Abstract group 1344.3787: $D_{84}.D_4$

• Label: 1344.3787
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $3$

Group information

Description:	$D_{84}.D_4$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{42}.(C_2^2\times C_6).C_2^5$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
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	8	12	14	21	24	28	42	56	84
Elements	1	95	2	144	22	6	272	120	66	12	112	24	132	288	48	1344
Conjugacy classes	1	4	1	4	5	3	7	4	12	3	4	6	21	12	6	93
Divisions	1	4	1	4	3	1	5	3	3	1	2	2	3	2	2	37
Autjugacy classes	1	4	1	4	3	1	5	3	3	1	2	2	3	2	2	37

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	8	38	38	0	9	0	0	0	0	93
Irr. rational chars.	8	8	5	4	1	4	1	5	1	37

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	96
Arbitrary	6	10	16

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{8}=d^{42}=[a,c]=[b,d]=1, a^{2}=c^{6}, b^{a}=bc^{4}d^{21}, d^{a}=c^{4}d^{29}, c^{b}=c^{3}, d^{c}=d^{13} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12,14,17,15,18,19,22)(13,16,21,25,20,24,26,23), (9,10) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{12}.D_4)$ $\,\rtimes\,$ $D_7$	$C_3$ $\,\rtimes\,$ $(D_{28}.D_4)$	$(C_{84}.D_4)$ $\,\rtimes\,$ $C_2$	$(D_{84}.C_4)$ $\,\rtimes\,$ $C_2$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{84}$ . $D_4$	$C_{28}$ . $(S_3\times D_4)$	$(C_{21}:Q_8)$ . $D_4$	$(C_{14}:Q_8)$ . $D_6$	all 35

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 1652 subgroups in 208 conjugacy classes, 54 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{42}:D_4$
Commutator:	$G' \simeq$ $C_2\times C_{84}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_{84}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_4:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8.D_4$
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

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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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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 1344: $D_{84}.D_4$

Order 672: $D_{84}:C_2^2$, $C_{42}:\SD_{16}$, $C_{12}.D_{28}$, $C_{12}.D_{28}$, $D_{84}.C_4$, $C_{84}.D_4$, $C_{28}.D_{12}$

Order 448: $D_{28}.D_4$

Order 336: $C_{12}.C_{28}$ x 2, $C_{21}:\SD_{16}$ x 2, $C_{21}:\OD_{16}$, $C_{14}:C_{24}$, $C_{21}:Q_{16}$, $C_{21}:\SD_{16}$, $C_{21}:\OD_{16}$, $D_{21}:C_8$, $D_4\times C_{42}$, $D_{84}:C_2$, $C_{42}:Q_8$, $C_{21}:\SD_{16}$, $C_{21}:D_8$

Order 224: $\OD_{16}:C_{14}$, $C_4.D_{28}$, $C_{28}.D_4$, $C_{14}:\SD_{16}$, $D_{28}:C_2^2$, $C_4.D_{28}$, $C_8.D_{14}$

Order 192: $C_{24}.D_4$

Order 168: $D_4\times C_{21}$ x 2, $C_7:C_{24}$ x 2, $C_3:C_{56}$ x 2, $C_{21}:Q_8$ x 2, $C_2^2\times C_{42}$, $C_{21}:C_8$, $C_2\times C_{84}$, $C_{21}:D_4$, $D_{84}$, $C_4\times D_{21}$, $C_{21}:Q_8$, $C_{14}:C_{12}$

Order 112: $C_7:\SD_{16}$ x 3, $C_7\times \OD_{16}$ x 2, $C_7:\OD_{16}$, $C_{14}:C_8$, $C_7:Q_{16}$, $C_{56}:C_2$, $C_{56}:C_2$, $D_4\times C_{14}$, $D_{28}:C_2$, $C_8\times D_7$, $C_{14}:Q_8$, $C_7:D_8$

Order 96: $C_{12}.D_4$, $C_{12}.D_4$, $C_{24}.C_4$, $C_6\times \SD_{16}$, $C_{12}.D_4$, $D_{12}:C_2^2$, $C_8.D_6$

Order 84: $C_2\times C_{42}$ x 3, $C_{84}$ x 2, $C_{21}:C_4$, $C_7:C_{12}$, $D_{42}$

Order 64: $C_8.D_4$

Order 56: $C_7:Q_8$ x 3, $C_7:C_8$ x 3, $C_7\times D_4$ x 2, $C_{56}$ x 2, $C_2\times C_{28}$, $C_7:D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_2^2\times C_{14}$

Order 48: $C_3:\OD_{16}$ x 3, $C_3\times \SD_{16}$ x 2, $C_{24}:C_2$, $C_6\times Q_8$, $C_6\times D_4$, $S_3\times C_8$, $D_{12}:C_2$, $C_2\times C_{24}$, $C_3:Q_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:D_8$

