Abstract group 1344.6795: $D_8:C_2\times F_7$

• Label: 1344.6795
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $56$
• Rank: $4$

Group information

Description:	$D_8:C_2\times F_7$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_7.(C_3\times D_4^2).C_2^2$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	7	8	12	14	24	28	56
Elements	1	127	14	64	434	6	64	224	90	224	48	48	1344
Conjugacy classes	1	11	2	6	22	1	4	12	5	8	3	2	77
Divisions	1	11	1	6	11	1	4	6	5	4	3	2	55
Autjugacy classes	1	8	2	6	16	1	2	12	4	4	3	1	60

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	48	12	6	8	0	2	1	77
Irr. rational chars.	16	20	6	8	2	2	1	55

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$56$
Rank:	$4$
Inequivalent generating quadruples:	$183859200$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	10	10	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{6}=d^{56}=[a,c]=[b,c]=1, b^{a}=bd^{28}, d^{a}=d^{29}, d^{b}=d^{15}, d^{c}=d^{33} \rangle$
Permutation group:	Degree $15$ $\langle(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15), (8,9)(11,13), (10,14)(11,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 49 \\ 0 & 15 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/56\Z)$
Direct product:	$(D_8:C_2)$ $\, \times\, $ $F_7$
Semidirect product:	$(D_7\times D_8)$ $\,\rtimes\,$ $C_6$ (2)	$D_7$ $\,\rtimes\,$ $(D_8:C_6)$ (2)	$(D_8\times F_7)$ $\,\rtimes\,$ $C_2$ (2)	$(D_8:F_7)$ $\,\rtimes\,$ $C_2$ (2)	all 84
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times F_7)$ . $D_4$ (2)	$C_4$ . $(D_4\times F_7)$	$D_{14}$ . $(C_6\times D_4)$ (2)	$C_{28}$ . $(C_6\times D_4)$	all 36
Aut. group:	$\Aut(D_{56})$	$\Aut(C_7:Q_{16})$	$\Aut(C_{56}:C_6)$	$\Aut(C_{56}.C_6)$	all 6

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 3892 subgroups in 596 conjugacy classes, 182 normal (90 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_4\times F_7$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^3\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $D_8:C_{14}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $D_8:C_2\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_{12}:C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:C_2^3$
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

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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: $D_8:C_2\times F_7$

Order 672: $D_{56}:C_6$ x 2, $\SD_{16}\times F_7$ x 2, $D_8:F_7$ x 2, $D_8\times F_7$ x 2, $C_{56}:(C_2\times C_6)$, $(C_2\times D_{28}):C_6$, $(D_4\times C_{14}):C_6$, $D_{56}:C_6$, $\OD_{16}\times F_7$, $D_4:C_2\times F_7$, $C_2\times D_4\times F_7$

Order 448: $D_8:D_{14}$

Order 336: $D_4\times F_7$ x 8, $D_{28}:C_6$ x 4, $C_{56}:C_6$ x 2, $C_{56}:C_6$ x 2, $C_8\times F_7$ x 2, $C_{56}:C_6$ x 2, $C_{56}:C_6$ x 2, $Q_8:F_7$ x 2, $D_4.F_7$ x 2, $D_4:F_7$ x 2, $D_{28}:C_6$ x 2, $D_{14}:C_{12}$ x 2, $C_{56}:C_6$ x 2, $C_{56}:C_6$, $C_2^3\times F_7$, $(C_2\times C_{28}):C_6$, $C_2\times C_{28}:C_6$, $C_2^3:F_7$, $C_{28}.C_{12}$, $D_{28}:C_6$, $Q_8\times F_7$, $D_{28}:C_6$

Order 224: $C_8:D_{14}$ x 2, $D_7\times \SD_{16}$ x 2, $C_8:D_{14}$ x 2, $D_7\times D_8$ x 2, $D_4:D_{14}$, $D_4\times D_{14}$, $D_8:C_{14}$, $D_4:D_{14}$, $D_{28}:C_2^2$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 192: $C_{24}:C_2^3$

Order 168: $C_2^2\times F_7$ x 11, $C_4\times F_7$ x 7, $D_{14}:C_6$ x 7, $C_{28}:C_6$ x 5, $C_{28}:C_6$ x 5, $C_{28}.C_6$ x 2, $C_7:C_{24}$ x 2, $C_{14}:C_{12}$ x 2, $C_{14}:C_{12}$ x 2, $C_7:C_{24}$ x 2, $C_2\times C_{14}:C_6$, $C_{28}.C_6$

