Abstract group 1344.8020: $(C_2\times C_{84}).C_2^3$

• Label: 1344.8020
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $672$
• Rank: $4$

Group information

Description:	$(C_2\times C_{84}).C_2^3$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2.(D_6\times C_2^6)$, 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	171	2	340	6	6	248	18	12	360	36	144	1344
Conjugacy classes	1	6	1	18	3	3	10	9	3	30	9	18	111
Divisions	1	6	1	14	3	1	7	3	1	7	3	3	50
Autjugacy classes	1	5	1	13	3	1	7	3	1	7	3	3	48

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	16	40	49	0	6	0	0	0	111
Irr. rational chars.	16	8	5	8	3	4	5	1	50

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:	$\langle a, b, c, d \mid b^{2}=c^{4}=d^{84}=[a,b]=1, a^{2}=d^{42}, c^{a}=cd^{42}, d^{a}=c^{2}d^{71}, c^{b}=cd^{42}, d^{b}=d^{83}, d^{c}=d^{43} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12)(13,15)(19,20,23,22)(21,24,25,26), (9,10)(14,16)(17,18) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_{42}:Q_8)$ $\,\rtimes\,$ $C_2$ (2)	$(D_{42}.D_4)$ $\,\rtimes\,$ $C_2$	$C_7$ $\,\rtimes\,$ $(C_4^2.D_6)$	$C_3$ $\,\rtimes\,$ $(C_4^2.D_{14})$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{14}$ . $(Q_8.D_6)$	$C_6$ . $(Q_8.D_{14})$	$(D_{42}.D_4)$ . $C_2$ (2)	$(C_{14}.D_4)$ . $D_6$ (4)	all 54

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 3332 subgroups in 428 conjugacy classes, 124 normal (120 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_6\times D_{14}$
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:	$\operatorname{Fit} \simeq$ $C_4:C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $(C_2\times C_{84}).C_2^3$	$G/R \simeq$ $C_1$
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: $(C_2\times C_{84}).C_2^3$

Order 672: $D_{42}.D_4$ x 2, $D_{42}:Q_8$ x 2, $D_{84}:C_4$, $C_6.(Q_8\times C_{14})$, $(C_2\times C_{12}).D_{14}$, $D_{42}.D_4$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_4\times D_{21}):C_4$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $C_{21}:(C_4\times Q_8)$

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

Order 336: $D_{42}:C_4$ x 6, $C_{42}.D_4$ x 4, $C_{42}.D_4$ x 4, $C_{42}.D_4$ x 3, $C_{21}:C_4^2$ x 2, $C_4\times D_{42}$ x 2, $D_{42}:C_4$ x 2, $D_{42}:C_4$ x 2, $C_{84}:C_4$, $C_{12}:C_{28}$, $C_{12}:C_{28}$, $C_{28}:C_{12}$, $C_{28}:C_{12}$, $C_6.D_{28}$, $C_{14}.D_{12}$, $C_2\times D_{84}$, $C_{42}:Q_8$, $C_4:C_{84}$

Order 224: $(C_2\times C_4).D_{14}$ x 2, $D_{14}:Q_8$ x 2, $D_{14}.D_4$ x 2, $(C_2\times C_4).D_{14}$ x 2, $C_4^2:D_7$ x 2, $D_{28}:C_4$, $(C_4\times D_7):C_4$, $D_{14}.D_4$, $C_7:(C_4\times Q_8)$, $C_{28}.Q_8$

Order 192: $C_4^2.D_6$

Order 168: $C_4\times D_{21}$ x 4, $C_6:C_{28}$ x 4, $C_{14}:C_{12}$ x 4, $D_{21}:C_4$ x 4, $C_2\times C_{84}$ x 3, $C_2\times D_{42}$ x 2, $C_{42}:C_4$ x 2, $D_{84}$ x 2, $C_{21}:Q_8$ x 2

Order 112: $D_{14}:C_4$ x 8, $C_{14}.D_4$ x 8, $C_4:C_{28}$ x 6, $C_4\times D_{14}$ x 4, $C_{28}:C_4$ x 4, $C_{28}:C_4$ x 2, $C_2\times D_{28}$, $C_{14}:Q_8$, $C_4\times C_{28}$

