Abstract group 5376.bl: $C_2^4:F_8:C_6$

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

Group information

Description:	$C_2^4:F_8:C_6$
Order:	\(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^4:F_8:C_6$, 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 and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	12	14
Elements	1	143	224	112	2464	384	896	1152	5376
Conjugacy classes	1	9	2	2	14	2	4	6	40
Divisions	1	9	1	2	7	1	2	3	26
Autjugacy classes	1	8	2	1	12	2	2	4	32

Dimension	1	2	3	6	7	14	28	42
Irr. complex chars.	12	0	8	0	12	6	0	2	40
Irr. rational chars.	4	4	0	4	4	6	2	2	26

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$1344$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{6}=b^{14}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,3,7,6,11,16,13,12,2,5,9,14,4,8)(10,15)(17,18), (1,2,4)(3,6,10,8,12,5)(9,13,11)(14,16)(17,18)\rangle$
Transitive group:	28T289	28T294			more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^3:F_8:C_6)$
Semidirect product:	$C_2^7$ $\,\rtimes\,$ $(C_7:C_6)$	$(C_2^5:F_8)$ $\,\rtimes\,$ $C_3$	$(C_2^4:F_8)$ $\,\rtimes\,$ $C_6$	$(C_2^4:F_8)$ $\,\rtimes\,$ $C_6$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(F_8:C_6)$	$C_2^3$ . $(C_2\times F_8:C_6)$			more information
Aut. group:	$\Aut(C_6^3:C_2^3)$	$\Aut(C_2^3:F_8:C_{12})$

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

Homology

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

Subgroups

There are 40450 subgroups in 1014 conjugacy classes, 22 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^3:F_8:C_6$
Commutator:	$G' \simeq$ $C_2^3:F_8$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2\times F_8:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5:D_4$	$G/\operatorname{Fit} \simeq$ $C_7:C_3$
Radical:	$R \simeq$ $C_2^4:F_8:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $F_8:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
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

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Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 5376: $C_2^4:F_8:C_6$

Order 2688: $C_2^3:F_8:C_6$ x 2, $C_2^3:F_8:C_6$

Order 1792: $C_2^5:F_8$

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

Order 896: $C_2^4:F_8$ x 2, $C_2^4:F_8$

Order 768: $C_2^7:C_6$

Order 672: $C_2\times F_8:C_6$

Order 448: $C_2^3:F_8$

Order 384: $C_2\wr C_6$ x 4, $C_2^5:C_{12}$, $C_2^5:A_4$, $C_2^5:A_4$

Order 336: $F_8:C_6$ x 4

Order 256: $C_2^5:D_4$

Order 224: $C_2^2\times F_8$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^4:A_4$ x 4, $C_2^4:C_{12}$ x 2, $C_2^5:C_6$

Order 168: $F_8:C_3$ x 2

Order 128: $C_2^3\wr C_2$ x 4, $C_2^7$, $C_2^4:D_4$, $C_2^5:C_4$

Order 112: $C_2\times F_8$ x 5

Order 96: $C_2^3:A_4$ x 5, $C_2^3\times A_4$ x 4, $D_4\times A_4$ x 4, $C_2^3:A_4$ x 2, $C_2^3:C_{12}$

Order 84: $C_{14}:C_6$

Order 64: $C_2^3:D_4$ x 8, $C_2^6$ x 7, $C_2^4:C_4$ x 4, $D_4\times C_2^3$, $C_2^4:C_4$

Order 56: $F_8$ x 3

Order 48: $C_2^2\times A_4$ x 23, $C_4\times A_4$ x 2, $C_2^2:A_4$, $C_6\times D_4$

Order 42: $C_7:C_6$ x 3

Order 32: $C_2^5$ x 72, $C_2^2\wr C_2$ x 12, $C_2^2\times D_4$ x 8, $C_2^3:C_4$ x 8, $C_2^3\times C_4$

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

Order 24: $C_2\times A_4$ x 21, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 305, $C_2\times D_4$ x 14, $C_2^2:C_4$ x 8, $C_2^2\times C_4$ x 4

Order 14: $C_{14}$ x 3

Order 12: $C_2\times C_6$ x 9, $A_4$ x 3, $C_{12}$ x 2

Order 8: $C_2^3$ x 305, $D_4$ x 8, $C_2\times C_4$ x 4

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 75, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5376: $C_2^4:F_8:C_6$

Order 2688: $C_2^3:F_8:C_6$, $C_2^3:F_8:C_6$

Order 1792: $C_2^5:F_8$

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

Order 896: $C_2^4:F_8$, $C_2^4:F_8$

Order 768: $C_2^7:C_6$

Order 672: $C_2\times F_8:C_6$

Order 448: $C_2^3:F_8$

Order 384: $C_2\wr C_6$ x 2, $C_2^5:C_{12}$, $C_2^5:A_4$, $C_2^5:A_4$

Order 336: $F_8:C_6$ x 3

Order 256: $C_2^5:D_4$

Order 224: $C_2^2\times F_8$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^4:A_4$ x 3, $C_2^4:C_{12}$, $C_2^5:C_6$