Order 112: $D_4\times D_7$ x 8, $C_7:D_8$ x 4, $D_{56}$ x 2, $C_{56}:C_2$ x 2, $C_{56}:C_2$ x 2, $D_4:D_7$ x 2, $D_{28}:C_2$ x 2, $C_8\times D_7$ x 2, $C_4\times D_{14}$ x 2, $C_7\times \SD_{16}$ x 2, $C_7\times D_8$ x 2, $Q_8:D_7$ x 2, $C_7:\SD_{16}$ x 2, $C_7:\OD_{16}$, $C_2^2\times D_{14}$, $D_4:C_{14}$, $D_4\times C_{14}$, $C_{14}:D_4$, $D_{28}:C_2$, $Q_8\times D_7$, $C_2\times D_{28}$, $C_7\times \OD_{16}$

Order 96: $D_8:C_6$ x 8, $C_6\times \SD_{16}$ x 2, $C_6\times D_8$ x 2, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_6\times \OD_{16}$

Order 84: $C_2\times F_7$ x 19, $C_{14}:C_6$ x 6, $C_7:C_{12}$ x 3, $C_7:C_{12}$ x 3

Order 64: $C_8:C_2^3$

Order 56: $C_2\times D_{14}$ x 11, $C_7:D_4$ x 7, $C_4\times D_7$ x 7, $C_7\times D_4$ x 5, $D_{28}$ x 5, $C_2\times C_{28}$ x 2, $C_{14}:C_4$ x 2, $C_7:Q_8$ x 2, $C_{56}$ x 2, $C_7:C_8$ x 2, $C_2^2\times C_{14}$, $C_7\times Q_8$

Order 48: $C_6\times D_4$ x 11, $C_3\times \SD_{16}$ x 8, $C_3\times D_8$ x 8, $D_4:C_6$ x 6, $C_3\times \OD_{16}$ x 4, $C_2^2\times C_{12}$ x 2, $C_2\times C_{24}$ x 2, $C_2^3\times C_6$, $C_6\times Q_8$

Order 42: $F_7$ x 6, $C_7:C_6$ x 5

Order 32: $D_8:C_2$ x 8, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \OD_{16}$

Order 28: $D_{14}$ x 19, $C_2\times C_{14}$ x 6, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 24: $C_3\times D_4$ x 17, $C_2^2\times C_6$ x 12, $C_2\times C_{12}$ x 11, $C_{24}$ x 4, $C_3\times Q_8$ x 3

Order 21: $C_7:C_3$

Order 16: $C_2\times D_4$ x 11, $\SD_{16}$ x 8, $D_8$ x 8, $D_4:C_2$ x 6, $\OD_{16}$ x 4, $C_2\times C_8$ x 2, $C_2^2\times C_4$ x 2, $C_2^4$, $C_2\times Q_8$

Order 14: $D_7$ x 6, $C_{14}$ x 5

Order 12: $C_2\times C_6$ x 25, $C_{12}$ x 6

Order 8: $D_4$ x 17, $C_2^3$ x 12, $C_2\times C_4$ x 11, $C_8$ x 4, $Q_8$ x 3

Order 7: $C_7$

Order 6: $C_6$ x 11

Order 4: $C_2^2$ x 25, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $D_8:C_2\times F_7$

Order 672: $C_{56}:(C_2\times C_6)$, $(C_2\times D_{28}):C_6$, $(D_4\times C_{14}):C_6$, $D_{56}:C_6$, $\SD_{16}\times F_7$, $D_8:F_7$, $D_8\times F_7$, $D_{56}:C_6$, $\OD_{16}\times F_7$, $D_4:C_2\times F_7$, $C_2\times D_4\times F_7$

Order 448: $D_8:D_{14}$

Order 336: $D_4\times F_7$ x 6, $D_{28}:C_6$ x 2, $D_4:F_7$ x 2, $D_{28}:C_6$ x 2, $D_{14}:C_{12}$ x 2, $C_{56}:C_6$, $C_{56}:C_6$, $C_8\times F_7$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{56}:C_6$, $C_2^3\times F_7$, $Q_8:F_7$, $D_4.F_7$, $(C_2\times C_{28}):C_6$, $C_2\times C_{28}:C_6$, $C_2^3:F_7$, $C_{28}.C_{12}$, $D_{28}:C_6$, $Q_8\times F_7$, $D_{28}:C_6$, $C_{56}:C_6$