Order 96: $(C_2\times C_4).D_6$ x 2, $C_4^2:S_3$ x 2, $(C_2\times C_4).D_6$ x 2, $D_6:Q_8$ x 2, $D_6.D_4$ x 2, $(C_4\times S_3):C_4$, $D_6.D_4$, $C_3:(C_4\times Q_8)$, $C_{12}.Q_8$, $D_{12}:C_4$

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

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

Order 56: $C_4\times D_7$ x 8, $C_2\times C_{28}$ x 7, $C_{14}:C_4$ x 6, $D_{28}$ x 2, $C_7:Q_8$ x 2, $C_2\times D_{14}$ x 2

Order 48: $D_6:C_4$ x 8, $C_6.D_4$ x 8, $C_4:C_{12}$ x 6, $C_4\times D_6$ x 4, $C_{12}:C_4$ x 4, $C_{12}:C_4$ x 2, $C_2\times D_{12}$, $C_6:Q_8$, $C_4\times C_{12}$

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

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

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

Order 24: $C_4\times S_3$ x 8, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 6, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_2\times D_6$ x 2

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 16, $C_2^2:C_4$ x 8, $C_4^2$ x 5, $C_2^2\times C_4$ x 4, $C_2\times Q_8$, $C_2\times D_4$

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

Order 12: $D_6$ x 7, $C_{12}$ x 7, $C_3:C_4$ x 7, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 21, $C_2^3$ x 2, $Q_8$ x 2, $D_4$ x 2

Order 7: $C_7$

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

Order 4: $C_4$ x 14, $C_2^2$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $(C_2\times C_{84}).C_2^3$

Order 672: $D_{42}.D_4$ x 2, $D_{42}:Q_8$ x 2, $D_{84}:C_4$, $C_6.(Q_8\times C_{14})$, $(C_2\times C_{12}).D_{14}$, $D_{42}.D_4$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_4\times D_{21}):C_4$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $(C_2\times C_{12}).D_{14}$, $C_{21}:(C_4\times Q_8)$

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

Order 336: $D_{42}:C_4$ x 6, $C_{42}.D_4$ x 4, $C_{42}.D_4$ x 4, $C_{42}.D_4$ x 3, $C_{21}:C_4^2$ x 2, $C_4\times D_{42}$ x 2, $D_{42}:C_4$ x 2, $D_{42}:C_4$ x 2, $C_{84}:C_4$, $C_{12}:C_{28}$, $C_{12}:C_{28}$, $C_{28}:C_{12}$, $C_{28}:C_{12}$, $C_6.D_{28}$, $C_{14}.D_{12}$, $C_2\times D_{84}$, $C_{42}:Q_8$, $C_4:C_{84}$

Order 224: $(C_2\times C_4).D_{14}$ x 2, $D_{14}:Q_8$ x 2, $D_{14}.D_4$ x 2, $(C_2\times C_4).D_{14}$ x 2, $C_4^2:D_7$ x 2, $D_{28}:C_4$, $(C_4\times D_7):C_4$, $D_{14}.D_4$, $C_7:(C_4\times Q_8)$, $C_{28}.Q_8$

Order 192: $C_4^2.D_6$

Order 168: $C_6:C_{28}$ x 4, $C_{14}:C_{12}$ x 4, $C_2\times C_{84}$ x 3, $C_2\times D_{42}$ x 2, $C_{42}:C_4$ x 2, $C_4\times D_{21}$ x 2, $D_{21}:C_4$ x 2, $D_{84}$, $C_{21}:Q_8$

Order 112: $D_{14}:C_4$ x 8, $C_{14}.D_4$ x 8, $C_4:C_{28}$ x 6, $C_4\times D_{14}$ x 4, $C_{28}:C_4$ x 4, $C_{28}:C_4$ x 2, $C_2\times D_{28}$, $C_{14}:Q_8$, $C_4\times C_{28}$

