Abstract group 1344.11567: $C_{84}.C_2^4$

• Label: 1344.11567
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $5$

Group information

Description:	$C_{84}.C_2^4$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^7.D_6^2$, of order \(96768\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 7 \)
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	12	14	21	28	42	84
Elements	1	239	2	272	70	6	184	258	12	120	84	96	1344
Conjugacy classes	1	16	1	17	6	3	10	30	3	18	12	15	132
Divisions	1	16	1	16	6	1	10	10	1	6	4	4	76
Autjugacy classes	1	6	1	6	3	1	3	3	1	3	2	2	32

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	32	64	26	0	10	0	0	0	132
Irr. rational chars.	32	16	0	16	2	8	1	1	76

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$5$
Inequivalent generating 5-tuples:	$13332480000$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	48
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{84}=[a,b]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12)(13,16)(14,17)(15,18)(19,23)(20,24)(21,25)(22,26) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8\times D_7)$ $\,\rtimes\,$ $D_6$	$(D_4:D_7)$ $\,\rtimes\,$ $D_6$	$(D_4:D_6)$ $\,\rtimes\,$ $D_7$	$(D_4.D_{14})$ $\,\rtimes\,$ $S_3$	all 58
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{84}$ . $C_2^3$	$D_{42}$ . $C_2^4$ (2)	$C_{84}$ . $C_2^4$ (2)	$C_{42}$ . $C_2^5$	all 68

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 11268 subgroups in 1620 conjugacy classes, 528 normal (36 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{21}:C_2^5$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{21}:C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{84}.C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.C_2^4$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1344: $C_{84}.C_2^4$

Order 672: $D_{12}:D_{14}$ x 6, $D_{12}:D_{14}$ x 6, $D_{12}:D_{14}$ x 3, $D_{12}:D_{14}$ x 3, $D_{12}.D_{14}$ x 3, $D_{12}:D_{14}$ x 3, $D_{12}:D_{14}$ x 2, $D_{12}.D_{14}$ x 2, $D_4:D_{42}$, $C_{84}.C_2^3$, $C_{84}.C_2^3$

Order 448: $C_{14}.C_2^5$

Order 336: $D_{12}:D_7$ x 21, $D_6:D_{14}$ x 12, $D_6.D_{14}$ x 12, $C_{28}:D_6$ x 6, $C_{12}.D_{14}$ x 6, $D_6:D_{14}$ x 6, $C_{14}:D_{12}$ x 6, $D_6.D_{14}$ x 6, $D_6.D_{14}$ x 6, $D_7\times D_{12}$ x 6, $C_{12}:D_{14}$ x 6, $D_{12}:D_7$ x 6, $C_{12}.D_{14}$ x 6, $D_{84}:C_2$ x 6, $C_{12}.D_{14}$ x 6, $D_4:D_{21}$ x 3, $D_4\times D_{21}$ x 3, $D_{84}:C_2$ x 3, $C_4\times D_{42}$ x 3, $D_{12}:C_{14}$ x 3, $C_{14}\times D_{12}$ x 3, $D_{28}:C_6$ x 3, $C_{42}:Q_8$ x 3, $C_{12}:D_{14}$ x 3, $D_{21}:Q_8$ x 3, $D_{12}:C_{14}$ x 2, $C_{12}.D_{14}$ x 2, $D_4:C_{42}$, $D_{84}:C_2$, $Q_8\times D_{21}$, $D_{28}:S_3$

Order 224: $D_4:D_{14}$ x 9, $D_4:D_{14}$ x 9, $D_4.D_{14}$ x 6, $D_4:D_{14}$ x 6, $C_{14}.C_2^4$