Order 224: $D_4:D_{14}$, $D_4\times D_{14}$, $D_8:C_{14}$, $D_4:D_{14}$, $D_{28}:C_2^2$, $C_8:D_{14}$, $D_7\times \SD_{16}$, $C_8:D_{14}$, $D_7\times D_8$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 192: $C_{24}:C_2^3$

Order 168: $C_4\times F_7$ x 7, $C_2^2\times F_7$ x 6, $D_{14}:C_6$ x 5, $C_{28}:C_6$ x 4, $C_{28}:C_6$ x 4, $C_{28}.C_6$ x 2, $C_{14}:C_{12}$ x 2, $C_{14}:C_{12}$ x 2, $C_2\times C_{14}:C_6$, $C_{28}.C_6$, $C_7:C_{24}$, $C_7:C_{24}$

Order 112: $D_4\times D_7$ x 6, $D_4:D_7$ x 2, $D_{28}:C_2$ x 2, $C_4\times D_{14}$ x 2, $C_7:D_8$ x 2, $C_7:\OD_{16}$, $D_{56}$, $C_{56}:C_2$, $C_2^2\times D_{14}$, $D_4:C_{14}$, $C_{56}:C_2$, $D_4\times C_{14}$, $C_{14}:D_4$, $D_{28}:C_2$, $Q_8\times D_7$, $C_8\times D_7$, $C_2\times D_{28}$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_7\times \OD_{16}$, $Q_8:D_7$, $C_7:\SD_{16}$

Order 96: $D_8:C_6$ x 6, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_6\times \SD_{16}$, $C_6\times D_8$, $C_6\times \OD_{16}$

Order 84: $C_2\times F_7$ x 9, $C_{14}:C_6$ x 4, $C_7:C_{12}$ x 3, $C_7:C_{12}$ x 3

Order 64: $C_8:C_2^3$

Order 56: $C_4\times D_7$ x 7, $C_2\times D_{14}$ x 6, $C_7:D_4$ x 5, $C_7\times D_4$ x 4, $D_{28}$ x 4, $C_2\times C_{28}$ x 2, $C_{14}:C_4$ x 2, $C_7:Q_8$ x 2, $C_{56}$, $C_2^2\times C_{14}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_6\times D_4$ x 9, $D_4:C_6$ x 6, $C_3\times \SD_{16}$ x 4, $C_3\times D_8$ x 4, $C_3\times \OD_{16}$ x 3, $C_2^2\times C_{12}$ x 2, $C_2^3\times C_6$, $C_6\times Q_8$, $C_2\times C_{24}$

Order 42: $C_7:C_6$ x 4, $F_7$ x 4

Order 32: $D_8:C_2$ x 6, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$

Order 28: $D_{14}$ x 9, $C_2\times C_{14}$ x 4, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 24: $C_3\times D_4$ x 13, $C_2\times C_{12}$ x 11, $C_2^2\times C_6$ x 7, $C_3\times Q_8$ x 3, $C_{24}$ x 2

Order 21: $C_7:C_3$

Order 16: $C_2\times D_4$ x 9, $D_4:C_2$ x 6, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 3, $C_2^2\times C_4$ x 2, $C_2\times C_8$, $C_2^4$, $C_2\times Q_8$

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

Order 12: $C_2\times C_6$ x 13, $C_{12}$ x 6

Order 8: $D_4$ x 13, $C_2\times C_4$ x 11, $C_2^3$ x 7, $Q_8$ x 3, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 8

Order 4: $C_2^2$ x 13, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $D_8:C_2\times F_7$ ($C_1$)

Order 672: $D_{56}:C_6$ ($C_2$) x 2, $\SD_{16}\times F_7$ ($C_2$) x 2, $D_8:F_7$ ($C_2$) x 2, $D_8\times F_7$ ($C_2$) x 2, $C_{56}:(C_2\times C_6)$ ($C_2$), $(C_2\times D_{28}):C_6$ ($C_2$), $(D_4\times C_{14}):C_6$ ($C_2$), $D_{56}:C_6$ ($C_2$), $\OD_{16}\times F_7$ ($C_2$), $D_4:C_2\times F_7$ ($C_2$), $C_2\times D_4\times F_7$ ($C_2$)

