Abstract group 4032.fk: $C_{28}:(C_6\times S_4)$

• Label: 4032.fk
• Order: \( 2^{6} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $84$
• Rank: $3$

Group information

Description:	$C_{28}:(C_6\times S_4)$
Order:	\(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$(C_{14}\times A_4).C_6.C_2^4$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2, $C_7$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	227	134	284	1618	6	1192	186	48	192	48	96	4032
Conjugacy classes	1	11	5	8	31	1	19	5	1	4	1	2	89
Divisions	1	11	3	8	17	1	10	5	1	4	1	1	63
Autjugacy classes	1	7	5	6	20	1	15	4	1	3	1	1	65

Dimension	1	2	3	4	6	8	12	18	24	36
Irr. complex chars.	24	18	24	3	10	0	5	4	0	1	89
Irr. rational chars.	8	14	8	7	14	1	5	4	1	1	63

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$84$
Rank:	$3$
Inequivalent generating triples:	$1048320$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	36	36
Arbitrary	11	11	11

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{84}=d^{2}=e^{2}=[a,b]=[a,d]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,4)(2,3)(6,7,8)(10,12,15)(11,13,14), (1,3)(2,4)(5,7), (1,2,4,3), (7,8) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4:S_4)$ $\,\rtimes\,$ $F_7$	$C_4$ $\,\rtimes\,$ $(S_4\times F_7)$	$(A_4:F_7)$ $\,\rtimes\,$ $D_4$ (2)	$A_4$ $\,\rtimes\,$ $(D_4\times F_7)$	all 46
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $(D_6\times F_7)$	$C_2$ . $(C_2\times S_4\times F_7)$	$C_{14}$ . $(C_2\times C_6\times S_4)$	$(C_7:C_6\times A_4)$ . $C_2^3$	all 11

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 15908 subgroups in 968 conjugacy classes, 74 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times S_4\times F_7$
Commutator:	$G' \simeq$ $A_4\times C_{14}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times S_4\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{28}$	$G/\operatorname{Fit} \simeq$ $C_6\times S_3$
Radical:	$R \simeq$ $C_{28}:(C_6\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_6\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 4032: $C_{28}:(C_6\times S_4)$

Order 2016: $(C_{14}\times S_4):C_6$ x 2, $C_2\times S_4\times F_7$ x 2, $C_4\times A_4:F_7$, $C_{28}:(C_3\times S_4)$, $A_4\times C_{28}:C_6$

Order 1344: $C_{28}:(C_6\times D_4)$, $D_{28}:S_4$

Order 1008: $S_4\times F_7$ x 4, $C_2\times A_4\times F_7$ x 2, $C_7:C_6\times S_4$ x 2, $C_2\times A_4:F_7$, $C_7:C_{12}\times A_4$, $D_{12}:F_7$, $(C_7\times A_4):C_{12}$

Order 672: $C_2\times D_4\times F_7$ x 3, $(D_4\times C_{14}):C_6$ x 2, $(C_2\times D_{28}):C_6$ x 2, $(C_2\times D_{28}):C_6$ x 2, $S_4\times D_{14}$ x 2, $C_7:\GL(2,\mathbb{Z}/4)$ x 2, $(D_4\times C_{14}):C_6$, $(C_2\times D_{28}):C_6$, $C_4\times D_{14}:C_6$, $(C_2\times D_{28}):C_6$, $D_{28}:C_{12}$, $D_{28}:A_4$, $A_4\times D_{28}$, $C_2\times D_{28}:C_6$, $C_{28}:S_4$, $C_{28}:S_4$

Order 576: $\GL(2,\mathbb{Z}/4):C_6$

Order 504: $D_6:F_7$ x 2, $D_6\times F_7$ x 2, $A_4\times F_7$ x 2, $A_4:F_7$ x 2, $C_7:C_3\times S_4$ x 2, $C_{84}:C_6$, $C_{84}:C_6$, $C_{84}:C_6$, $C_7:C_6\times A_4$

