Abstract group 1344.3194: $C_{84}:C_4^2$

• Label: 1344.3194
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $1344$
• Rank: $3$

Group information

Description:	$C_{84}:C_4^2$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{42}.(C_2^5\times C_6).C_2^5$, of order \(258048\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 7 \)
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	7	2	504	14	6	240	42	12	336	84	96	1344
Conjugacy classes	1	7	1	32	7	3	12	21	3	36	21	24	168
Divisions	1	7	1	18	7	1	8	7	1	8	7	4	70
Autjugacy classes	1	7	1	5	7	1	2	7	1	2	7	1	42

Dimension	1	2	4	6	12	24
Irr. complex chars.	32	72	64	0	0	0	168
Irr. rational chars.	8	24	10	8	16	4	70

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$1344$
Rank:	$3$
Inequivalent generating triples:	$2688$

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:	$\langle a, b, c \mid a^{4}=b^{4}=c^{84}=[a,b]=1, c^{a}=c^{71}, c^{b}=c^{55} \rangle$
Permutation group:	Degree $22$ $\langle(2,3)(4,5)(6,7)(11,12)(13,14)(15,16,18,17), (9,10)(11,12)(13,14)(15,17,18,16) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{84}$ $\,\rtimes\,$ $C_4^2$ (4)	$(C_{84}:C_4)$ $\,\rtimes\,$ $C_4$ (4)	$(C_{28}:C_{12})$ $\,\rtimes\,$ $C_4$ (4)	$(C_{12}:C_{28})$ $\,\rtimes\,$ $C_4$ (4)	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{42}:C_4)$ . $Q_8$ (2)	$C_{42}$ . $(C_4\times Q_8)$ (2)	$C_{14}$ . $(D_6.D_4)$	$C_6$ . $(D_{14}.D_4)$	all 84

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

Homology

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

Subgroups

There are 2036 subgroups in 388 conjugacy classes, 176 normal (76 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $S_3\times D_{14}$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2\times C_4^2$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{84}:C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:C_4^2$
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: $C_{84}:C_4^2$

Order 672: $C_{42}:C_4^2$ x 2, $C_{42}.C_4^2$ x 2, $C_{42}:C_4^2$, $C_2\times C_{12}:C_{28}$, $C_2\times C_{28}:C_{12}$

Order 448: $C_{28}:C_4^2$

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

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

Order 192: $C_{12}:C_4^2$

Order 168: $C_{42}:C_4$ x 10, $C_6:C_{28}$ x 8, $C_{14}:C_{12}$ x 8, $C_2\times C_{84}$ x 6, $C_2^2\times C_{42}$

Order 112: $C_{28}:C_4$ x 8, $C_2^2.D_{14}$ x 4, $C_4:C_{28}$ x 4, $C_{28}:C_4$ x 4, $C_2^2\times C_{28}$ x 3

Order 96: $C_6:C_4^2$ x 3, $C_6.C_4^2$ x 2, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$

Order 84: $C_2\times C_{42}$ x 7, $C_{21}:C_4$ x 6, $C_{84}$ x 4, $C_7:C_{12}$ x 4, $C_3:C_{28}$ x 4

Order 64: $C_4:C_4^2$

Order 56: $C_{14}:C_4$ x 18, $C_2\times C_{28}$ x 14, $C_2^2\times C_{14}$

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

Order 42: $C_{42}$ x 7

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

Order 28: $C_7:C_4$ x 10, $C_{28}$ x 8, $C_2\times C_{14}$ x 7

Order 24: $C_6:C_4$ x 18, $C_2\times C_{12}$ x 14, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 8, $C_4^2$ x 8, $C_2^2\times C_4$ x 7

Order 14: $C_{14}$ x 7

Order 12: $C_3:C_4$ x 10, $C_{12}$ x 8, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 32, $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 18, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_{84}:C_4^2$

