Abstract group 5376.bk: $C_2\wr C_7.C_6$

• Label: 5376.bk
• Order: \( 2^{8} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $18$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_2\wr C_7.C_6$
Order:	\(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2\wr F_7$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 8, $C_3$, $C_7$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	7	12	14	28
Elements	1	127	224	128	1568	384	1792	384	768	5376
Conjugacy classes	1	11	2	12	14	2	16	2	4	64
Divisions	1	11	1	6	7	1	4	1	1	33
Autjugacy classes	1	9	2	5	10	1	6	1	1	36

Dimension	1	2	3	4	6	7	12	14	21	28	42
Irr. complex chars.	12	0	8	0	0	36	0	0	8	0	0	64
Irr. rational chars.	2	3	0	1	2	6	1	9	4	3	2	33

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$2352$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	7	14	14
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{12}=b^{7}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(2,4,8,13,3,6)(5,9,12,7,11,10)(15,17,16,18), (1,2,3,5,6,10,11)(4,7,8,12,9,14,13)(15,16)(17,18)\rangle$
Transitive group:	28T286				more information
Direct product:	$C_4$ $\, \times\, $ $(C_2^3:F_8:C_3)$
Semidirect product:	$(C_2^6:C_{28})$ $\,\rtimes\,$ $C_3$	$C_2^6$ $\,\rtimes\,$ $(C_7:C_{12})$	$(C_2^3:F_8)$ $\,\rtimes\,$ $C_{12}$	$C_2^3$ $\,\rtimes\,$ $(F_8:C_{12})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\wr C_7)$ . $C_6$	$C_2^7$ . $(C_7:C_6)$	$C_2^4$ . $(F_8:C_6)$	$(C_2\wr C_7:C_3)$ . $C_2$	all 5

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 63078 subgroups in 2990 conjugacy classes, 18 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_2^3:F_8:C_3$
Commutator:	$G' \simeq$ $C_2^3:F_8$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\wr C_7:C_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6\times C_4$	$G/\operatorname{Fit} \simeq$ $C_7:C_3$
Radical:	$R \simeq$ $C_2\wr C_7.C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7$	$G/\operatorname{soc} \simeq$ $C_7:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6\times C_4$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 5376: $C_2\wr C_7.C_6$

Order 2688: $C_2\wr C_7:C_3$

Order 1792: $C_2^6:C_{28}$

Order 1344: $C_2^3:F_8:C_3$

Order 896: $C_2\wr C_7$

Order 768: $C_2^6:C_{12}$

Order 672: $F_8:C_{12}$ x 2

Order 448: $C_2^3:F_8$

Order 384: $C_2^5:C_{12}$ x 6, $C_2^5:A_4$

Order 336: $F_8:C_6$ x 2

Order 256: $C_2^6\times C_4$

Order 224: $C_4\times F_8$ x 2

Order 192: $C_2^2\wr C_3$ x 7, $C_2^4:C_{12}$ x 5, $C_2^4:C_{12}$ x 4

Order 168: $F_8:C_3$ x 2

Order 128: $C_2^5\times C_4$ x 10, $C_2^7$

Order 112: $C_2\times F_8$ x 2

Order 96: $C_2^3:C_{12}$ x 30, $C_2^3:A_4$ x 7, $C_2^3\times A_4$ x 5

Order 84: $C_7:C_{12}$

Order 64: $C_2^4\times C_4$ x 130, $C_2^6$ x 11

Order 56: $F_8$ x 2

Order 48: $C_2^2\times A_4$ x 35, $C_4\times A_4$ x 20, $C_2^2:A_4$, $C_2^2\times C_{12}$

Order 42: $C_7:C_6$

Order 32: $C_2^3\times C_4$ x 552, $C_2^5$ x 135

Order 28: $C_{28}$

Order 24: $C_2\times A_4$ x 35, $C_2\times C_{12}$ x 6, $C_2^2\times C_6$

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 587, $C_2^2\times C_4$ x 510

Order 14: $C_{14}$

Order 12: $C_2\times C_6$ x 7, $A_4$ x 5, $C_{12}$ x 4

Order 8: $C_2^3$ x 587, $C_2\times C_4$ x 100

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 135, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5376: $C_2\wr C_7.C_6$