Order 192: $C_{12}.C_2^4$

Order 168: $C_7:D_{12}$ x 18, $C_6.D_{14}$ x 18, $C_4\times D_{21}$ x 10, $C_7\times D_{12}$ x 9, $C_{21}:Q_8$ x 9, $D_6\times C_{14}$ x 6, $S_3\times D_{14}$ x 6, $C_{21}:D_4$ x 6, $D_6:C_{14}$ x 6, $S_3\times C_{28}$ x 6, $C_{21}:D_4$ x 6, $C_{14}:C_{12}$ x 6, $C_{12}\times D_7$ x 6, $C_{21}:Q_8$ x 6, $C_{21}:D_4$ x 6, $D_{21}:C_4$ x 6, $C_2\times D_{42}$ x 3, $D_4\times C_{21}$ x 3, $C_2\times C_{84}$ x 3, $C_{42}:C_4$ x 3, $D_{84}$ x 3, $C_{21}:Q_8$ x 3, $C_3:D_{28}$ x 2, $C_6.D_{14}$ x 2, $Q_8\times C_{21}$, $C_{21}:Q_8$, $C_3\times D_{28}$

Order 112: $D_4:D_7$ x 54, $C_{14}:D_4$ x 18, $D_4\times D_7$ x 18, $D_{28}:C_2$ x 18, $D_4\times C_{14}$ x 9, $C_4\times D_{14}$ x 9, $C_{14}:Q_8$ x 9, $D_4:C_{14}$ x 6, $C_2^2.D_{14}$ x 6, $Q_8\times D_7$ x 6, $D_{28}:C_2$ x 2

Order 96: $D_4:D_6$ x 10, $D_4:D_6$ x 10, $Q_8.D_6$ x 5, $Q_8:D_6$ x 5, $C_6.C_2^4$

Order 84: $S_3\times C_{14}$ x 12, $D_{42}$ x 7, $S_3\times D_7$ x 6, $C_7:C_{12}$ x 6, $C_{84}$ x 4, $C_{21}:C_4$ x 4, $C_2\times C_{42}$ x 3, $C_3:C_{28}$ x 2, $C_3\times D_{14}$ x 2

Order 64: $C_4.C_2^4$

Order 56: $C_7:D_4$ x 36, $C_{14}:C_4$ x 27, $C_4\times D_7$ x 24, $C_7\times D_4$ x 18, $C_7:Q_8$ x 18, $C_2\times C_{28}$ x 9, $C_2\times D_{14}$ x 9, $D_{28}$ x 6, $C_2^2\times C_{14}$ x 6, $C_7\times Q_8$ x 2

Order 48: $D_{12}:C_2$ x 30, $S_3\times D_4$ x 30, $D_{12}:C_2$ x 30, $C_2\times D_{12}$ x 15, $C_4\times D_6$ x 15, $D_4:C_6$ x 10, $S_3\times Q_8$ x 10, $D_4:S_3$ x 10, $C_6\times Q_8$ x 5

Order 42: $S_3\times C_7$ x 6, $C_{42}$ x 4, $D_{21}$ x 4, $C_3\times D_7$ x 2

Order 32: $D_4:C_2^2$ x 15, $D_4:C_2^2$ x 10, $C_4.C_2^3$ x 6

Order 28: $C_2\times C_{14}$ x 15, $D_{14}$ x 15, $C_7:C_4$ x 10, $C_{28}$ x 6

Order 24: $C_4\times S_3$ x 40, $D_{12}$ x 30, $C_3:D_4$ x 20, $C_2\times C_{12}$ x 15, $C_2\times D_6$ x 15, $C_3:Q_8$ x 10, $C_3\times Q_8$ x 10, $C_3\times D_4$ x 10, $C_6:C_4$ x 5

Order 21: $C_{21}$

Order 16: $D_4:C_2$ x 80, $C_2\times D_4$ x 45, $C_2\times Q_8$ x 15, $C_2^2\times C_4$ x 15

Order 14: $C_{14}$ x 10, $D_7$ x 6

Order 12: $D_6$ x 25, $C_{12}$ x 10, $C_3:C_4$ x 6, $C_2\times C_6$ x 5

Order 8: $D_4$ x 60, $C_2\times C_4$ x 60, $Q_8$ x 20, $C_2^3$ x 15

Order 7: $C_7$

Order 6: $S_3$ x 10, $C_6$ x 6

Order 4: $C_2^2$ x 30, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 16

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_{84}.C_2^4$

Order 672: $D_4:D_{42}$, $C_{84}.C_2^3$, $C_{84}.C_2^3$, $D_{12}:D_{14}$, $D_{12}.D_{14}$, $D_{12}:D_{14}$, $D_{12}:D_{14}$, $D_{12}:D_{14}$, $D_{12}:D_{14}$, $D_{12}.D_{14}$, $D_{12}:D_{14}$