Order 672: $C_{42}:C_4^2$, $C_2\times C_{12}:C_{28}$, $C_2\times C_{28}:C_{12}$, $C_{42}:C_4^2$, $C_{42}.C_4^2$

Order 448: $C_{28}:C_4^2$

Order 336: $C_2^2.D_{42}$ x 2, $C_{84}:C_4$, $C_{12}:C_{28}$, $C_{28}:C_{12}$, $C_{21}:C_4^2$, $C_2^2\times C_{84}$, $C_{42}.C_2^3$, $C_{42}.C_2^3$

Order 224: $C_{14}:C_4^2$ x 2, $C_{14}.C_4^2$, $C_2\times C_4:C_{28}$, $C_2^2.D_{28}$

Order 192: $C_{12}:C_4^2$

Order 168: $C_{42}:C_4$ x 6, $C_2\times C_{84}$ x 3, $C_6:C_{28}$ x 3, $C_{14}:C_{12}$ x 3, $C_2^2\times C_{42}$

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

Order 96: $C_6:C_4^2$ x 2, $C_6.C_4^2$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$

Order 84: $C_2\times C_{42}$ x 7, $C_{21}:C_4$ x 2, $C_{84}$, $C_7:C_{12}$, $C_3:C_{28}$

Order 64: $C_4:C_4^2$

Order 56: $C_{14}:C_4$ x 9, $C_2\times C_{28}$ x 6, $C_2^2\times C_{14}$

Order 48: $C_6.C_2^3$ x 3, $C_2^2\times C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_4:C_{12}$, $C_{12}:C_4$

Order 42: $C_{42}$ x 7

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

Order 28: $C_2\times C_{14}$ x 7, $C_7:C_4$ x 3, $C_{28}$ x 2

Order 24: $C_6:C_4$ x 9, $C_2\times C_{12}$ x 6, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $C_2^2\times C_4$ x 5, $C_4:C_4$ x 2, $C_4^2$ x 2

Order 14: $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 7, $C_3:C_4$ x 3, $C_{12}$ x 2

Order 8: $C_2\times C_4$ x 15, $C_2^3$

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 7, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_{84}:C_4^2$ ($C_1$)

Order 672: $C_{42}:C_4^2$ ($C_2$) x 2, $C_{42}.C_4^2$ ($C_2$) x 2, $C_{42}:C_4^2$ ($C_2$), $C_2\times C_{12}:C_{28}$ ($C_2$), $C_2\times C_{28}:C_{12}$ ($C_2$)

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

Order 224: $C_2^2.D_{28}$ ($S_3$)

Order 168: $C_2\times C_{84}$ ($C_2\times C_4$) x 6, $C_{42}:C_4$ ($C_2\times C_4$) x 4, $C_6:C_{28}$ ($C_2\times C_4$) x 4, $C_{14}:C_{12}$ ($C_2\times C_4$) x 4, $C_{42}:C_4$ ($Q_8$) x 2, $C_{42}:C_4$ ($D_4$) x 2, $C_2^2\times C_{42}$ ($C_2^3$)

Order 112: $C_{28}:C_4$ ($C_3:C_4$) x 4, $C_2^2.D_{14}$ ($D_6$) x 2, $C_2^2\times C_{28}$ ($D_6$)

Order 96: $C_2^2.D_{12}$ ($D_7$)

Order 84: $C_{84}$ ($C_4^2$) x 4, $C_{21}:C_4$ ($C_4:C_4$) x 4, $C_2\times C_{42}$ ($C_2^2\times C_4$) x 3, $C_2\times C_{42}$ ($D_4:C_2$) x 2, $C_2\times C_{42}$ ($C_2\times Q_8$), $C_2\times C_{42}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($C_4\times S_3$) x 4, $C_{14}:C_4$ ($C_6:C_4$) x 4, $C_2\times C_{28}$ ($C_6:C_4$) x 2, $C_2^2\times C_{14}$ ($C_2\times D_6$)

