Abstract group 672.1215: $C_{84}.C_2^3$

• Label: 672.1215
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• 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:	$(C_{14}\times A_4).C_6.C_2^4$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
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	35	2	92	70	6	184	42	12	48	84	96	672
Conjugacy classes	1	6	2	10	12	3	20	12	6	15	24	30	141
Divisions	1	6	1	10	6	1	10	4	1	4	4	4	52
Autjugacy classes	1	3	1	3	3	1	3	2	1	2	2	2	24

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	48	72	21	0	0	0	0	0	141
Irr. rational chars.	16	16	1	8	1	8	1	1	52

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$4$
Inequivalent generating quadruples:	$3830400$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{4}=d^{42}=[a,b]=[a,c]=[c,d]=1, a^{2}=c^{2}, d^{a}=c^{2}d, c^{b}=c^{3}, d^{b}=c^{2}d^{13} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12)(13,17)(14,18)(15,23)(16,19)(20,24)(21,25)(22,26) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(D_4.D_{14})$
Semidirect product:	$(Q_8\times D_7)$ $\,\rtimes\,$ $C_6$	$(D_4:D_7)$ $\,\rtimes\,$ $C_6$	$(D_4:C_6)$ $\,\rtimes\,$ $D_7$	$(C_{42}:Q_8)$ $\,\rtimes\,$ $C_2$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{84}$ . $C_2^3$ (2)	$C_{42}$ . $C_2^4$	$(C_3\times Q_8)$ . $D_{14}$	$Q_8$ . $(C_3\times D_{14})$	all 38

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

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 972 subgroups in 292 conjugacy classes, 170 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2^2\times D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{42}:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

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

Order 336: $C_{12}.D_{14}$ x 6, $D_{28}:C_6$ x 3, $C_{42}:Q_8$ x 3, $C_{12}.D_{14}$ x 2, $D_4:C_{42}$

Order 224: $D_4.D_{14}$

Order 168: $C_{21}:Q_8$ x 9, $C_{21}:D_4$ x 6, $C_{14}:C_{12}$ x 6, $C_{12}\times D_7$ x 6, $D_4\times C_{21}$ x 3, $C_2\times C_{84}$ x 3, $Q_8\times C_{21}$, $C_3\times D_{28}$

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

Order 96: $C_6.C_2^4$

Order 84: $C_7:C_{12}$ x 6, $C_{84}$ x 4, $C_2\times C_{42}$ x 3, $C_3\times D_{14}$ x 2

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

Order 48: $D_4:C_6$ x 10, $C_6\times Q_8$ x 5

Order 42: $C_{42}$ x 4, $C_3\times D_7$ x 2

Order 32: $C_4.C_2^3$

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

Order 24: $C_2\times C_{12}$ x 15, $C_3\times Q_8$ x 10, $C_3\times D_4$ x 10

Order 21: $C_{21}$

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

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

Order 12: $C_{12}$ x 10, $C_2\times C_6$ x 5

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

Order 7: $C_7$

Order 6: $C_6$ x 6

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: $D_4:C_{42}$, $C_{12}.D_{14}$, $C_{12}.D_{14}$, $D_{28}:C_6$, $C_{42}:Q_8$

Order 224: $D_4.D_{14}$

Order 168: $C_{21}:Q_8$ x 2, $Q_8\times C_{21}$, $D_4\times C_{21}$, $C_2\times C_{84}$, $C_{21}:D_4$, $C_{14}:C_{12}$, $C_3\times D_{28}$, $C_{12}\times D_7$

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

Order 96: $C_6.C_2^4$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$, $C_2\times C_{42}$, $C_3\times D_{14}$

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

Order 48: $D_4:C_6$ x 3, $C_6\times Q_8$ x 2

Order 42: $C_{42}$ x 2, $C_3\times D_7$

Order 32: $C_4.C_2^3$

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

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

Order 21: $C_{21}$

Order 16: $D_4:C_2$ x 3, $C_2\times Q_8$ x 2

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

Order 12: $C_{12}$ x 3, $C_2\times C_6$ x 2

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

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 336: $C_{12}.D_{14}$ ($C_2$) x 6, $D_{28}:C_6$ ($C_2$) x 3, $C_{42}:Q_8$ ($C_2$) x 3, $C_{12}.D_{14}$ ($C_2$) x 2, $D_4:C_{42}$ ($C_2$)

Order 224: $D_4.D_{14}$ ($C_3$)

Order 168: $C_{21}:Q_8$ ($C_2^2$) x 9, $C_{21}:D_4$ ($C_2^2$) x 6, $C_{14}:C_{12}$ ($C_2^2$) x 6, $C_{12}\times D_7$ ($C_2^2$) x 6, $D_4\times C_{21}$ ($C_2^2$) x 3, $C_2\times C_{84}$ ($C_2^2$) x 3, $Q_8\times C_{21}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$)