Order 448: $C_{14}.C_2^5$

Order 336: $D_{12}:D_7$ x 4, $D_4:C_{42}$, $D_{84}:C_2$, $Q_8\times D_{21}$, $D_4:D_{21}$, $D_4\times D_{21}$, $D_{84}:C_2$, $C_4\times D_{42}$, $D_{12}:C_{14}$, $C_{28}:D_6$, $D_{12}:C_{14}$, $C_{14}\times D_{12}$, $C_{12}.D_{14}$, $C_{12}.D_{14}$, $D_{28}:C_6$, $C_{42}:Q_8$, $D_6:D_{14}$, $D_6:D_{14}$, $C_{14}:D_{12}$, $D_6.D_{14}$, $D_6.D_{14}$, $D_6.D_{14}$, $C_{12}:D_{14}$, $D_7\times D_{12}$, $C_{12}:D_{14}$, $D_{12}:D_7$, $C_{12}.D_{14}$, $D_{21}:Q_8$, $D_{84}:C_2$, $C_{12}.D_{14}$, $D_{28}:S_3$

Order 224: $D_4.D_{14}$ x 3, $D_4:D_{14}$ x 3, $D_4:D_{14}$ x 2, $D_4:D_{14}$ x 2, $C_{14}.C_2^4$

Order 192: $C_{12}.C_2^4$

Order 168: $C_4\times D_{21}$ x 4, $C_7\times D_{12}$ x 2, $C_{21}:Q_8$ x 2, $C_7:D_{12}$ x 2, $C_6.D_{14}$ x 2, $C_2\times D_{42}$, $D_6\times C_{14}$, $S_3\times D_{14}$, $Q_8\times C_{21}$, $D_4\times C_{21}$, $C_2\times C_{84}$, $C_{21}:D_4$, $C_{42}:C_4$, $D_{84}$, $C_{21}:Q_8$, $D_6:C_{14}$, $S_3\times C_{28}$, $C_{21}:Q_8$, $C_{21}:D_4$, $C_{14}:C_{12}$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_{21}:Q_8$, $C_3:D_{28}$, $C_{21}:D_4$, $D_{21}:C_4$, $C_6.D_{14}$

Order 112: $D_4:D_7$ x 9, $D_4\times D_7$ x 4, $D_{28}:C_2$ x 4, $D_4:C_{14}$ x 3, $Q_8\times D_7$ x 3, $D_4\times C_{14}$ x 2, $C_{14}:D_4$ x 2, $D_{28}:C_2$ x 2, $C_4\times D_{14}$ x 2, $C_{14}:Q_8$ x 2, $C_2^2.D_{14}$

Order 96: $D_4:D_6$ x 3, $D_4:D_6$ x 3, $Q_8.D_6$ x 2, $Q_8:D_6$ x 2, $C_6.C_2^4$

Order 84: $D_{42}$ x 3, $C_{84}$ x 2, $C_{21}:C_4$ x 2, $S_3\times C_{14}$ x 2, $S_3\times D_7$, $C_7:C_{12}$, $C_3:C_{28}$, $C_2\times C_{42}$, $C_3\times D_{14}$

Order 64: $C_4.C_2^4$

Order 56: $C_4\times D_7$ x 7, $C_7:D_4$ x 5, $C_7\times D_4$ x 4, $C_{14}:C_4$ x 4, $C_7:Q_8$ x 4, $D_{28}$ x 3, $C_2\times C_{28}$ x 2, $C_2\times D_{14}$ x 2, $C_7\times Q_8$ x 2, $C_2^2\times C_{14}$

