Abstract group 1344.7166: $(D_7\times C_4^2):S_3$

• Label: 1344.7166
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $672$
• Rank: $4$

Group information

Description:	$(D_7\times C_4^2):S_3$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	Group of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \)
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	12	14	21	28	42	84
Elements	1	127	2	384	62	6	192	90	12	288	36	144	1344
Conjugacy classes	1	7	1	20	7	3	24	15	3	42	9	36	168
Divisions	1	7	1	14	5	1	8	4	1	7	3	4	56
Autjugacy classes	1	6	1	12	4	1	6	4	1	6	3	3	48

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	16	92	60	0	0	0	0	0	168
Irr. rational chars.	16	8	6	8	6	4	6	2	56

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$672$
Rank:	$4$
Inequivalent generating quadruples:	$14938560$

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:	$\langle a, b, c, d \mid a^{2}=b^{14}=c^{4}=d^{12}=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{a}=b^{13}d^{6}, d^{b}=c^{2}d^{5} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(12,15)(16,18)(19,20)(22,23), (9,10)(11,12)(13,16)(14,15) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{84}:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_{84}:D_4)$ $\,\rtimes\,$ $C_2$	$(D_7\times C_4^2)$ $\,\rtimes\,$ $S_3$	$(C_4^2:S_3)$ $\,\rtimes\,$ $D_7$	all 19
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times D_{14})$ . $D_6$ (3)	$(C_4\times D_6)$ . $D_{14}$	$(D_6:C_4)$ . $D_{14}$ (2)	$(C_6.D_4)$ . $D_{14}$ (2)	all 63

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

Homology

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

Subgroups

There are 3364 subgroups in 468 conjugacy classes, 136 normal (120 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $S_3\times D_{14}$
Commutator:	$G' \simeq$ $C_2\times C_{42}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $(D_7\times C_4^2):S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^2$
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

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1344: $(D_7\times C_4^2):S_3$

Order 672: $C_4^2:D_{21}$, $C_4^2:S_3\times C_7$, $C_4\times C_{12}\times D_7$, $C_{84}:Q_8$, $D_{14}:D_{12}$, $(C_{12}\times D_{14}):C_2$, $C_{28}:D_{12}$, $C_{12}:D_{28}$, $C_{84}:D_4$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_3\times D_{14}):Q_8$, $(C_{12}\times D_{14}):C_2$

Order 448: $C_4^2.D_{14}$

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

Order 224: $C_{28}:D_4$ x 2, $(C_2\times C_4).D_{14}$, $D_{14}:Q_8$, $D_{14}.D_4$, $(C_2\times C_4).D_{14}$, $C_2^3.D_{14}$, $D_{14}:D_4$, $D_{14}.D_4$, $C_2^3.D_{14}$, $C_4\times D_{28}$, $C_4^2:D_7$, $D_7\times C_4^2$, $C_{28}:Q_8$, $C_4^2:C_{14}$

Order 192: $C_4^2.D_6$

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

Order 112: $D_{14}:C_4$ x 6, $C_{14}.D_4$ x 6, $C_4\times D_{14}$ x 4, $C_{28}:C_4$ x 4, $C_{14}:D_4$ x 2, $C_4:C_{28}$ x 2, $C_2^2:C_{28}$ x 2, $C_4\times C_{28}$ x 2, $C_{14}.D_4$ x 2, $C_{28}:C_4$ x 2, $C_2^2\times C_{28}$, $C_2\times D_{28}$, $C_{14}:Q_8$

Order 96: $C_4^2:S_3$ x 2, $C_4^2:S_3$ x 2, $C_2^3.D_6$ x 2, $C_{12}:D_4$ x 2, $C_4^2:S_3$, $C_4\times D_{12}$, $C_{12}.Q_8$, $C_{12}:Q_8$, $C_2\times C_4\times C_{12}$, $C_{12}:D_4$, $C_{12}.D_4$