Order 48: $C_{12}:C_4$ ($C_7:C_4$) x 4, $C_6.C_2^3$ ($D_{14}$) x 2, $C_2^2\times C_{12}$ ($D_{14}$)

Order 42: $C_{42}$ ($C_4\times Q_8$) x 2, $C_{42}$ ($C_4\times D_4$) x 2, $C_{42}$ ($C_4^2:C_2$), $C_{42}$ ($C_2^2.D_4$), $C_{42}$ ($C_2\times C_4^2$)

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

Order 24: $C_2\times C_{12}$ ($C_4\times D_7$) x 4, $C_6:C_4$ ($C_{14}:C_4$) x 4, $C_2\times C_{12}$ ($C_{14}:C_4$) x 2, $C_2^2\times C_6$ ($C_2\times D_{14}$)

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

Order 16: $C_2^2\times C_4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($(C_4\times S_3):C_4$), $C_{14}$ ($D_6.D_4$), $C_{14}$ ($C_3:(C_4\times Q_8)$), $C_{14}$ ($C_3:C_4\times Q_8$), $C_{14}$ ($C_2^3.D_6$), $C_{14}$ ($C_6:C_4^2$), $C_{14}$ ($D_{12}:C_4$)

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

Order 8: $C_2\times C_4$ ($D_{21}:C_4$) x 2, $C_2\times C_4$ ($C_6.D_{14}$) x 2, $C_2\times C_4$ ($C_6.D_{14}$) x 2, $C_2^3$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_{12}:C_4^2$)

Order 6: $C_6$ ($D_{28}:C_4$), $C_6$ ($(C_4\times D_7):C_4$), $C_6$ ($D_{14}.D_4$), $C_6$ ($C_7:(C_4\times Q_8)$), $C_6$ ($C_7:C_4\times Q_8$), $C_6$ ($C_2^3.D_{14}$), $C_6$ ($C_{14}:C_4^2$)

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

Order 3: $C_3$ ($C_{28}:C_4^2$)

Order 2: $C_2$ ($C_{42}:C_4^2$), $C_2$ ($D_{42}.D_4$), $C_2$ ($C_{84}:(C_2\times C_4)$), $C_2$ ($C_{84}:(C_2\times C_4)$), $C_2$ ($(C_4\times D_{21}):C_4$), $C_2$ ($C_{21}:(C_4\times Q_8)$), $C_2$ ($C_{21}:(C_4\times Q_8)$)

Order 1: $C_1$ ($C_{84}:C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_{84}:C_4^2$ ($C_1$)

Order 672: $C_{42}:C_4^2$ ($C_2$), $C_2\times C_{12}:C_{28}$ ($C_2$), $C_2\times C_{28}:C_{12}$ ($C_2$), $C_{42}:C_4^2$ ($C_2$), $C_{42}.C_4^2$ ($C_2$)

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

Order 224: $C_2^2.D_{28}$ ($S_3$)

Order 168: $C_2\times C_{84}$ ($C_2\times C_4$) x 3, $C_{42}:C_4$ ($C_2\times C_4$) x 2, $C_2^2\times C_{42}$ ($C_2^3$), $C_{42}:C_4$ ($Q_8$), $C_{42}:C_4$ ($D_4$), $C_6:C_{28}$ ($C_2\times C_4$), $C_{14}:C_{12}$ ($C_2\times C_4$)

Order 112: $C_2^2\times C_{28}$ ($D_6$), $C_2^2.D_{14}$ ($D_6$), $C_{28}:C_4$ ($C_3:C_4$)

Order 96: $C_2^2.D_{12}$ ($D_7$)

Order 84: $C_2\times C_{42}$ ($C_2^2\times C_4$) x 3, $C_2\times C_{42}$ ($D_4:C_2$) x 2, $C_{84}$ ($C_4^2$), $C_{21}:C_4$ ($C_4:C_4$), $C_2\times C_{42}$ ($C_2\times Q_8$), $C_2\times C_{42}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($C_4\times S_3$) x 2, $C_2\times C_{28}$ ($C_6:C_4$), $C_{14}:C_4$ ($C_6:C_4$), $C_2^2\times C_{14}$ ($C_2\times D_6$)