Order 48: $D_{12}:C_2$ x 7, $S_3\times D_4$ x 5, $D_{12}:C_2$ x 5, $D_4:C_6$ x 3, $S_3\times Q_8$ x 3, $D_4:S_3$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3, $C_6\times Q_8$ x 2

Order 42: $C_{42}$ x 2, $D_{21}$ x 2, $C_3\times D_7$, $S_3\times C_7$

Order 32: $D_4:C_2^2$ x 5, $C_4.C_2^3$ x 3, $D_4:C_2^2$ x 3

Order 28: $D_{14}$ x 5, $C_2\times C_{14}$ x 3, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 24: $C_4\times S_3$ x 8, $D_{12}$ x 5, $C_3:D_4$ x 4, $C_2\times C_{12}$ x 3, $C_3:Q_8$ x 3, $C_2\times D_6$ x 3, $C_3\times Q_8$ x 3, $C_3\times D_4$ x 3, $C_6:C_4$ x 2

Order 21: $C_{21}$

Order 16: $D_4:C_2$ x 18, $C_2\times D_4$ x 8, $C_2\times Q_8$ x 5, $C_2^2\times C_4$ x 3

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

Order 12: $D_6$ x 6, $C_{12}$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$ x 2

Order 8: $C_2\times C_4$ x 13, $D_4$ x 12, $Q_8$ x 6, $C_2^3$ x 3

Order 7: $C_7$

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

Order 4: $C_2^2$ x 8, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_{84}.C_2^4$ ($C_1$)

Order 672: $D_{12}:D_{14}$ ($C_2$) x 6, $D_{12}:D_{14}$ ($C_2$) x 6, $D_{12}:D_{14}$ ($C_2$) x 3, $D_{12}:D_{14}$ ($C_2$) x 3, $D_{12}.D_{14}$ ($C_2$) x 3, $D_{12}:D_{14}$ ($C_2$) x 3, $D_{12}:D_{14}$ ($C_2$) x 2, $D_{12}.D_{14}$ ($C_2$) x 2, $D_4:D_{42}$ ($C_2$), $C_{84}.C_2^3$ ($C_2$), $C_{84}.C_2^3$ ($C_2$)

Order 336: $D_{12}:D_7$ ($C_2^2$) x 21, $D_6:D_{14}$ ($C_2^2$) x 12, $D_6.D_{14}$ ($C_2^2$) x 12, $C_{28}:D_6$ ($C_2^2$) x 6, $C_{12}.D_{14}$ ($C_2^2$) x 6, $D_6:D_{14}$ ($C_2^2$) x 6, $C_{14}:D_{12}$ ($C_2^2$) x 6, $D_6.D_{14}$ ($C_2^2$) x 6, $D_6.D_{14}$ ($C_2^2$) x 6, $D_7\times D_{12}$ ($C_2^2$) x 6, $C_{12}:D_{14}$ ($C_2^2$) x 6, $D_{12}:D_7$ ($C_2^2$) x 6, $C_{12}.D_{14}$ ($C_2^2$) x 6, $D_{84}:C_2$ ($C_2^2$) x 6, $C_{12}.D_{14}$ ($C_2^2$) x 6, $D_4:D_{21}$ ($C_2^2$) x 3, $D_4\times D_{21}$ ($C_2^2$) x 3, $D_{84}:C_2$ ($C_2^2$) x 3, $C_4\times D_{42}$ ($C_2^2$) x 3, $D_{12}:C_{14}$ ($C_2^2$) x 3, $C_{14}\times D_{12}$ ($C_2^2$) x 3, $D_{28}:C_6$ ($C_2^2$) x 3, $C_{42}:Q_8$ ($C_2^2$) x 3, $C_{12}:D_{14}$ ($C_2^2$) x 3, $D_{21}:Q_8$ ($C_2^2$) x 3, $D_{12}:C_{14}$ ($C_2^2$) x 2, $C_{12}.D_{14}$ ($C_2^2$) x 2, $D_4:C_{42}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $Q_8\times D_{21}$ ($C_2^2$), $D_{28}:S_3$ ($C_2^2$)

Order 224: $D_4.D_{14}$ ($S_3$)