Order 448: $D_8:D_{14}$ ($C_3$)

Order 336: $D_4\times F_7$ ($C_2^2$) x 5, $D_{28}:C_6$ ($C_2^2$) x 4, $C_{56}:C_6$ ($C_2^2$) x 2, $C_{56}:C_6$ ($C_2^2$) x 2, $C_8\times F_7$ ($C_2^2$) x 2, $C_{56}:C_6$ ($C_2^2$) x 2, $C_{56}:C_6$ ($C_2^2$) x 2, $Q_8:F_7$ ($C_2^2$) x 2, $D_4.F_7$ ($C_2^2$) x 2, $C_{56}:C_6$ ($C_2^2$) x 2, $C_{56}:C_6$ ($C_2^2$), $(C_2\times C_{28}):C_6$ ($C_2^2$), $C_2\times C_{28}:C_6$ ($C_2^2$), $C_{28}.C_{12}$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $Q_8\times F_7$ ($C_2^2$), $D_4:F_7$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $D_{14}:C_{12}$ ($C_2^2$)

Order 224: $C_8:D_{14}$ ($C_6$) x 2, $D_7\times \SD_{16}$ ($C_6$) x 2, $C_8:D_{14}$ ($C_6$) x 2, $D_7\times D_8$ ($C_6$) x 2, $D_4:D_{14}$ ($C_6$), $D_4\times D_{14}$ ($C_6$), $D_8:C_{14}$ ($C_6$), $D_4:D_{14}$ ($C_6$), $D_{28}:C_2^2$ ($C_6$), $C_8:D_{14}$ ($C_6$), $D_7\times \OD_{16}$ ($C_6$)

Order 168: $C_{28}:C_6$ ($C_2^3$) x 3, $C_{28}:C_6$ ($C_2^3$) x 3, $C_4\times F_7$ ($C_2^3$) x 2, $C_4\times F_7$ ($D_4$) x 2, $C_7:C_{24}$ ($C_2^3$) x 2, $C_7:C_{24}$ ($C_2^3$) x 2, $C_{28}.C_6$ ($C_2^3$), $C_2^2\times F_7$ ($D_4$), $C_{28}.C_6$ ($C_2^3$), $C_{14}:C_{12}$ ($C_2^3$), $C_{14}:C_{12}$ ($D_4$)

Order 112: $D_4\times D_7$ ($C_2\times C_6$) x 5, $C_7:D_8$ ($C_2\times C_6$) x 4, $D_{56}$ ($C_2\times C_6$) x 2, $C_{56}:C_2$ ($C_2\times C_6$) x 2, $C_{56}:C_2$ ($C_2\times C_6$) x 2, $C_8\times D_7$ ($C_2\times C_6$) x 2, $C_7\times \SD_{16}$ ($C_2\times C_6$) x 2, $C_7\times D_8$ ($C_2\times C_6$) x 2, $Q_8:D_7$ ($C_2\times C_6$) x 2, $C_7:\SD_{16}$ ($C_2\times C_6$) x 2, $C_7:\OD_{16}$ ($C_2\times C_6$), $D_4:C_{14}$ ($C_2\times C_6$), $D_4\times C_{14}$ ($C_2\times C_6$), $D_{28}:C_2$ ($C_2\times C_6$), $Q_8\times D_7$ ($C_2\times C_6$), $D_4:D_7$ ($C_2\times C_6$), $D_{28}:C_2$ ($C_2\times C_6$), $C_2\times D_{28}$ ($C_2\times C_6$), $C_4\times D_{14}$ ($C_2\times C_6$), $C_7\times \OD_{16}$ ($C_2\times C_6$)

Order 84: $C_2\times F_7$ ($C_2\times D_4$) x 2, $C_7:C_{12}$ ($C_2\times D_4$) x 2, $C_{14}:C_6$ ($C_2\times D_4$), $C_7:C_{12}$ ($C_2^4$), $C_7:C_{12}$ ($C_2\times D_4$)

