_{21}$ ($C_2^2$)

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

Order 7: $C_7$ ($D_6$)

Order 6: $C_6$ ($D_7$)

Order 3: $C_3$ ($D_{14}$)

Order 2: $C_2$ ($D_{21}$)

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

Series

Derived series	$D_{42}$	$\rhd$	$C_{21}$	$\rhd$	$C_1$
Chief series	$D_{42}$	$\rhd$	$D_{21}$	$\rhd$	$C_{21}$	$\rhd$	$C_7$	$\rhd$	$C_1$
Lower central series	$D_{42}$	$\rhd$	$C_{21}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 52 larger groups in the database.

This group is a maximal quotient of 44 larger groups in the database.

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	6A	7A	14A	21A	42A
	Size	1	1	21	21	2	2	6	6	12	12
	2 P	1A	1A	1A	1A	3A	3A	7A	7A	21A	21A
	3 P	1A	2A	2B	2C	1A	2A	7A	14A	7A	14A
	7 P	1A	2A	2B	2C	3A	6A	1A	2A	3A	6A
84.14.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
84.14.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
84.14.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
84.14.1d		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
84.14.2a		$2$	$2$	$0$	$0$	$-1$	$-1$	$2$	$2$	$-1$	$-1$
84.14.2b		$2$	$-2$	$0$	$0$	$-1$	$1$	$2$	$-2$	$-1$	$1$
84.14.2c		$6$	$6$	$0$	$0$	$6$	$6$	$-1$	$-1$	$-1$	$-1$
84.14.2d		$6$	$-6$	$0$	$0$	$6$	$-6$	$-1$	$1$	$-1$	$1$
84.14.2e		$12$	$12$	$0$	$0$	$-6$	$-6$	$-2$	$-2$	$1$	$1$
84.14.2f		$12$	$-12$	$0$	$0$	$-6$	$6$	$-2$	$2$	$1$	$-1$