Abstract group 672.1158: $C_{84}.C_2^3$

• Label: 672.1158
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $4$

Group information

Description:	$C_{84}.C_2^3$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$(C_{21}\times A_4).C_6.C_2^4$, of order \(24192\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7 \)
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	147	2	108	30	6	96	42	12	144	12	72	672
Conjugacy classes	1	6	1	10	3	3	6	9	3	18	3	9	72
Divisions	1	6	1	10	2	1	6	2	1	6	1	3	40
Autjugacy classes	1	4	1	4	2	1	2	2	1	2	1	1	22

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	16	32	21	0	3	0	0	72
Irr. rational chars.	16	8	1	8	1	4	2	40

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$4$
Inequivalent generating quadruples:	$2489760$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{6}=d^{28}=[a,b]=[a,c]=[b,d]=1, a^{2}=d^{14}, d^{a}=d^{15}, c^{b}=c^{5}d^{14}, d^{c}=d^{13} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(12,16)(17,23)(18,24)(25,26), (9,10)(11,12)(13,17)(14,18) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8\times D_7)$ $\,\rtimes\,$ $S_3$	$(S_3\times Q_8)$ $\,\rtimes\,$ $D_7$	$C_7$ $\,\rtimes\,$ $(Q_8.D_6)$	$(D_{21}:Q_8)$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{84}$ . $C_2^2$	$D_{42}$ . $C_2^3$	$C_{84}$ . $C_2^3$	$C_{42}$ . $C_2^4$	all 37

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 1808 subgroups in 292 conjugacy classes, 108 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6\times D_{14}$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{84}.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.C_2^3$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 672: $C_{84}.C_2^3$

Order 336: $D_{84}:C_2$ x 3, $C_{12}.D_{14}$ x 3, $D_{21}:Q_8$ x 3, $D_{84}:C_2$ x 3, $D_{84}:C_2$, $C_{28}.D_6$, $C_{12}.D_{14}$

Order 224: $Q_8.D_{14}$

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

Order 112: $D_{28}:C_2$ x 6, $D_{28}:C_2$ x 4, $Q_8\times D_7$ x 4, $Q_8\times C_{14}$

Order 96: $Q_8.D_6$

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

Order 56: $C_4\times D_7$ x 12, $D_{28}$ x 6, $C_7:Q_8$ x 6, $C_7:D_4$ x 4, $C_7\times Q_8$ x 4, $C_2\times C_{28}$ x 3

Order 48: $D_{12}:C_2$ x 6, $D_{12}:C_2$ x 4, $S_3\times Q_8$ x 4, $C_6\times Q_8$

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

Order 32: $C_4.C_2^3$

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

Order 24: $C_4\times S_3$ x 12, $D_{12}$ x 6, $C_3:Q_8$ x 6, $C_3:D_4$ x 4, $C_3\times Q_8$ x 4, $C_2\times C_{12}$ x 3

Order 21: $C_{21}$

Order 16: $D_4:C_2$ x 10, $C_2\times Q_8$ x 5

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

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

Order 8: $C_2\times C_4$ x 15, $Q_8$ x 10, $D_4$ x 10

Order 7: $C_7$

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

Order 4: $C_4$ x 10, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $C_{84}.C_2^3$

Order 336: $D_{84}:C_2$, $C_{28}.D_6$, $C_{12}.D_{14}$, $D_{84}:C_2$, $C_{12}.D_{14}$, $D_{21}:Q_8$, $D_{84}:C_2$

Order 224: $Q_8.D_{14}$

Order 168: $Q_8\times C_{21}$, $D_{84}$, $C_4\times D_{21}$, $S_3\times C_{28}$, $C_{21}:Q_8$, $C_{12}\times D_7$, $C_{21}:Q_8$, $C_{21}:Q_8$, $C_7:D_{12}$, $C_3:D_{28}$, $C_{21}:D_4$, $D_{21}:C_4$

Order 112: $D_{28}:C_2$ x 2, $Q_8\times D_7$ x 2, $D_{28}:C_2$ x 2, $Q_8\times C_{14}$

Order 96: $Q_8.D_6$

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

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

Order 48: $D_{12}:C_2$ x 2, $S_3\times Q_8$ x 2, $D_{12}:C_2$ x 2, $C_6\times Q_8$