Order 168: $C_7:D_{12}$ ($C_2^3$) x 18, $C_6.D_{14}$ ($C_2^3$) x 18, $C_4\times D_{21}$ ($C_2^3$) x 10, $C_7\times D_{12}$ ($C_2^3$) x 9, $C_{21}:Q_8$ ($C_2^3$) x 9, $D_6\times C_{14}$ ($C_2^3$) x 6, $S_3\times D_{14}$ ($C_2^3$) x 6, $C_{21}:D_4$ ($C_2^3$) x 6, $D_6:C_{14}$ ($C_2^3$) x 6, $S_3\times C_{28}$ ($C_2^3$) x 6, $C_{21}:D_4$ ($C_2^3$) x 6, $C_{14}:C_{12}$ ($C_2^3$) x 6, $C_{12}\times D_7$ ($C_2^3$) x 6, $C_{21}:Q_8$ ($C_2^3$) x 6, $C_{21}:D_4$ ($C_2^3$) x 6, $D_{21}:C_4$ ($C_2^3$) x 6, $C_2\times D_{42}$ ($C_2^3$) x 3, $D_4\times C_{21}$ ($C_2^3$) x 3, $C_2\times C_{84}$ ($C_2^3$) x 3, $C_{42}:C_4$ ($C_2^3$) x 3, $D_{84}$ ($C_2^3$) x 3, $C_{21}:Q_8$ ($C_2^3$) x 3, $C_3:D_{28}$ ($C_2^3$) x 2, $C_6.D_{14}$ ($C_2^3$) x 2, $Q_8\times C_{21}$ ($C_2^3$), $C_{21}:Q_8$ ($C_2^3$), $C_3\times D_{28}$ ($C_2^3$)

Order 112: $D_4:D_7$ ($D_6$) x 6, $D_{28}:C_2$ ($D_6$) x 3, $C_{14}:Q_8$ ($D_6$) x 3, $Q_8\times D_7$ ($D_6$) x 2, $D_4:C_{14}$ ($D_6$)

Order 96: $D_4:D_6$ ($D_7$)

Order 84: $C_7:C_{12}$ ($C_2^4$) x 6, $S_3\times C_{14}$ ($C_2^4$) x 6, $C_{84}$ ($C_2^4$) x 4, $C_{21}:C_4$ ($C_2^4$) x 4, $D_{42}$ ($C_2^4$) x 4, $C_2\times C_{42}$ ($C_2^4$) x 3, $C_3:C_{28}$ ($C_2^4$) x 2, $C_3\times D_{14}$ ($C_2^4$) x 2

Order 56: $C_7:Q_8$ ($C_2\times D_6$) x 9, $C_7:D_4$ ($C_2\times D_6$) x 6, $C_{14}:C_4$ ($C_2\times D_6$) x 6, $C_4\times D_7$ ($C_2\times D_6$) x 6, $C_7\times D_4$ ($C_2\times D_6$) x 3, $C_2\times C_{28}$ ($C_2\times D_6$) x 3, $D_{28}$ ($C_2\times D_6$), $C_7\times Q_8$ ($C_2\times D_6$)

Order 48: $S_3\times D_4$ ($D_{14}$) x 6, $D_{12}:C_2$ ($D_{14}$) x 3, $C_2\times D_{12}$ ($D_{14}$) x 3, $D_{12}:C_2$ ($D_{14}$) x 2, $D_4:C_6$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^5$)

Order 28: $C_7:C_4$ ($C_2^2\times D_6$) x 6, $C_{28}$ ($C_2^2\times D_6$) x 4, $C_2\times C_{14}$ ($C_2^2\times D_6$) x 3, $D_{14}$ ($C_2^2\times D_6$) x 2

Order 24: $D_{12}$ ($C_2\times D_{14}$) x 9, $C_3:D_4$ ($C_2\times D_{14}$) x 6, $C_4\times S_3$ ($C_2\times D_{14}$) x 6, $C_2\times D_6$ ($C_2\times D_{14}$) x 6, $C_2\times C_{12}$ ($C_2\times D_{14}$) x 3, $C_3\times D_4$ ($C_2\times D_{14}$) x 3, $C_3:Q_8$ ($C_2\times D_{14}$), $C_3\times Q_8$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($C_4.C_2^4$)

