Abstract group 1344.11371: $C_2^4:C_{84}$

• Label: 1344.11371
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $168$
• Rank: $2$

Group information

Description:	$C_2^4:C_{84}$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$D_4\times F_8:C_6$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	31	2	32	62	48	64	144	96	192	288	384	1344
Conjugacy classes	1	7	2	8	14	6	16	18	12	24	36	48	192
Divisions	1	7	1	4	7	1	4	3	1	2	3	2	36
Autjugacy classes	1	5	1	2	5	2	2	4	2	2	4	2	32

Dimension	1	2	4	6	7	12	14	24	28
Irr. complex chars.	168	0	0	0	24	0	0	0	0	192
Irr. rational chars.	4	6	2	4	4	6	6	2	2	36

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$168$
Rank:	$2$
Inequivalent generating pairs:	$192$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{28}=c^{2}=d^{2}=e^{6}=[a,b]=[a,c]=[a,d]=[a,e]=[c,d]=[c,e]=[d,e]=1, c^{b}=d, d^{b}=cde^{3}, e^{b}=cde^{4} \rangle$
Permutation group:	Degree $17$ $\langle(12,13), (12,13)(14,15,16,17), (9,10,11), (2,3,7,8,4,5,6), (14,16)(15,17), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,8)(6,7), (1,4)(2,6)(3,8)(5,7)\rangle$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $F_8$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $C_{84}$	$(C_2^4\times C_{12})$ $\,\rtimes\,$ $C_7$	$(C_2^4\times C_4)$ $\,\rtimes\,$ $C_{21}$	$(C_2^3\times C_6)$ $\,\rtimes\,$ $C_{28}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_{42}$	$(C_6\times F_8)$ . $C_2^2$	$(C_2^2\times F_8)$ . $C_6$	$C_6$ . $(C_2^2\times F_8)$	all 12

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

Homology

Abelianization:	$C_{2} \times C_{84} \simeq C_{2} \times C_{4} \times C_{3} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 1506 subgroups in 254 conjugacy classes, 48 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{12}$	$G/Z \simeq$ $F_8$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2\times C_{84}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^4:C_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_7$
Radical:	$R \simeq$ $C_2^4:C_{84}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_6$	$G/\operatorname{soc} \simeq$ $C_{14}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4\times C_4$
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 1344: $C_2^4:C_{84}$

Order 672: $C_{12}\times F_8$ x 2, $C_2^4:C_{42}$

Order 448: $C_2^4:C_{28}$

Order 336: $C_6\times F_8$ x 3

Order 224: $C_4\times F_8$ x 2, $C_2^2\times F_8$

Order 192: $C_2^4\times C_{12}$

Order 168: $C_3\times F_8$, $C_2\times C_{84}$

Order 112: $C_2\times F_8$ x 3

Order 96: $C_2^3\times C_{12}$ x 6, $C_2^4\times C_6$

Order 84: $C_{84}$ x 2, $C_2\times C_{42}$

Order 64: $C_2^4\times C_4$

Order 56: $C_2\times C_{28}$, $F_8$

Order 48: $C_2^2\times C_{12}$ x 20, $C_2^3\times C_6$ x 7

Order 42: $C_{42}$ x 3

Order 32: $C_2^3\times C_4$ x 6, $C_2^5$

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

Order 24: $C_2^2\times C_6$ x 23, $C_2\times C_{12}$ x 18

Order 21: $C_{21}$

Order 16: $C_2^2\times C_4$ x 20, $C_2^4$ x 7

Order 14: $C_{14}$ x 3

Order 12: $C_2\times C_6$ x 23, $C_{12}$ x 4

Order 8: $C_2^3$ x 23, $C_2\times C_4$ x 18

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 23, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_2^4:C_{84}$

Order 672: $C_2^4:C_{42}$, $C_{12}\times F_8$

Order 448: $C_2^4:C_{28}$

Order 336: $C_6\times F_8$ x 2

Order 224: $C_2^2\times F_8$, $C_4\times F_8$

Order 192: $C_2^4\times C_{12}$

Order 168: $C_3\times F_8$, $C_2\times C_{84}$

Order 112: $C_2\times F_8$ x 2

Order 96: $C_2^3\times C_{12}$ x 4, $C_2^4\times C_6$

Order 84: $C_{84}$, $C_2\times C_{42}$

Order 64: $C_2^4\times C_4$

Order 56: $C_2\times C_{28}$, $F_8$

Order 48: $C_2^2\times C_{12}$ x 6, $C_2^3\times C_6$ x 5

Order 42: $C_{42}$ x 2

Order 32: $C_2^3\times C_4$ x 4, $C_2^5$

Order 28: $C_2\times C_{14}$, $C_{28}$

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

Order 21: $C_{21}$

Order 16: $C_2^2\times C_4$ x 6, $C_2^4$ x 5

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 9, $C_{12}$ x 2

Order 8: $C_2^3$ x 9, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_2^4:C_{84}$ ($C_1$)