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

Order 32: $C_4.C_2^3$

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

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

Order 21: $C_{21}$

Order 16: $D_4:C_2$ x 4, $C_2\times Q_8$ x 3

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

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

Order 8: $Q_8$ x 4, $D_4$ x 4, $C_2\times C_4$ x 4

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $C_{84}.C_2^3$ ($C_1$)

Order 336: $D_{84}:C_2$ ($C_2$) x 3, $C_{12}.D_{14}$ ($C_2$) x 3, $D_{21}:Q_8$ ($C_2$) x 3, $D_{84}:C_2$ ($C_2$) x 3, $D_{84}:C_2$ ($C_2$), $C_{28}.D_6$ ($C_2$), $C_{12}.D_{14}$ ($C_2$)

Order 168: $D_{21}:C_4$ ($C_2^2$) x 6, $D_{84}$ ($C_2^2$) x 3, $C_4\times D_{21}$ ($C_2^2$) x 3, $S_3\times C_{28}$ ($C_2^2$) x 3, $C_{21}:Q_8$ ($C_2^2$) x 3, $C_{12}\times D_7$ ($C_2^2$) x 3, $C_{21}:Q_8$ ($C_2^2$) x 3, $C_{21}:Q_8$ ($C_2^2$) x 3, $C_7:D_{12}$ ($C_2^2$) x 3, $C_3:D_{28}$ ($C_2^2$) x 3, $Q_8\times C_{21}$ ($C_2^2$), $C_{21}:D_4$ ($C_2^2$)

Order 112: $Q_8\times D_7$ ($S_3$)

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

Order 56: $C_4\times D_7$ ($D_6$) x 3, $C_7:Q_8$ ($D_6$) x 3, $C_7\times Q_8$ ($D_6$)

Order 48: $S_3\times Q_8$ ($D_7$)

Order 42: $C_{42}$ ($C_2^4$)

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

Order 24: $C_4\times S_3$ ($D_{14}$) x 3, $C_3:Q_8$ ($D_{14}$) x 3, $C_3\times Q_8$ ($D_{14}$)

Order 21: $C_{21}$ ($C_4.C_2^3$)

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

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

Order 8: $Q_8$ ($S_3\times D_7$)

Order 7: $C_7$ ($Q_8.D_6$)

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

Order 4: $C_4$ ($S_3\times D_{14}$) x 3

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $C_{84}.C_2^3$ ($C_1$)

Order 336: $D_{84}:C_2$ ($C_2$), $C_{28}.D_6$ ($C_2$), $C_{12}.D_{14}$ ($C_2$), $D_{84}:C_2$ ($C_2$), $C_{12}.D_{14}$ ($C_2$), $D_{21}:Q_8$ ($C_2$), $D_{84}:C_2$ ($C_2$)

Order 168: $Q_8\times C_{21}$ ($C_2^2$), $D_{84}$ ($C_2^2$), $C_4\times D_{21}$ ($C_2^2$), $S_3\times C_{28}$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$), $C_{12}\times D_7$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$), $C_7:D_{12}$ ($C_2^2$), $C_3:D_{28}$ ($C_2^2$), $C_{21}:D_4$ ($C_2^2$), $D_{21}:C_4$ ($C_2^2$)

Order 112: $Q_8\times D_7$ ($S_3$)

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

Order 56: $C_4\times D_7$ ($D_6$), $C_7:Q_8$ ($D_6$), $C_7\times Q_8$ ($D_6$)

Order 48: $S_3\times Q_8$ ($D_7$)

Order 42: $C_{42}$ ($C_2^4$)

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

Order 24: $C_4\times S_3$ ($D_{14}$), $C_3:Q_8$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$)

Order 21: $C_{21}$ ($C_4.C_2^3$)

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

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

Order 8: $Q_8$ ($S_3\times D_7$)

Order 7: $C_7$ ($Q_8.D_6$)

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

Order 4: $C_4$ ($S_3\times D_{14}$)

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

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

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

Series

Derived series

$C_{84}.C_2^3$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_{84}.C_2^3$

$\rhd$

$C_{28}.D_6$

$\rhd$

$Q_8\times C_{21}$

$\rhd$

$C_{84}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{84}.C_2^3$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$Q_8$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 120 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.