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

Order 14: $C_{14}$ ($C_2^3\times D_6$)

Order 12: $D_6$ ($C_2^2\times D_{14}$) x 6, $C_{12}$ ($C_2^2\times D_{14}$) x 4, $C_2\times C_6$ ($C_2^2\times D_{14}$) x 3, $C_3:C_4$ ($C_2^2\times D_{14}$) x 2

Order 8: $D_4$ ($S_3\times D_{14}$) x 3, $C_2\times C_4$ ($S_3\times D_{14}$) x 3, $Q_8$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_{12}.C_2^4$)

Order 6: $C_6$ ($C_2^3\times D_{14}$)

Order 4: $C_4$ ($D_6\times D_{14}$) x 4, $C_2^2$ ($D_6\times D_{14}$) x 3

Order 3: $C_3$ ($C_{14}.C_2^5$)

Order 2: $C_2$ ($C_{21}:C_2^5$)

Order 1: $C_1$ ($C_{84}.C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_{84}.C_2^4$ ($C_1$)

Order 672: $D_4:D_{42}$ ($C_2$), $C_{84}.C_2^3$ ($C_2$), $C_{84}.C_2^3$ ($C_2$), $D_{12}:D_{14}$ ($C_2$), $D_{12}.D_{14}$ ($C_2$), $D_{12}:D_{14}$ ($C_2$), $D_{12}:D_{14}$ ($C_2$), $D_{12}:D_{14}$ ($C_2$), $D_{12}:D_{14}$ ($C_2$), $D_{12}.D_{14}$ ($C_2$), $D_{12}:D_{14}$ ($C_2$)

Order 336: $D_{12}:D_7$ ($C_2^2$) x 4, $D_4:C_{42}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $Q_8\times D_{21}$ ($C_2^2$), $D_4:D_{21}$ ($C_2^2$), $D_4\times D_{21}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $C_4\times D_{42}$ ($C_2^2$), $D_{12}:C_{14}$ ($C_2^2$), $C_{28}:D_6$ ($C_2^2$), $D_{12}:C_{14}$ ($C_2^2$), $C_{14}\times D_{12}$ ($C_2^2$), $C_{12}.D_{14}$ ($C_2^2$), $C_{12}.D_{14}$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $D_6:D_{14}$ ($C_2^2$), $D_6:D_{14}$ ($C_2^2$), $C_{14}:D_{12}$ ($C_2^2$), $D_6.D_{14}$ ($C_2^2$), $D_6.D_{14}$ ($C_2^2$), $D_6.D_{14}$ ($C_2^2$), $C_{12}:D_{14}$ ($C_2^2$), $D_7\times D_{12}$ ($C_2^2$), $C_{12}:D_{14}$ ($C_2^2$), $D_{12}:D_7$ ($C_2^2$), $C_{12}.D_{14}$ ($C_2^2$), $D_{21}:Q_8$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $C_{12}.D_{14}$ ($C_2^2$), $D_{28}:S_3$ ($C_2^2$)

Order 224: $D_4.D_{14}$ ($S_3$)

Order 168: $C_4\times D_{21}$ ($C_2^3$) x 4, $C_7\times D_{12}$ ($C_2^3$) x 2, $C_{21}:Q_8$ ($C_2^3$) x 2, $C_7:D_{12}$ ($C_2^3$) x 2, $C_6.D_{14}$ ($C_2^3$) x 2, $C_2\times D_{42}$ ($C_2^3$), $D_6\times C_{14}$ ($C_2^3$), $S_3\times D_{14}$ ($C_2^3$), $Q_8\times C_{21}$ ($C_2^3$), $D_4\times C_{21}$ ($C_2^3$), $C_2\times C_{84}$ ($C_2^3$), $C_{21}:D_4$ ($C_2^3$), $C_{42}:C_4$ ($C_2^3$), $D_{84}$ ($C_2^3$), $C_{21}:Q_8$ ($C_2^3$), $D_6:C_{14}$ ($C_2^3$), $S_3\times C_{28}$ ($C_2^3$), $C_{21}:Q_8$ ($C_2^3$), $C_{21}:D_4$ ($C_2^3$), $C_{14}:C_{12}$ ($C_2^3$), $C_3\times D_{28}$ ($C_2^3$), $C_{12}\times D_7$ ($C_2^3$), $C_{21}:Q_8$ ($C_2^3$), $C_3:D_{28}$ ($C_2^3$), $C_{21}:D_4$ ($C_2^3$), $D_{21}:C_4$ ($C_2^3$), $C_6.D_{14}$ ($C_2^3$)