Order 96: $(C_2\times C_4).D_6$ x 2, $C_4^2:S_3$ x 2, $(C_2\times C_4).D_6$ x 2, $D_6:Q_8$ x 2, $D_6.D_4$ x 2, $(C_4\times S_3):C_4$, $D_6.D_4$, $C_3:(C_4\times Q_8)$, $C_{12}.Q_8$, $D_{12}:C_4$

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

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, $C_2\times D_{14}$ x 2, $D_{28}$, $C_7:Q_8$

Order 48: $D_6:C_4$ x 8, $C_6.D_4$ x 8, $C_4:C_{12}$ x 6, $C_4\times D_6$ x 4, $C_{12}:C_4$ x 4, $C_{12}:C_4$ x 2, $C_2\times D_{12}$, $C_6:Q_8$, $C_4\times C_{12}$

Order 42: $C_{42}$ x 3, $D_{21}$ x 2

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

Order 28: $C_{28}$ x 7, $D_{14}$ x 6, $C_7:C_4$ x 6, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 7, $C_6:C_4$ x 6, $C_4\times S_3$ x 4, $C_2\times D_6$ x 2, $D_{12}$, $C_3:Q_8$

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 16, $C_2^2:C_4$ x 8, $C_4^2$ x 5, $C_2^2\times C_4$ x 4, $C_2\times Q_8$, $C_2\times D_4$

Order 14: $C_{14}$ x 3, $D_7$ x 2

Order 12: $C_{12}$ x 7, $D_6$ x 6, $C_3:C_4$ x 6, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 17, $C_2^3$ x 2, $Q_8$, $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 3, $S_3$ x 2

Order 4: $C_4$ x 13, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $(C_2\times C_{84}).C_2^3$ ($C_1$)

Order 672: $D_{42}.D_4$ ($C_2$) x 2, $D_{42}:Q_8$ ($C_2$) x 2, $D_{84}:C_4$ ($C_2$), $C_6.(Q_8\times C_{14})$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $D_{42}.D_4$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_4\times D_{21}):C_4$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $C_{21}:(C_4\times Q_8)$ ($C_2$)

Order 336: $D_{42}:C_4$ ($C_2^2$) x 6, $C_{42}.D_4$ ($C_2^2$) x 4, $C_{42}.D_4$ ($C_2^2$) x 4, $C_{42}.D_4$ ($C_2^2$) x 3, $C_{21}:C_4^2$ ($C_2^2$) x 2, $C_4\times D_{42}$ ($C_2^2$) x 2, $D_{42}:C_4$ ($C_2^2$) x 2, $D_{42}:C_4$ ($C_2^2$) x 2, $C_{84}:C_4$ ($C_2^2$), $C_{12}:C_{28}$ ($C_2^2$), $C_{12}:C_{28}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $C_6.D_{28}$ ($C_2^2$), $C_{14}.D_{12}$ ($C_2^2$), $C_2\times D_{84}$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_4:C_{84}$ ($C_2^2$)

Order 224: $(C_2\times C_4).D_{14}$ ($S_3$)

Order 168: $C_6:C_{28}$ ($C_2^3$) x 4, $C_{14}:C_{12}$ ($C_2^3$) x 4, $C_2\times C_{84}$ ($C_2^3$) x 3, $C_2\times D_{42}$ ($C_2^3$) x 2, $C_{42}:C_4$ ($C_2^3$) x 2

Order 112: $C_{14}.D_4$ ($D_6$) x 4, $C_4:C_{28}$ ($D_6$), $C_{28}:C_4$ ($D_6$), $C_{28}:C_4$ ($D_6$)

Order 96: $(C_2\times C_4).D_6$ ($D_7$)

Order 84: $C_{21}:C_4$ ($D_4:C_2$) x 2, $D_{42}$ ($D_4:C_2$) x 2, $C_2\times C_{42}$ ($C_2^4$)

Order 56: $C_{14}:C_4$ ($C_2\times D_6$) x 4, $C_2\times C_{28}$ ($C_2\times D_6$) x 3

Order 48: $C_6.D_4$ ($D_{14}$) x 4, $C_4:C_{12}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($D_4:C_2^2$) x 2, $C_{42}$ ($C_4.C_2^3$)