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

Order 32: $C_4.D_4$, $\OD_{16}:C_2$, $Q_{16}:C_2$, $D_8:C_2$, $C_2\times \SD_{16}$, $\OD_{16}:C_2$, $C_8.C_4$

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

Order 24: $C_3:C_8$ x 3, $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3\times D_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $\SD_{16}$ x 4, $\OD_{16}$ x 4, $C_2\times C_8$ x 2, $Q_{16}$, $D_8$, $D_4:C_2$, $C_2\times Q_8$, $C_2\times D_4$

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

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

Order 8: $C_8$ x 5, $D_4$ x 4, $Q_8$ x 3, $C_2\times C_4$ x 3, $C_2^3$

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $D_{84}.D_4$

Order 672: $D_{84}:C_2^2$, $C_{42}:\SD_{16}$, $C_{12}.D_{28}$, $C_{12}.D_{28}$, $D_{84}.C_4$, $C_{84}.D_4$, $C_{28}.D_{12}$

Order 448: $D_{28}.D_4$

Order 336: $C_{12}.C_{28}$ x 2, $C_{21}:\SD_{16}$ x 2, $C_{21}:\OD_{16}$, $C_{14}:C_{24}$, $C_{21}:Q_{16}$, $C_{21}:\SD_{16}$, $C_{21}:\OD_{16}$, $D_{21}:C_8$, $D_4\times C_{42}$, $D_{84}:C_2$, $C_{42}:Q_8$, $C_{21}:\SD_{16}$, $C_{21}:D_8$

Order 224: $\OD_{16}:C_{14}$, $C_4.D_{28}$, $C_{28}.D_4$, $C_{14}:\SD_{16}$, $D_{28}:C_2^2$, $C_4.D_{28}$, $C_8.D_{14}$

Order 192: $C_{24}.D_4$

Order 168: $D_4\times C_{21}$ x 2, $C_7:C_{24}$ x 2, $C_3:C_{56}$ x 2, $C_{21}:Q_8$ x 2, $C_2^2\times C_{42}$, $C_{21}:C_8$, $C_2\times C_{84}$, $C_{21}:D_4$, $D_{84}$, $C_4\times D_{21}$, $C_{21}:Q_8$, $C_{14}:C_{12}$

Order 112: $C_7:\SD_{16}$ x 3, $C_7\times \OD_{16}$ x 2, $C_7:\OD_{16}$, $C_{14}:C_8$, $C_7:Q_{16}$, $C_{56}:C_2$, $C_{56}:C_2$, $D_4\times C_{14}$, $D_{28}:C_2$, $C_8\times D_7$, $C_{14}:Q_8$, $C_7:D_8$

Order 96: $C_{12}.D_4$, $C_{12}.D_4$, $C_{24}.C_4$, $C_6\times \SD_{16}$, $C_{12}.D_4$, $D_{12}:C_2^2$, $C_8.D_6$

Order 84: $C_2\times C_{42}$ x 3, $C_{84}$ x 2, $C_{21}:C_4$, $C_7:C_{12}$, $D_{42}$

Order 64: $C_8.D_4$

Order 56: $C_7:Q_8$ x 3, $C_7:C_8$ x 3, $C_7\times D_4$ x 2, $C_{56}$ x 2, $C_2\times C_{28}$, $C_7:D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_2^2\times C_{14}$

Order 48: $C_3:\OD_{16}$ x 3, $C_3\times \SD_{16}$ x 2, $C_{24}:C_2$, $C_6\times Q_8$, $C_6\times D_4$, $S_3\times C_8$, $D_{12}:C_2$, $C_2\times C_{24}$, $C_3:Q_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:D_8$

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

Order 32: $C_4.D_4$, $\OD_{16}:C_2$, $Q_{16}:C_2$, $D_8:C_2$, $C_2\times \SD_{16}$, $\OD_{16}:C_2$, $C_8.C_4$

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

Order 24: $C_3:C_8$ x 3, $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3\times D_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $\SD_{16}$ x 4, $\OD_{16}$ x 4, $C_2\times C_8$ x 2, $Q_{16}$, $D_8$, $D_4:C_2$, $C_2\times Q_8$, $C_2\times D_4$

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

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

Order 8: $C_8$ x 5, $D_4$ x 4, $Q_8$ x 3, $C_2\times C_4$ x 3, $C_2^3$

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $D_{84}.D_4$ ($C_1$)

Order 672: $D_{84}:C_2^2$ ($C_2$), $C_{42}:\SD_{16}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $D_{84}.C_4$ ($C_2$), $C_{84}.D_4$ ($C_2$), $C_{28}.D_{12}$ ($C_2$)