Order 48: $C_2^2\times C_{12}$ ($D_{14}$), $C_6.C_2^3$ ($D_{14}$), $C_{12}:C_4$ ($C_7:C_4$)

Order 42: $C_{42}$ ($C_4\times Q_8$) x 2, $C_{42}$ ($C_4\times D_4$) x 2, $C_{42}$ ($C_4^2:C_2$), $C_{42}$ ($C_2^2.D_4$), $C_{42}$ ($C_2\times C_4^2$)

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

Order 24: $C_2\times C_{12}$ ($C_4\times D_7$) x 2, $C_2\times C_{12}$ ($C_{14}:C_4$), $C_6:C_4$ ($C_{14}:C_4$), $C_2^2\times C_6$ ($C_2\times D_{14}$)

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

Order 16: $C_2^2\times C_4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($(C_4\times S_3):C_4$), $C_{14}$ ($D_6.D_4$), $C_{14}$ ($C_3:(C_4\times Q_8)$), $C_{14}$ ($C_3:C_4\times Q_8$), $C_{14}$ ($C_2^3.D_6$), $C_{14}$ ($C_6:C_4^2$), $C_{14}$ ($D_{12}:C_4$)

Order 12: $C_2\times C_6$ ($C_4\times D_{14}$) x 2, $C_2\times C_6$ ($C_2^2.D_{14}$), $C_2\times C_6$ ($D_{28}:C_2$), $C_2\times C_6$ ($Q_8\times D_7$), $C_2\times C_6$ ($D_4:D_7$), $C_2\times C_6$ ($D_4\times D_7$), $C_{12}$ ($C_{28}:C_4$)

Order 8: $C_2^3$ ($S_3\times D_{14}$), $C_2\times C_4$ ($D_{21}:C_4$), $C_2\times C_4$ ($C_6.D_{14}$), $C_2\times C_4$ ($C_6.D_{14}$)

Order 7: $C_7$ ($C_{12}:C_4^2$)

Order 6: $C_6$ ($D_{28}:C_4$), $C_6$ ($(C_4\times D_7):C_4$), $C_6$ ($D_{14}.D_4$), $C_6$ ($C_7:(C_4\times Q_8)$), $C_6$ ($C_7:C_4\times Q_8$), $C_6$ ($C_2^3.D_{14}$), $C_6$ ($C_{14}:C_4^2$)

Order 4: $C_2^2$ ($D_{42}:C_4$), $C_2^2$ ($D_6.D_{14}$), $C_2^2$ ($D_{14}.D_6$), $C_2^2$ ($C_{12}:D_{14}$), $C_2^2$ ($D_{21}:Q_8$), $C_2^2$ ($D_{12}:D_7$), $C_2^2$ ($D_{28}:S_3$), $C_4$ ($C_{21}:C_4^2$)

Order 3: $C_3$ ($C_{28}:C_4^2$)

Order 2: $C_2$ ($C_{42}:C_4^2$), $C_2$ ($D_{42}.D_4$), $C_2$ ($C_{84}:(C_2\times C_4)$), $C_2$ ($C_{84}:(C_2\times C_4)$), $C_2$ ($(C_4\times D_{21}):C_4$), $C_2$ ($C_{21}:(C_4\times Q_8)$), $C_2$ ($C_{21}:(C_4\times Q_8)$)

Order 1: $C_1$ ($C_{84}:C_4^2$)

Series

Derived series

$C_{84}:C_4^2$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_{84}:C_4^2$

$\rhd$

$C_{42}:C_4^2$

$\rhd$

$C_{42}.C_2^3$

$\rhd$

$C_2^2\times C_{42}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{84}:C_4^2$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

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 $70 \times 70$ rational character table.