Order 28: $C_2\times C_{14}$ ($C_2^2\times D_6$)

Order 24: $C_6:C_4$ ($C_2\times D_{14}$) x 4, $C_2\times C_{12}$ ($C_2\times D_{14}$) x 3

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

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

Order 14: $C_{14}$ ($D_4:D_6$) x 2, $C_{14}$ ($Q_8.D_6$)

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

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, $C_6$ ($Q_8.D_{14}$)

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

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

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

Order 1: $C_1$ ($(C_2\times C_{84}).C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $(C_2\times C_{84}).C_2^3$ ($C_1$)

Order 672: $D_{42}.D_4$ ($C_2$) x 2, $D_{42}:Q_8$ ($C_2$) x 2, $D_{84}:C_4$ ($C_2$), $C_6.(Q_8\times C_{14})$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $D_{42}.D_4$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_4\times D_{21}):C_4$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $(C_2\times C_{12}).D_{14}$ ($C_2$), $C_{21}:(C_4\times Q_8)$ ($C_2$)

Order 336: $D_{42}:C_4$ ($C_2^2$) x 6, $C_{42}.D_4$ ($C_2^2$) x 4, $C_{42}.D_4$ ($C_2^2$) x 4, $C_{42}.D_4$ ($C_2^2$) x 3, $C_{21}:C_4^2$ ($C_2^2$) x 2, $C_4\times D_{42}$ ($C_2^2$) x 2, $D_{42}:C_4$ ($C_2^2$) x 2, $D_{42}:C_4$ ($C_2^2$) x 2, $C_{84}:C_4$ ($C_2^2$), $C_{12}:C_{28}$ ($C_2^2$), $C_{12}:C_{28}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $C_6.D_{28}$ ($C_2^2$), $C_{14}.D_{12}$ ($C_2^2$), $C_2\times D_{84}$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_4:C_{84}$ ($C_2^2$)

Order 224: $(C_2\times C_4).D_{14}$ ($S_3$)

Order 168: $C_6:C_{28}$ ($C_2^3$) x 4, $C_{14}:C_{12}$ ($C_2^3$) x 4, $C_2\times C_{84}$ ($C_2^3$) x 3, $C_2\times D_{42}$ ($C_2^3$) x 2, $C_{42}:C_4$ ($C_2^3$) x 2

Order 112: $C_{14}.D_4$ ($D_6$) x 4, $C_4:C_{28}$ ($D_6$), $C_{28}:C_4$ ($D_6$), $C_{28}:C_4$ ($D_6$)

Order 96: $(C_2\times C_4).D_6$ ($D_7$)

Order 84: $C_{21}:C_4$ ($D_4:C_2$), $C_2\times C_{42}$ ($C_2^4$), $D_{42}$ ($D_4:C_2$)

Order 56: $C_{14}:C_4$ ($C_2\times D_6$) x 4, $C_2\times C_{28}$ ($C_2\times D_6$) x 3

Order 48: $C_6.D_4$ ($D_{14}$) x 4, $C_4:C_{12}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($D_4:C_2^2$) x 2, $C_{42}$ ($C_4.C_2^3$)

Order 28: $C_2\times C_{14}$ ($C_2^2\times D_6$)

Order 24: $C_6:C_4$ ($C_2\times D_{14}$) x 4, $C_2\times C_{12}$ ($C_2\times D_{14}$) x 3

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

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

Order 14: $C_{14}$ ($D_4:D_6$) x 2, $C_{14}$ ($Q_8.D_6$)

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

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, $C_6$ ($Q_8.D_{14}$)

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

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

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

Order 1: $C_1$ ($(C_2\times C_{84}).C_2^3$)

Series

Derived series

$(C_2\times C_{84}).C_2^3$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_1$

Chief series

$(C_2\times C_{84}).C_2^3$

$\rhd$

$(C_2\times C_{12}).D_{14}$

$\rhd$

$D_{42}:C_4$

$\rhd$

$C_6:C_{28}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_{84}).C_2^3$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4:C_4$

Character theory

Complex character table

See the $111 \times 111$ 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.