Order 336: $C_{12}.C_{28}$ ($C_2^2$) x 2, $C_{21}:\OD_{16}$ ($C_2^2$), $C_{14}:C_{24}$ ($C_2^2$), $D_4\times C_{42}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$)

Order 224: $C_{14}:\SD_{16}$ ($S_3$)

Order 168: $C_7:C_{24}$ ($D_4$) x 2, $C_2\times C_{84}$ ($C_2^3$), $D_{84}$ ($D_4$), $C_{21}:Q_8$ ($D_4$)

Order 112: $C_{14}:C_8$ ($D_6$), $D_4\times C_{14}$ ($D_6$), $C_{14}:Q_8$ ($D_6$)

Order 96: $C_{12}.D_4$ ($D_7$)

Order 84: $C_{84}$ ($C_2\times D_4$) x 2, $C_2\times C_{42}$ ($D_4:C_2$)

Order 56: $C_7:C_8$ ($C_3:D_4$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$)

Order 48: $C_3:\OD_{16}$ ($D_{14}$) x 2, $C_6\times D_4$ ($D_{14}$)

Order 42: $C_{42}$ ($C_4:D_4$)

Order 28: $C_2\times C_{14}$ ($D_4:S_3$), $C_{28}$ ($C_6:D_4$), $C_{28}$ ($S_3\times D_4$)

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

Order 21: $C_{21}$ ($C_8.D_4$)

Order 16: $C_2\times D_4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_{12}:D_4$)

Order 12: $C_{12}$ ($D_4\times D_7$) x 2, $C_2\times C_6$ ($D_{28}:C_2$)

Order 8: $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_{24}.D_4$)

Order 6: $C_6$ ($D_{14}:D_4$)

Order 4: $C_2^2$ ($D_{14}.D_6$), $C_4$ ($D_6:D_{14}$), $C_4$ ($D_6:D_{14}$)

Order 3: $C_3$ ($D_{28}.D_4$)

Order 2: $C_2$ ($D_{42}:D_4$)

Order 1: $C_1$ ($D_{84}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $D_{84}.D_4$ ($C_1$)

Order 672: $D_{84}:C_2^2$ ($C_2$), $C_{42}:\SD_{16}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $D_{84}.C_4$ ($C_2$), $C_{84}.D_4$ ($C_2$), $C_{28}.D_{12}$ ($C_2$)

Order 336: $C_{12}.C_{28}$ ($C_2^2$) x 2, $C_{21}:\OD_{16}$ ($C_2^2$), $C_{14}:C_{24}$ ($C_2^2$), $D_4\times C_{42}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$)

Order 224: $C_{14}:\SD_{16}$ ($S_3$)

Order 168: $C_7:C_{24}$ ($D_4$) x 2, $C_2\times C_{84}$ ($C_2^3$), $D_{84}$ ($D_4$), $C_{21}:Q_8$ ($D_4$)

Order 112: $C_{14}:C_8$ ($D_6$), $D_4\times C_{14}$ ($D_6$), $C_{14}:Q_8$ ($D_6$)

Order 96: $C_{12}.D_4$ ($D_7$)

Order 84: $C_{84}$ ($C_2\times D_4$) x 2, $C_2\times C_{42}$ ($D_4:C_2$)

Order 56: $C_7:C_8$ ($C_3:D_4$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$)

Order 48: $C_3:\OD_{16}$ ($D_{14}$) x 2, $C_6\times D_4$ ($D_{14}$)

Order 42: $C_{42}$ ($C_4:D_4$)

Order 28: $C_2\times C_{14}$ ($D_4:S_3$), $C_{28}$ ($C_6:D_4$), $C_{28}$ ($S_3\times D_4$)

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

Order 21: $C_{21}$ ($C_8.D_4$)

Order 16: $C_2\times D_4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_{12}:D_4$)

Order 12: $C_{12}$ ($D_4\times D_7$) x 2, $C_2\times C_6$ ($D_{28}:C_2$)

Order 8: $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_{24}.D_4$)

Order 6: $C_6$ ($D_{14}:D_4$)

Order 4: $C_2^2$ ($D_{14}.D_6$), $C_4$ ($D_6:D_{14}$), $C_4$ ($D_6:D_{14}$)

Order 3: $C_3$ ($D_{28}.D_4$)

Order 2: $C_2$ ($D_{42}:D_4$)

Order 1: $C_1$ ($D_{84}.D_4$)

Series

Derived series

$D_{84}.D_4$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_1$

Chief series

$D_{84}.D_4$

$\rhd$

$C_{84}.D_4$

$\rhd$

$C_{12}.C_{28}$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{84}.D_4$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times D_4$

Character theory

Complex character table

See the $93 \times 93$ 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.