Order 56: $C_7\times D_4$ ($C_2^2\times C_6$) x 3, $D_{28}$ ($C_2^2\times C_6$) x 3, $C_4\times D_7$ ($C_2^2\times C_6$) x 2, $C_4\times D_7$ ($C_3\times D_4$) x 2, $C_{56}$ ($C_2^2\times C_6$) x 2, $C_7:C_8$ ($C_2^2\times C_6$) x 2, $C_2\times C_{28}$ ($C_2^2\times C_6$), $C_{14}:C_4$ ($C_3\times D_4$), $C_7:Q_8$ ($C_2^2\times C_6$), $C_2\times D_{14}$ ($C_3\times D_4$), $C_7\times Q_8$ ($C_2^2\times C_6$)

Order 42: $F_7$ ($D_8:C_2$) x 2, $C_7:C_6$ ($C_2^2\times D_4$)

Order 32: $D_8:C_2$ ($F_7$)

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

Order 21: $C_7:C_3$ ($C_8:C_2^3$)

Order 16: $\SD_{16}$ ($C_2\times F_7$) x 2, $D_8$ ($C_2\times F_7$) x 2, $\OD_{16}$ ($C_2\times F_7$), $D_4:C_2$ ($C_2\times F_7$), $C_2\times D_4$ ($C_2\times F_7$)

Order 14: $D_7$ ($D_8:C_6$) x 2, $C_{14}$ ($C_{12}:C_2^3$)

Order 8: $D_4$ ($C_2^2\times F_7$) x 3, $C_8$ ($C_2^2\times F_7$) x 2, $Q_8$ ($C_2^2\times F_7$), $C_2\times C_4$ ($C_2^2\times F_7$)

Order 7: $C_7$ ($C_{24}:C_2^3$)

Order 4: $C_2^2$ ($D_4\times F_7$), $C_4$ ($C_2^3\times F_7$), $C_4$ ($D_4\times F_7$)

Order 2: $C_2$ ($C_2\times D_4\times F_7$)

Order 1: $C_1$ ($D_8:C_2\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $D_8:C_2\times F_7$ ($C_1$)

Order 672: $C_{56}:(C_2\times C_6)$ ($C_2$), $(C_2\times D_{28}):C_6$ ($C_2$), $(D_4\times C_{14}):C_6$ ($C_2$), $D_{56}:C_6$ ($C_2$), $\SD_{16}\times F_7$ ($C_2$), $D_8:F_7$ ($C_2$), $D_8\times F_7$ ($C_2$), $D_{56}:C_6$ ($C_2$), $\OD_{16}\times F_7$ ($C_2$), $D_4:C_2\times F_7$ ($C_2$), $C_2\times D_4\times F_7$ ($C_2$)

Order 448: $D_8:D_{14}$ ($C_3$)

Order 336: $D_4\times F_7$ ($C_2^2$) x 3, $D_{28}:C_6$ ($C_2^2$) x 2, $C_{56}:C_6$ ($C_2^2$), $C_{56}:C_6$ ($C_2^2$), $C_8\times F_7$ ($C_2^2$), $C_{56}:C_6$ ($C_2^2$), $C_{56}:C_6$ ($C_2^2$), $C_{56}:C_6$ ($C_2^2$), $Q_8:F_7$ ($C_2^2$), $D_4.F_7$ ($C_2^2$), $(C_2\times C_{28}):C_6$ ($C_2^2$), $C_2\times C_{28}:C_6$ ($C_2^2$), $C_{28}.C_{12}$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $Q_8\times F_7$ ($C_2^2$), $D_4:F_7$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $D_{14}:C_{12}$ ($C_2^2$), $C_{56}:C_6$ ($C_2^2$)

Order 224: $D_4:D_{14}$ ($C_6$), $D_4\times D_{14}$ ($C_6$), $D_8:C_{14}$ ($C_6$), $D_4:D_{14}$ ($C_6$), $D_{28}:C_2^2$ ($C_6$), $C_8:D_{14}$ ($C_6$), $D_7\times \SD_{16}$ ($C_6$), $C_8:D_{14}$ ($C_6$), $D_7\times D_8$ ($C_6$), $C_8:D_{14}$ ($C_6$), $D_7\times \OD_{16}$ ($C_6$)