Order 168: $F_8:C_3$ x 2

Order 128: $C_2^3\wr C_2$ x 2, $C_2^7$, $C_2^4:D_4$, $C_2^5:C_4$

Order 112: $C_2\times F_8$ x 4

Order 96: $C_2^3:A_4$ x 5, $C_2^3\times A_4$ x 4, $D_4\times A_4$ x 2, $C_2^3:A_4$, $C_2^3:C_{12}$

Order 84: $C_{14}:C_6$

Order 64: $C_2^6$ x 7, $C_2^3:D_4$ x 6, $C_2^4:C_4$ x 2, $D_4\times C_2^3$, $C_2^4:C_4$

Order 56: $F_8$ x 3

Order 48: $C_2^2\times A_4$ x 22, $C_2^2:A_4$, $C_6\times D_4$, $C_4\times A_4$

Order 42: $C_7:C_6$ x 2

Order 32: $C_2^5$ x 72, $C_2^2\times D_4$ x 6, $C_2^2\wr C_2$ x 6, $C_2^3:C_4$ x 6, $C_2^3\times C_4$

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

Order 24: $C_2\times A_4$ x 20, $C_2^2\times C_6$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 304, $C_2\times D_4$ x 10, $C_2^2:C_4$ x 4, $C_2^2\times C_4$ x 3

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 8, $A_4$ x 3, $C_{12}$

Order 8: $C_2^3$ x 304, $D_4$ x 4, $C_2\times C_4$ x 3

Order 7: $C_7$

Order 6: $C_6$ x 6

Order 4: $C_2^2$ x 74, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5376: $C_2^4:F_8:C_6$ ($C_1$)

Order 2688: $C_2^3:F_8:C_6$ ($C_2$) x 2, $C_2^3:F_8:C_6$ ($C_2$)

Order 1792: $C_2^5:F_8$ ($C_3$)

Order 1344: $C_2^3:F_8:C_3$ ($C_2^2$)

Order 896: $C_2^4:F_8$ ($C_6$) x 2, $C_2^4:F_8$ ($C_6$)

Order 448: $C_2^3:F_8$ ($C_2\times C_6$)

Order 256: $C_2^5:D_4$ ($C_7:C_3$)

Order 128: $C_2^3\wr C_2$ ($C_7:C_6$) x 2, $C_2^7$ ($C_7:C_6$)

Order 64: $C_2^6$ ($C_{14}:C_6$)

Order 32: $C_2^5$ ($F_8:C_3$)

Order 16: $C_2^4$ ($F_8:C_6$) x 3

Order 8: $C_2^3$ ($C_2\times F_8:C_6$)

Order 2: $C_2$ ($C_2^3:F_8:C_6$)

Order 1: $C_1$ ($C_2^4:F_8:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5376: $C_2^4:F_8:C_6$ ($C_1$)

Order 2688: $C_2^3:F_8:C_6$ ($C_2$), $C_2^3:F_8:C_6$ ($C_2$)

Order 1792: $C_2^5:F_8$ ($C_3$)

Order 1344: $C_2^3:F_8:C_3$ ($C_2^2$)

Order 896: $C_2^4:F_8$ ($C_6$), $C_2^4:F_8$ ($C_6$)

Order 448: $C_2^3:F_8$ ($C_2\times C_6$)

Order 256: $C_2^5:D_4$ ($C_7:C_3$)

Order 128: $C_2^7$ ($C_7:C_6$), $C_2^3\wr C_2$ ($C_7:C_6$)

Order 64: $C_2^6$ ($C_{14}:C_6$)

Order 32: $C_2^5$ ($F_8:C_3$)

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

Order 8: $C_2^3$ ($C_2\times F_8:C_6$)

Order 2: $C_2$ ($C_2^3:F_8:C_6$)

Order 1: $C_1$ ($C_2^4:F_8:C_6$)

Series

Derived series

$C_2^4:F_8:C_6$

$\rhd$

$C_2^4:F_8:C_6$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_2^4:F_8:C_6$

$\rhd$

$C_2^4:F_8:C_6$

$\rhd$

$C_2^3:F_8:C_6$

$\rhd$

$C_2^3:F_8:C_6$

$\rhd$

$C_2^3:F_8:C_3$

$\rhd$

$C_2^3:F_8:C_3$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^6$

$\rhd$

$C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^4:F_8:C_6$

$\rhd$

$C_2^4:F_8:C_6$

$\rhd$

$C_2^3:F_8$

$\rhd$

$C_2^3:F_8$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 6 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 $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $26 \times 26$ rational character table.