Abstract group 672.1240: $Q_8\times D_{42}$

• Label: 672.1240
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $336$
• Rank: $4$

Group information

Description:	$Q_8\times D_{42}$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^4.(C_3\times C_{42}).C_6.C_2^4$, of order \(193536\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	87	2	264	6	6	24	18	12	72	36	144	672
Conjugacy classes	1	7	1	12	3	3	6	9	6	18	18	36	120
Divisions	1	7	1	12	3	1	6	3	1	6	3	6	50
Autjugacy classes	1	3	1	2	2	1	1	2	1	1	2	1	18

Dimension	1	2	4	6	12	24
Irr. complex chars.	16	84	20	0	0	0	120
Irr. rational chars.	16	12	2	8	10	2	50

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$336$
Rank:	$4$
Inequivalent generating quadruples:	$311220$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=d^{84}=[a,b]=[a,c]=[b,c]=[b,d]=1, c^{2}=d^{42}, d^{a}=d^{41}, d^{c}=d^{43} \rangle$
Permutation group:	Degree $20$ $\langle(2,3)(4,5)(6,7)(9,10), (11,12)(13,15)(14,16)(17,18)(19,20), (11,13,12,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 28 \\ 28 & 29 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 29 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 21 & 22 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	$C_2$ $\, \times\, $ $Q_8$ $\, \times\, $ $D_{21}$
Semidirect product:	$(C_7\times Q_8)$ $\,\rtimes\,$ $D_6$	$(C_6\times Q_8)$ $\,\rtimes\,$ $D_7$	$C_7$ $\,\rtimes\,$ $(Q_8\times D_6)$	$C_6$ $\,\rtimes\,$ $(Q_8\times D_7)$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{42}$ . $C_2^3$	$C_{84}$ . $C_2^3$	$C_{42}$ . $C_2^4$	$(C_4\times D_{42})$ . $C_2$	all 22

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 1964 subgroups in 312 conjugacy classes, 135 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_{42}$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times D_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $Q_8\times D_{42}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times Q_8$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 672: $Q_8\times D_{42}$

Order 336: $Q_8\times D_{21}$ x 8, $C_4\times D_{42}$ x 3, $C_{42}:Q_8$ x 3, $Q_8\times C_{42}$

Order 224: $Q_8\times D_{14}$

Order 168: $C_4\times D_{21}$ x 12, $C_{21}:Q_8$ x 12, $Q_8\times C_{21}$ x 4, $C_2\times C_{84}$ x 3, $C_{42}:C_4$ x 3, $C_2\times D_{42}$

Order 112: $Q_8\times D_7$ x 8, $C_4\times D_{14}$ x 3, $C_{14}:Q_8$ x 3, $Q_8\times C_{14}$

Order 96: $Q_8\times D_6$

Order 84: $C_{84}$ x 6, $C_{21}:C_4$ x 6, $D_{42}$ x 6, $C_2\times C_{42}$

Order 56: $C_4\times D_7$ x 12, $C_7:Q_8$ x 12, $C_7\times Q_8$ x 4, $C_2\times C_{28}$ x 3, $C_{14}:C_4$ x 3, $C_2\times D_{14}$

Order 48: $S_3\times Q_8$ x 8, $C_4\times D_6$ x 3, $C_6:Q_8$ x 3, $C_6\times Q_8$

Order 42: $D_{21}$ x 4, $C_{42}$ x 3

Order 32: $C_2^2\times Q_8$

Order 28: $D_{14}$ x 6, $C_{28}$ x 6, $C_7:C_4$ x 6, $C_2\times C_{14}$

Order 24: $C_4\times S_3$ x 12, $C_3:Q_8$ x 12, $C_3\times Q_8$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_2\times D_6$

Order 21: $C_{21}$

Order 16: $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 3

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

Order 12: $D_6$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 6, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 18, $Q_8$ x 16, $C_2^3$

Order 7: $C_7$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $Q_8\times D_{42}$

Order 336: $Q_8\times C_{42}$, $Q_8\times D_{21}$, $C_4\times D_{42}$, $C_{42}:Q_8$

Order 224: $Q_8\times D_{14}$

Order 168: $C_2\times D_{42}$, $Q_8\times C_{21}$, $C_2\times C_{84}$, $C_{42}:C_4$, $C_4\times D_{21}$, $C_{21}:Q_8$

Order 112: $Q_8\times C_{14}$, $Q_8\times D_7$, $C_4\times D_{14}$, $C_{14}:Q_8$

Order 96: $Q_8\times D_6$

Order 84: $D_{42}$ x 2, $C_{84}$, $C_{21}:C_4$, $C_2\times C_{42}$

Order 56: $C_2\times C_{28}$, $C_{14}:C_4$, $C_4\times D_7$, $C_7:Q_8$, $C_2\times D_{14}$, $C_7\times Q_8$

Order 48: $C_6\times Q_8$, $S_3\times Q_8$, $C_4\times D_6$, $C_6:Q_8$

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

Order 32: $C_2^2\times Q_8$

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

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $C_3:Q_8$, $C_2\times D_6$, $C_3\times Q_8$

Order 21: $C_{21}$

Order 16: $C_2\times Q_8$ x 3, $C_2^2\times C_4$

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

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

Order 8: $C_2\times C_4$ x 3, $Q_8$ x 2, $C_2^3$

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $Q_8\times D_{42}$ ($C_1$)