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

Order 64: $C_4^2.C_2^2$

Order 56: $C_2\times C_{28}$ x 8, $C_4\times D_7$ x 8, $C_{14}:C_4$ x 6, $C_7:D_4$ x 4, $D_{28}$ x 2, $C_7:Q_8$ x 2, $C_2\times D_{14}$ x 2, $C_2^2\times C_{14}$

Order 48: $D_6:C_4$ x 8, $C_6.D_4$ x 8, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3, $C_6:D_4$ x 2, $C_4\times D_6$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_{12}:C_4$ x 2, $C_2\times D_{12}$, $C_6:Q_8$

Order 42: $C_{42}$ x 3, $C_3\times D_7$ x 2, $D_{21}$, $S_3\times C_7$

Order 32: $C_4\times D_4$ x 3, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_4^2:C_2$ x 2, $C_4.Q_8$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4:D_4$, $C_4\times Q_8$, $C_2\times C_4^2$

Order 28: $D_{14}$ x 7, $C_{28}$ x 7, $C_7:C_4$ x 7, $C_2\times C_{14}$ x 4

Order 24: $C_2\times C_{12}$ x 12, $C_6:C_4$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_2\times D_6$ x 2, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 10, $C_2^2:C_4$ x 10, $C_4^2$ x 6, $C_2^2\times C_4$ x 5, $C_2\times D_4$ x 3, $C_2\times Q_8$

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

Order 12: $C_{12}$ x 8, $D_6$ x 6, $C_3:C_4$ x 6, $C_2\times C_6$ x 5

Order 8: $C_2\times C_4$ x 22, $D_4$ x 6, $C_2^3$ x 3, $Q_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_4$ x 14, $C_2^2$ x 11

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $(D_7\times C_4^2):S_3$

Order 672: $C_4^2:D_{21}$, $C_4^2:S_3\times C_7$, $C_4\times C_{12}\times D_7$, $C_{84}:Q_8$, $D_{14}:D_{12}$, $(C_{12}\times D_{14}):C_2$, $C_{28}:D_{12}$, $C_{12}:D_{28}$, $C_{84}:D_4$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_3\times D_{14}):Q_8$, $(C_{12}\times D_{14}):C_2$

Order 448: $C_4^2.D_{14}$

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

Order 224: $C_{28}:D_4$ x 2, $(C_2\times C_4).D_{14}$, $D_{14}:Q_8$, $D_{14}.D_4$, $(C_2\times C_4).D_{14}$, $C_2^3.D_{14}$, $D_{14}:D_4$, $D_{14}.D_4$, $C_2^3.D_{14}$, $C_4\times D_{28}$, $C_4^2:D_7$, $D_7\times C_4^2$, $C_{28}:Q_8$, $C_4^2:C_{14}$

Order 192: $C_4^2.D_6$

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

Order 112: $D_{14}:C_4$ x 6, $C_{14}.D_4$ x 6, $C_4\times D_{14}$ x 4, $C_{28}:C_4$ x 4, $C_{14}:D_4$ x 2, $C_4:C_{28}$ x 2, $C_2^2:C_{28}$ x 2, $C_4\times C_{28}$ x 2, $C_{14}.D_4$ x 2, $C_{28}:C_4$ x 2, $C_2^2\times C_{28}$, $C_2\times D_{28}$, $C_{14}:Q_8$

Order 96: $C_4^2:S_3$ x 2, $C_4^2:S_3$ x 2, $C_2^3.D_6$ x 2, $C_{12}:D_4$ x 2, $C_4^2:S_3$, $C_4\times D_{12}$, $C_{12}.Q_8$, $C_{12}:Q_8$, $C_2\times C_4\times C_{12}$, $C_{12}:D_4$, $C_{12}.D_4$

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

Order 64: $C_4^2.C_2^2$

Order 56: $C_2\times C_{28}$ x 7, $C_{14}:C_4$ x 6, $C_4\times D_7$ x 4, $C_7:D_4$ x 2, $C_2\times D_{14}$ x 2, $D_{28}$, $C_7:Q_8$, $C_2^2\times C_{14}$