Order 672: $C_{12}\times F_8$ ($C_2$) x 2, $C_2^4:C_{42}$ ($C_2$)

Order 448: $C_2^4:C_{28}$ ($C_3$)

Order 336: $C_6\times F_8$ ($C_4$) x 2, $C_6\times F_8$ ($C_2^2$)

Order 224: $C_4\times F_8$ ($C_6$) x 2, $C_2^2\times F_8$ ($C_6$)

Order 192: $C_2^4\times C_{12}$ ($C_7$)

Order 168: $C_3\times F_8$ ($C_2\times C_4$)

Order 112: $C_2\times F_8$ ($C_{12}$) x 2, $C_2\times F_8$ ($C_2\times C_6$)

Order 96: $C_2^3\times C_{12}$ ($C_{14}$) x 2, $C_2^4\times C_6$ ($C_{14}$)

Order 64: $C_2^4\times C_4$ ($C_{21}$)

Order 56: $F_8$ ($C_2\times C_{12}$)

Order 48: $C_2^3\times C_6$ ($C_{28}$) x 2, $C_2^3\times C_6$ ($C_2\times C_{14}$)

Order 32: $C_2^3\times C_4$ ($C_{42}$) x 2, $C_2^5$ ($C_{42}$)

Order 24: $C_2\times C_{12}$ ($F_8$), $C_2^2\times C_6$ ($C_2\times C_{28}$)

Order 16: $C_2^4$ ($C_{84}$) x 2, $C_2^4$ ($C_2\times C_{42}$)

Order 12: $C_{12}$ ($C_2\times F_8$) x 2, $C_2\times C_6$ ($C_2\times F_8$)

Order 8: $C_2^3$ ($C_2\times C_{84}$), $C_2\times C_4$ ($C_3\times F_8$)

Order 6: $C_6$ ($C_4\times F_8$) x 2, $C_6$ ($C_2^2\times F_8$)

Order 4: $C_4$ ($C_6\times F_8$) x 2, $C_2^2$ ($C_6\times F_8$)

Order 3: $C_3$ ($C_2^4:C_{28}$)

Order 2: $C_2$ ($C_{12}\times F_8$) x 2, $C_2$ ($C_2^4:C_{42}$)

Order 1: $C_1$ ($C_2^4:C_{84}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_2^4:C_{84}$ ($C_1$)

Order 672: $C_2^4:C_{42}$ ($C_2$), $C_{12}\times F_8$ ($C_2$)

Order 448: $C_2^4:C_{28}$ ($C_3$)

Order 336: $C_6\times F_8$ ($C_2^2$), $C_6\times F_8$ ($C_4$)

Order 224: $C_2^2\times F_8$ ($C_6$), $C_4\times F_8$ ($C_6$)

Order 192: $C_2^4\times C_{12}$ ($C_7$)

Order 168: $C_3\times F_8$ ($C_2\times C_4$)

Order 112: $C_2\times F_8$ ($C_2\times C_6$), $C_2\times F_8$ ($C_{12}$)

Order 96: $C_2^4\times C_6$ ($C_{14}$), $C_2^3\times C_{12}$ ($C_{14}$)

Order 64: $C_2^4\times C_4$ ($C_{21}$)

Order 56: $F_8$ ($C_2\times C_{12}$)

Order 48: $C_2^3\times C_6$ ($C_2\times C_{14}$), $C_2^3\times C_6$ ($C_{28}$)

Order 32: $C_2^5$ ($C_{42}$), $C_2^3\times C_4$ ($C_{42}$)

Order 24: $C_2\times C_{12}$ ($F_8$), $C_2^2\times C_6$ ($C_2\times C_{28}$)

Order 16: $C_2^4$ ($C_{84}$), $C_2^4$ ($C_2\times C_{42}$)

Order 12: $C_2\times C_6$ ($C_2\times F_8$), $C_{12}$ ($C_2\times F_8$)

Order 8: $C_2^3$ ($C_2\times C_{84}$), $C_2\times C_4$ ($C_3\times F_8$)

Order 6: $C_6$ ($C_2^2\times F_8$), $C_6$ ($C_4\times F_8$)

Order 4: $C_2^2$ ($C_6\times F_8$), $C_4$ ($C_6\times F_8$)

Order 3: $C_3$ ($C_2^4:C_{28}$)

Order 2: $C_2$ ($C_2^4:C_{42}$), $C_2$ ($C_{12}\times F_8$)

Order 1: $C_1$ ($C_2^4:C_{84}$)

Series

Derived series

$C_2^4:C_{84}$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^4:C_{84}$

$\rhd$

$C_2^4:C_{42}$

$\rhd$

$C_6\times F_8$

$\rhd$

$C_3\times F_8$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^4:C_{84}$

$\rhd$

$C_2^3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{12}$

Character theory

Complex character table

See the $192 \times 192$ 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 $36 \times 36$ rational character table.