Order 2688: $C_2\wr C_7:C_3$

Order 1792: $C_2^6:C_{28}$

Order 1344: $C_2^3:F_8:C_3$

Order 896: $C_2\wr C_7$

Order 768: $C_2^6:C_{12}$

Order 672: $F_8:C_{12}$

Order 448: $C_2^3:F_8$

Order 384: $C_2^5:C_{12}$ x 4, $C_2^5:A_4$

Order 336: $F_8:C_6$

Order 256: $C_2^6\times C_4$

Order 224: $C_4\times F_8$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^4:C_{12}$ x 3, $C_2^4:C_{12}$ x 3

Order 168: $F_8:C_3$

Order 128: $C_2^5\times C_4$ x 8, $C_2^7$

Order 112: $C_2\times F_8$

Order 96: $C_2^3:C_{12}$ x 16, $C_2^3:A_4$ x 5, $C_2^3\times A_4$ x 3

Order 84: $C_7:C_{12}$

Order 64: $C_2^4\times C_4$ x 80, $C_2^6$ x 9

Order 56: $F_8$

Order 48: $C_2^2\times A_4$ x 19, $C_4\times A_4$ x 11, $C_2^2:A_4$, $C_2^2\times C_{12}$

Order 42: $C_7:C_6$

Order 32: $C_2^3\times C_4$ x 306, $C_2^5$ x 84

Order 28: $C_{28}$

Order 24: $C_2\times A_4$ x 19, $C_2\times C_{12}$ x 4, $C_2^2\times C_6$

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 330, $C_2^2\times C_4$ x 284

Order 14: $C_{14}$

Order 12: $C_2\times C_6$ x 5, $A_4$ x 3, $C_{12}$ x 3

Order 8: $C_2^3$ x 330, $C_2\times C_4$ x 60

Order 7: $C_7$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 84, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5376: $C_2\wr C_7.C_6$ ($C_1$)

Order 2688: $C_2\wr C_7:C_3$ ($C_2$)

Order 1792: $C_2^6:C_{28}$ ($C_3$)

Order 1344: $C_2^3:F_8:C_3$ ($C_4$)

Order 896: $C_2\wr C_7$ ($C_6$)

Order 448: $C_2^3:F_8$ ($C_{12}$)

Order 256: $C_2^6\times C_4$ ($C_7:C_3$)

Order 128: $C_2^7$ ($C_7:C_6$)

Order 64: $C_2^6$ ($C_7:C_{12}$)

Order 32: $C_2^3\times C_4$ ($F_8:C_3$) x 2

Order 16: $C_2^4$ ($F_8:C_6$) x 2

Order 8: $C_2^3$ ($F_8:C_{12}$) x 2

Order 4: $C_4$ ($C_2^3:F_8:C_3$)

Order 2: $C_2$ ($C_2\wr C_7:C_3$)

Order 1: $C_1$ ($C_2\wr C_7.C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5376: $C_2\wr C_7.C_6$ ($C_1$)

Order 2688: $C_2\wr C_7:C_3$ ($C_2$)

Order 1792: $C_2^6:C_{28}$ ($C_3$)

Order 1344: $C_2^3:F_8:C_3$ ($C_4$)

Order 896: $C_2\wr C_7$ ($C_6$)

Order 448: $C_2^3:F_8$ ($C_{12}$)

Order 256: $C_2^6\times C_4$ ($C_7:C_3$)

Order 128: $C_2^7$ ($C_7:C_6$)

Order 64: $C_2^6$ ($C_7:C_{12}$)

Order 32: $C_2^3\times C_4$ ($F_8:C_3$)

Order 16: $C_2^4$ ($F_8:C_6$)

Order 8: $C_2^3$ ($F_8:C_{12}$)

Order 4: $C_4$ ($C_2^3:F_8:C_3$)

Order 2: $C_2$ ($C_2\wr C_7:C_3$)

Order 1: $C_1$ ($C_2\wr C_7.C_6$)

Series

Derived series

$C_2\wr C_7.C_6$

$\rhd$

$C_2\wr C_7.C_6$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_2\wr C_7.C_6$

$\rhd$

$C_2\wr C_7.C_6$

$\rhd$

$C_2\wr C_7:C_3$

$\rhd$

$C_2\wr C_7:C_3$

$\rhd$

$C_2^3:F_8:C_3$

$\rhd$

$C_2^3:F_8:C_3$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^6$

$\rhd$

$C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2\wr C_7.C_6$

$\rhd$

$C_2\wr C_7.C_6$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^3:F_8$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.