Order 48: $D_6:C_4$ x 8, $C_6.D_4$ x 8, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3, $C_6:D_4$ x 2, $C_4\times D_6$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_{12}:C_4$ x 2, $C_2\times D_{12}$, $C_6:Q_8$

Order 42: $C_{42}$ x 3, $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 32: $C_4\times D_4$ x 3, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_4^2:C_2$ x 2, $C_4.Q_8$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4:D_4$, $C_4\times Q_8$, $C_2\times C_4^2$

Order 28: $D_{14}$ x 6, $C_{28}$ x 6, $C_7:C_4$ x 6, $C_2\times C_{14}$ x 4

Order 24: $C_2\times C_{12}$ x 9, $C_6:C_4$ x 6, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_3:Q_8$, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 10, $C_2^2:C_4$ x 10, $C_4^2$ x 6, $C_2^2\times C_4$ x 5, $C_2\times D_4$ x 3, $C_2\times Q_8$

Order 14: $C_{14}$ x 4, $D_7$ x 2

Order 12: $D_6$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 6, $C_2\times C_6$ x 4

Order 8: $C_2\times C_4$ x 17, $C_2^3$ x 3, $D_4$ x 3, $Q_8$

Order 7: $C_7$

Order 6: $C_6$ x 4, $S_3$ x 2

Order 4: $C_4$ x 12, $C_2^2$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $(D_7\times C_4^2):S_3$ ($C_1$)

Order 672: $C_4^2:D_{21}$ ($C_2$), $C_4^2:S_3\times C_7$ ($C_2$), $C_4\times C_{12}\times D_7$ ($C_2$), $C_{84}:Q_8$ ($C_2$), $D_{14}:D_{12}$ ($C_2$), $(C_{12}\times D_{14}):C_2$ ($C_2$), $C_{28}:D_{12}$ ($C_2$), $C_{12}:D_{28}$ ($C_2$), $C_{84}:D_4$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_3\times D_{14}):Q_8$ ($C_2$), $(C_{12}\times D_{14}):C_2$ ($C_2$)

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

Order 224: $D_7\times C_4^2$ ($S_3$)

Order 168: $C_2\times C_{84}$ ($C_2^3$) x 3, $C_{42}:C_4$ ($C_2^3$) x 3, $C_6:C_{28}$ ($C_2^3$) x 3, $C_{14}:C_{12}$ ($C_2^3$) x 3, $C_2\times D_{42}$ ($C_2^3$), $D_6\times C_{14}$ ($C_2^3$), $C_6\times D_{14}$ ($C_2^3$)

Order 112: $C_4\times D_{14}$ ($D_6$) x 3, $C_{28}:C_4$ ($D_6$) x 3, $C_4\times C_{28}$ ($D_6$)

Order 96: $C_4^2:S_3$ ($D_7$)

Order 84: $C_{84}$ ($D_4:C_2$) x 2, $C_7:C_{12}$ ($D_4:C_2$) x 2, $C_3\times D_{14}$ ($D_4:C_2$) x 2, $C_2\times C_{42}$ ($C_2^4$)

Order 56: $C_2\times C_{28}$ ($C_2\times D_6$) x 3, $C_{14}:C_4$ ($C_2\times D_6$) x 3, $C_2\times D_{14}$ ($C_2\times D_6$)

Order 48: $D_6:C_4$ ($D_{14}$) x 2, $C_6.D_4$ ($D_{14}$) x 2, $C_4\times D_6$ ($D_{14}$), $C_4\times C_{12}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($D_4:C_2^2$) x 3

Order 28: $D_{14}$ ($D_{12}:C_2$) x 2, $C_{28}$ ($D_{12}:C_2$) x 2, $C_7:C_4$ ($D_{12}:C_2$) x 2, $C_2\times C_{14}$ ($C_2^2\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_2\times D_{14}$) x 3, $C_6:C_4$ ($C_2\times D_{14}$) x 3, $C_2\times D_6$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($C_4^2.C_2^2$)

Order 16: $C_4^2$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($D_{12}:C_2^2$) x 3

Order 12: $C_{12}$ ($D_{28}:C_2$) x 2, $C_2\times C_6$ ($C_2^2\times D_{14}$)