Order 112: $D_4:C_{14}$ ($D_6$), $Q_8\times D_7$ ($D_6$), $D_4:D_7$ ($D_6$), $D_{28}:C_2$ ($D_6$), $C_{14}:Q_8$ ($D_6$)

Order 96: $D_4:D_6$ ($D_7$)

Order 84: $C_{84}$ ($C_2^4$) x 2, $C_{21}:C_4$ ($C_2^4$) x 2, $D_{42}$ ($C_2^4$) x 2, $C_7:C_{12}$ ($C_2^4$), $C_3:C_{28}$ ($C_2^4$), $C_2\times C_{42}$ ($C_2^4$), $S_3\times C_{14}$ ($C_2^4$), $C_3\times D_{14}$ ($C_2^4$)

Order 56: $C_7:Q_8$ ($C_2\times D_6$) x 2, $C_7\times D_4$ ($C_2\times D_6$), $C_2\times C_{28}$ ($C_2\times D_6$), $C_7:D_4$ ($C_2\times D_6$), $C_{14}:C_4$ ($C_2\times D_6$), $D_{28}$ ($C_2\times D_6$), $C_4\times D_7$ ($C_2\times D_6$), $C_7\times Q_8$ ($C_2\times D_6$)

Order 48: $D_4:C_6$ ($D_{14}$), $D_{12}:C_2$ ($D_{14}$), $S_3\times D_4$ ($D_{14}$), $D_{12}:C_2$ ($D_{14}$), $C_2\times D_{12}$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^5$)

Order 28: $C_{28}$ ($C_2^2\times D_6$) x 2, $C_2\times C_{14}$ ($C_2^2\times D_6$), $D_{14}$ ($C_2^2\times D_6$), $C_7:C_4$ ($C_2^2\times D_6$)

Order 24: $D_{12}$ ($C_2\times D_{14}$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_3:D_4$ ($C_2\times D_{14}$), $C_4\times S_3$ ($C_2\times D_{14}$), $C_3:Q_8$ ($C_2\times D_{14}$), $C_2\times D_6$ ($C_2\times D_{14}$), $C_3\times Q_8$ ($C_2\times D_{14}$), $C_3\times D_4$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($C_4.C_2^4$)

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

Order 14: $C_{14}$ ($C_2^3\times D_6$)

Order 12: $C_{12}$ ($C_2^2\times D_{14}$) x 2, $C_2\times C_6$ ($C_2^2\times D_{14}$), $D_6$ ($C_2^2\times D_{14}$), $C_3:C_4$ ($C_2^2\times D_{14}$)

Order 8: $Q_8$ ($S_3\times D_{14}$), $D_4$ ($S_3\times D_{14}$), $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_{12}.C_2^4$)

Order 6: $C_6$ ($C_2^3\times D_{14}$)

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

Order 3: $C_3$ ($C_{14}.C_2^5$)

Order 2: $C_2$ ($C_{21}:C_2^5$)

Order 1: $C_1$ ($C_{84}.C_2^4$)

Series

Derived series

$C_{84}.C_2^4$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_{84}.C_2^4$

$\rhd$

$D_{12}:D_{14}$

$\rhd$

$C_{28}:D_6$

$\rhd$

$D_6\times C_{14}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{84}.C_2^4$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4:C_2$

Character theory

Complex character table

See the $132 \times 132$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $76 \times 76$ rational character table.