Order 336: $Q_8\times D_{21}$ ($C_2$) x 8, $C_4\times D_{42}$ ($C_2$) x 3, $C_{42}:Q_8$ ($C_2$) x 3, $Q_8\times C_{42}$ ($C_2$)

Order 168: $C_4\times D_{21}$ ($C_2^2$) x 12, $C_{21}:Q_8$ ($C_2^2$) x 12, $Q_8\times C_{21}$ ($C_2^2$) x 4, $C_2\times C_{84}$ ($C_2^2$) x 3, $C_{42}:C_4$ ($C_2^2$) x 3, $C_2\times D_{42}$ ($C_2^2$)

Order 112: $Q_8\times C_{14}$ ($S_3$)

Order 84: $C_{84}$ ($C_2^3$) x 6, $C_{21}:C_4$ ($C_2^3$) x 6, $D_{42}$ ($Q_8$) x 4, $D_{42}$ ($C_2^3$) x 2, $C_2\times C_{42}$ ($C_2^3$)

Order 56: $C_7\times Q_8$ ($D_6$) x 4, $C_2\times C_{28}$ ($D_6$) x 3

Order 48: $C_6\times Q_8$ ($D_7$)

Order 42: $D_{21}$ ($C_2\times Q_8$) x 4, $C_{42}$ ($C_2\times Q_8$) x 2, $C_{42}$ ($C_2^4$)

Order 28: $C_{28}$ ($C_2\times D_6$) x 6, $C_2\times C_{14}$ ($C_2\times D_6$)

Order 24: $C_3\times Q_8$ ($D_{14}$) x 4, $C_2\times C_{12}$ ($D_{14}$) x 3

Order 21: $C_{21}$ ($C_2^2\times Q_8$)

Order 16: $C_2\times Q_8$ ($D_{21}$)

Order 14: $C_{14}$ ($S_3\times Q_8$) x 2, $C_{14}$ ($C_2^2\times D_6$)

Order 12: $C_{12}$ ($C_2\times D_{14}$) x 6, $C_2\times C_6$ ($C_2\times D_{14}$)

Order 8: $Q_8$ ($D_{42}$) x 4, $C_2\times C_4$ ($D_{42}$) x 3

Order 7: $C_7$ ($Q_8\times D_6$)

Order 6: $C_6$ ($Q_8\times D_7$) x 2, $C_6$ ($C_2^2\times D_{14}$)

Order 4: $C_4$ ($C_2\times D_{42}$) x 6, $C_2^2$ ($C_2\times D_{42}$)

Order 3: $C_3$ ($Q_8\times D_{14}$)

Order 2: $C_2$ ($Q_8\times D_{21}$) x 2, $C_2$ ($C_2^2\times D_{42}$)

Order 1: $C_1$ ($Q_8\times D_{42}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $Q_8\times D_{42}$ ($C_1$)

Order 336: $Q_8\times C_{42}$ ($C_2$), $Q_8\times D_{21}$ ($C_2$), $C_4\times D_{42}$ ($C_2$), $C_{42}:Q_8$ ($C_2$)

Order 168: $C_2\times D_{42}$ ($C_2^2$), $Q_8\times C_{21}$ ($C_2^2$), $C_2\times C_{84}$ ($C_2^2$), $C_{42}:C_4$ ($C_2^2$), $C_4\times D_{21}$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$)

Order 112: $Q_8\times C_{14}$ ($S_3$)

Order 84: $C_{84}$ ($C_2^3$), $C_{21}:C_4$ ($C_2^3$), $C_2\times C_{42}$ ($C_2^3$), $D_{42}$ ($C_2^3$), $D_{42}$ ($Q_8$)

Order 56: $C_2\times C_{28}$ ($D_6$), $C_7\times Q_8$ ($D_6$)

Order 48: $C_6\times Q_8$ ($D_7$)

Order 42: $C_{42}$ ($C_2^4$), $C_{42}$ ($C_2\times Q_8$), $D_{21}$ ($C_2\times Q_8$)

Order 28: $C_2\times C_{14}$ ($C_2\times D_6$), $C_{28}$ ($C_2\times D_6$)

Order 24: $C_2\times C_{12}$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$)

Order 21: $C_{21}$ ($C_2^2\times Q_8$)

Order 16: $C_2\times Q_8$ ($D_{21}$)

Order 14: $C_{14}$ ($C_2^2\times D_6$), $C_{14}$ ($S_3\times Q_8$)

Order 12: $C_2\times C_6$ ($C_2\times D_{14}$), $C_{12}$ ($C_2\times D_{14}$)

Order 8: $Q_8$ ($D_{42}$), $C_2\times C_4$ ($D_{42}$)

Order 7: $C_7$ ($Q_8\times D_6$)

Order 6: $C_6$ ($C_2^2\times D_{14}$), $C_6$ ($Q_8\times D_7$)

Order 4: $C_2^2$ ($C_2\times D_{42}$), $C_4$ ($C_2\times D_{42}$)

Order 3: $C_3$ ($Q_8\times D_{14}$)

Order 2: $C_2$ ($C_2^2\times D_{42}$), $C_2$ ($Q_8\times D_{21}$)

Order 1: $C_1$ ($Q_8\times D_{42}$)

Series

Derived series

$Q_8\times D_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$Q_8\times D_{42}$

$\rhd$

$C_4\times D_{42}$

$\rhd$

$C_2\times D_{42}$

$\rhd$

$D_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$Q_8\times D_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times Q_8$

Supergroups

This group is a maximal subgroup of 66 larger groups in the database.

This group is a maximal quotient of 37 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $50 \times 50$ rational character table.