Order 168: $C_{28}:C_6$ ($C_2^3$) x 2, $C_4\times F_7$ ($C_2^3$) x 2, $C_4\times F_7$ ($D_4$) x 2, $C_{28}:C_6$ ($C_2^3$) x 2, $C_{28}.C_6$ ($C_2^3$), $C_2^2\times F_7$ ($D_4$), $C_{28}.C_6$ ($C_2^3$), $C_7:C_{24}$ ($C_2^3$), $C_{14}:C_{12}$ ($C_2^3$), $C_{14}:C_{12}$ ($D_4$), $C_7:C_{24}$ ($C_2^3$)

Order 112: $D_4\times D_7$ ($C_2\times C_6$) x 3, $C_7:D_8$ ($C_2\times C_6$) x 2, $C_7:\OD_{16}$ ($C_2\times C_6$), $D_{56}$ ($C_2\times C_6$), $C_{56}:C_2$ ($C_2\times C_6$), $D_4:C_{14}$ ($C_2\times C_6$), $C_{56}:C_2$ ($C_2\times C_6$), $D_4\times C_{14}$ ($C_2\times C_6$), $D_{28}:C_2$ ($C_2\times C_6$), $Q_8\times D_7$ ($C_2\times C_6$), $D_4:D_7$ ($C_2\times C_6$), $D_{28}:C_2$ ($C_2\times C_6$), $C_8\times D_7$ ($C_2\times C_6$), $C_2\times D_{28}$ ($C_2\times C_6$), $C_4\times D_{14}$ ($C_2\times C_6$), $C_7\times \SD_{16}$ ($C_2\times C_6$), $C_7\times D_8$ ($C_2\times C_6$), $C_7\times \OD_{16}$ ($C_2\times C_6$), $Q_8:D_7$ ($C_2\times C_6$), $C_7:\SD_{16}$ ($C_2\times C_6$)

Order 84: $C_2\times F_7$ ($C_2\times D_4$) x 2, $C_7:C_{12}$ ($C_2\times D_4$) x 2, $C_{14}:C_6$ ($C_2\times D_4$), $C_7:C_{12}$ ($C_2^4$), $C_7:C_{12}$ ($C_2\times D_4$)

Order 56: $C_7\times D_4$ ($C_2^2\times C_6$) x 2, $D_{28}$ ($C_2^2\times C_6$) x 2, $C_4\times D_7$ ($C_2^2\times C_6$) x 2, $C_4\times D_7$ ($C_3\times D_4$) x 2, $C_2\times C_{28}$ ($C_2^2\times C_6$), $C_{14}:C_4$ ($C_3\times D_4$), $C_7:Q_8$ ($C_2^2\times C_6$), $C_{56}$ ($C_2^2\times C_6$), $C_2\times D_{14}$ ($C_3\times D_4$), $C_7\times Q_8$ ($C_2^2\times C_6$), $C_7:C_8$ ($C_2^2\times C_6$)

Order 42: $C_7:C_6$ ($C_2^2\times D_4$), $F_7$ ($D_8:C_2$)

Order 32: $D_8:C_2$ ($F_7$)

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

Order 21: $C_7:C_3$ ($C_8:C_2^3$)

Order 16: $\SD_{16}$ ($C_2\times F_7$), $D_8$ ($C_2\times F_7$), $\OD_{16}$ ($C_2\times F_7$), $D_4:C_2$ ($C_2\times F_7$), $C_2\times D_4$ ($C_2\times F_7$)

Order 14: $C_{14}$ ($C_{12}:C_2^3$), $D_7$ ($D_8:C_6$)

Order 8: $D_4$ ($C_2^2\times F_7$) x 2, $Q_8$ ($C_2^2\times F_7$), $C_2\times C_4$ ($C_2^2\times F_7$), $C_8$ ($C_2^2\times F_7$)

Order 7: $C_7$ ($C_{24}:C_2^3$)

Order 4: $C_2^2$ ($D_4\times F_7$), $C_4$ ($C_2^3\times F_7$), $C_4$ ($D_4\times F_7$)

Order 2: $C_2$ ($C_2\times D_4\times F_7$)

Order 1: $C_1$ ($D_8:C_2\times F_7$)

Series

Derived series

$D_8:C_2\times F_7$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$D_8:C_2\times F_7$

$\rhd$

$C_{56}:(C_2\times C_6)$

$\rhd$

$C_{56}:C_6$

$\rhd$

$C_7:C_{24}$

$\rhd$

$C_{56}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_8:C_2\times F_7$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_8:C_2$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $77 \times 77$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $55 \times 55$ rational character table.