Order 112: $D_4:D_7$ ($C_6$) x 6, $D_{28}:C_2$ ($C_6$) x 3, $C_{14}:Q_8$ ($C_6$) x 3, $Q_8\times D_7$ ($C_6$) x 2, $D_4:C_{14}$ ($C_6$)

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

Order 56: $C_7:Q_8$ ($C_2\times C_6$) x 9, $C_7:D_4$ ($C_2\times C_6$) x 6, $C_{14}:C_4$ ($C_2\times C_6$) x 6, $C_4\times D_7$ ($C_2\times C_6$) x 6, $C_7\times D_4$ ($C_2\times C_6$) x 3, $C_2\times C_{28}$ ($C_2\times C_6$) x 3, $D_{28}$ ($C_2\times C_6$), $C_7\times Q_8$ ($C_2\times C_6$)

Order 48: $D_4:C_6$ ($D_7$)

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

Order 28: $C_7:C_4$ ($C_2^2\times C_6$) x 6, $C_{28}$ ($C_2^2\times C_6$) x 4, $C_2\times C_{14}$ ($C_2^2\times C_6$) x 3, $D_{14}$ ($C_2^2\times C_6$) x 2

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

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

Order 16: $D_4:C_2$ ($C_3\times D_7$)

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

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

Order 8: $D_4$ ($C_3\times D_{14}$) x 3, $C_2\times C_4$ ($C_3\times D_{14}$) x 3, $Q_8$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($C_6.C_2^4$)

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

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

Order 3: $C_3$ ($D_4.D_{14}$)

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

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: $D_4:C_{42}$ ($C_2$), $C_{12}.D_{14}$ ($C_2$), $C_{12}.D_{14}$ ($C_2$), $D_{28}:C_6$ ($C_2$), $C_{42}:Q_8$ ($C_2$)

Order 224: $D_4.D_{14}$ ($C_3$)

Order 168: $C_{21}:Q_8$ ($C_2^2$) x 2, $Q_8\times C_{21}$ ($C_2^2$), $D_4\times C_{21}$ ($C_2^2$), $C_2\times C_{84}$ ($C_2^2$), $C_{21}:D_4$ ($C_2^2$), $C_{14}:C_{12}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$), $C_{12}\times D_7$ ($C_2^2$)

Order 112: $D_4:C_{14}$ ($C_6$), $Q_8\times D_7$ ($C_6$), $D_4:D_7$ ($C_6$), $D_{28}:C_2$ ($C_6$), $C_{14}:Q_8$ ($C_6$)

Order 84: $C_{84}$ ($C_2^3$) x 2, $C_7:C_{12}$ ($C_2^3$), $C_2\times C_{42}$ ($C_2^3$), $C_3\times D_{14}$ ($C_2^3$)

Order 56: $C_7:Q_8$ ($C_2\times C_6$) x 2, $C_7\times D_4$ ($C_2\times C_6$), $C_2\times C_{28}$ ($C_2\times C_6$), $C_7:D_4$ ($C_2\times C_6$), $C_{14}:C_4$ ($C_2\times C_6$), $D_{28}$ ($C_2\times C_6$), $C_4\times D_7$ ($C_2\times C_6$), $C_7\times Q_8$ ($C_2\times C_6$)

Order 48: $D_4:C_6$ ($D_7$)

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

Order 28: $C_{28}$ ($C_2^2\times C_6$) x 2, $C_2\times C_{14}$ ($C_2^2\times C_6$), $D_{14}$ ($C_2^2\times C_6$), $C_7:C_4$ ($C_2^2\times C_6$)

Order 24: $C_2\times C_{12}$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$), $C_3\times D_4$ ($D_{14}$)

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

Order 16: $D_4:C_2$ ($C_3\times D_7$)

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

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

Order 8: $Q_8$ ($C_3\times D_{14}$), $D_4$ ($C_3\times D_{14}$), $C_2\times C_4$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($C_6.C_2^4$)

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

Order 4: $C_4$ ($C_6\times D_{14}$) x 2, $C_2^2$ ($C_6\times D_{14}$)

Order 3: $C_3$ ($D_4.D_{14}$)

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

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

Series

Derived series

$C_{84}.C_2^3$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_{84}.C_2^3$

$\rhd$

$D_{28}:C_6$

$\rhd$

$C_{12}\times D_7$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{84}.C_2^3$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$D_4:C_6$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 72 larger groups in the database.

Character theory

Complex character table

See the $141 \times 141$ 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 $52 \times 52$ rational character table.