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

Order 7: $C_7$ ($C_4^2.D_6$)

Order 6: $C_6$ ($D_4:D_{14}$) x 2, $C_6$ ($D_{28}:C_2^2$)

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

Order 3: $C_3$ ($C_4^2.D_{14}$)

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

Order 1: $C_1$ ($(D_7\times C_4^2):S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $(D_7\times C_4^2):S_3$ ($C_1$)

Order 672: $C_4^2:D_{21}$ ($C_2$), $C_4^2:S_3\times C_7$ ($C_2$), $C_4\times C_{12}\times D_7$ ($C_2$), $C_{84}:Q_8$ ($C_2$), $D_{14}:D_{12}$ ($C_2$), $(C_{12}\times D_{14}):C_2$ ($C_2$), $C_{28}:D_{12}$ ($C_2$), $C_{12}:D_{28}$ ($C_2$), $C_{84}:D_4$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_3\times D_{14}):Q_8$ ($C_2$), $(C_{12}\times D_{14}):C_2$ ($C_2$)

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

Order 224: $D_7\times C_4^2$ ($S_3$)

Order 168: $C_2\times C_{84}$ ($C_2^3$) x 3, $C_{42}:C_4$ ($C_2^3$) x 3, $C_6:C_{28}$ ($C_2^3$) x 3, $C_{14}:C_{12}$ ($C_2^3$) x 3, $C_2\times D_{42}$ ($C_2^3$), $D_6\times C_{14}$ ($C_2^3$), $C_6\times D_{14}$ ($C_2^3$)

Order 112: $C_4\times D_{14}$ ($D_6$) x 3, $C_{28}:C_4$ ($D_6$) x 3, $C_4\times C_{28}$ ($D_6$)

Order 96: $C_4^2:S_3$ ($D_7$)

Order 84: $C_{84}$ ($D_4:C_2$), $C_7:C_{12}$ ($D_4:C_2$), $C_2\times C_{42}$ ($C_2^4$), $C_3\times D_{14}$ ($D_4:C_2$)

Order 56: $C_2\times C_{28}$ ($C_2\times D_6$) x 3, $C_{14}:C_4$ ($C_2\times D_6$) x 3, $C_2\times D_{14}$ ($C_2\times D_6$)

Order 48: $D_6:C_4$ ($D_{14}$) x 2, $C_6.D_4$ ($D_{14}$) x 2, $C_4\times D_6$ ($D_{14}$), $C_4\times C_{12}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($D_4:C_2^2$) x 3

Order 28: $C_2\times C_{14}$ ($C_2^2\times D_6$), $D_{14}$ ($D_{12}:C_2$), $C_{28}$ ($D_{12}:C_2$), $C_7:C_4$ ($D_{12}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_2\times D_{14}$) x 3, $C_6:C_4$ ($C_2\times D_{14}$) x 3, $C_2\times D_6$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($C_4^2.C_2^2$)

Order 16: $C_4^2$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($D_{12}:C_2^2$) x 3

Order 12: $C_2\times C_6$ ($C_2^2\times D_{14}$), $C_{12}$ ($D_{28}:C_2$)

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

Order 7: $C_7$ ($C_4^2.D_6$)

Order 6: $C_6$ ($D_4:D_{14}$) x 2, $C_6$ ($D_{28}:C_2^2$)

Order 4: $C_2^2$ ($D_6\times D_{14}$), $C_4$ ($C_{12}.D_{14}$)

Order 3: $C_3$ ($C_4^2.D_{14}$)

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

Order 1: $C_1$ ($(D_7\times C_4^2):S_3$)

Series

Derived series

$(D_7\times C_4^2):S_3$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_1$

Chief series

$(D_7\times C_4^2):S_3$

$\rhd$

$C_4\times C_{12}\times D_7$

$\rhd$

$C_4\times C_{84}$

$\rhd$

$C_4\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(D_7\times C_4^2):S_3$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

Character theory

Complex character table

See the $168 \times 168$ 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 $56 \times 56$ rational character table.