Order 448: $D_{28}:D_4$

Order 336: $D_4\times F_7$ x 12, $C_2^3:F_7$ x 7, $D_{28}:C_6$ x 6, $D_{14}:C_{12}$ x 5, $C_2^3\times F_7$ x 4, $D_7\times S_4$ x 4, $C_2\times C_{28}:C_6$ x 3, $D_{14}:C_{12}$ x 3, $(C_2\times C_{14}):C_{12}$ x 2, $D_{14}:A_4$ x 2, $A_4\times D_{14}$ x 2, $C_{14}\times S_4$ x 2, $C_{28}:C_{12}$, $C_2^3.F_7$, $C_{14}:S_4$, $C_{28}:A_4$, $A_4\times C_{28}$, $C_2\times C_{14}:C_{12}$, $C_{12}:D_{14}$, $C_{14}.(C_2\times C_{12})$, $C_{28}:C_{12}$, $C_{14}.S_4$

Order 288: $C_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_6\times S_4$ x 2, $C_2^3.C_6^2$, $C_{12}:S_4$, $C_{12}\times S_4$

Order 252: $S_3\times F_7$ x 4, $C_{42}:C_6$ x 2, $C_6\times F_7$ x 2, $C_{42}:C_6$, $A_4\times C_7:C_3$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 224: $D_4\times D_{14}$ x 3, $D_{14}:D_4$ x 2, $C_2^2:D_{28}$ x 2, $D_{14}:D_4$ x 2, $C_4:D_{28}$, $D_{28}:C_4$, $C_2^2\times D_{28}$, $C_{28}:D_4$, $C_{28}:D_4$, $C_{28}:D_4$

Order 192: $D_4\times S_4$, $C_3\times D_4^2$

Order 168: $C_2^2\times F_7$ x 19, $D_{14}:C_6$ x 14, $C_{28}:C_6$ x 10, $C_4\times F_7$ x 6, $C_{28}:C_6$ x 6, $C_{14}:C_{12}$ x 4, $C_2\times C_{14}:C_6$ x 3, $C_{14}:C_{12}$ x 3, $S_3\times D_{14}$ x 2, $D_7:A_4$ x 2, $A_4\times D_7$ x 2, $C_7:S_4$ x 2, $C_7\times S_4$ x 2, $C_{21}:D_4$ x 2, $C_{14}:A_4$, $A_4\times C_{14}$, $C_4\times D_{21}$, $C_7\times D_{12}$, $C_3\times D_{28}$

Order 144: $C_6\times S_4$ x 7, $C_2^2:C_6^2$ x 2, $C_{12}:D_6$, $C_{12}\times A_4$, $A_4:C_{12}$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$ x 2, $C_3\times F_7$ x 2, $C_{21}:C_6$

Order 112: $D_4\times D_7$ x 12, $C_{14}:D_4$ x 7, $C_2\times D_{28}$ x 6, $D_{14}:C_4$ x 5, $C_2^2\times D_{14}$ x 4, $D_4\times C_{14}$ x 3, $C_4\times D_{14}$ x 3, $C_2^2:C_{28}$ x 2, $C_2^2\times C_{28}$, $C_4:C_{28}$, $C_{14}.D_4$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $C_{12}:C_2^3$ x 4, $C_{12}:D_4$ x 4, $C_2^4:C_6$ x 4, $C_2^2\times S_4$ x 2, $D_4\times A_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $D_4\times C_{12}$ x 2, $C_4:S_4$, $C_4\times S_4$, $C_{12}:D_4$

Order 84: $C_2\times F_7$ x 26, $C_{14}:C_6$ x 9, $C_7:C_{12}$ x 5, $S_3\times D_7$ x 4, $C_7:C_{12}$ x 4, $S_3\times C_{14}$ x 2, $C_3\times D_{14}$ x 2, $C_{84}$, $C_{21}:C_4$, $D_{42}$, $C_7:A_4$, $C_7\times A_4$

Order 72: $C_3\times S_4$ x 4, $C_6\times A_4$ x 3, $C_6\times D_6$ x 2, $C_6\wr C_2$ x 2, $D_4\times C_3^2$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $D_4^2$

Order 63: $C_{21}:C_3$

Order 56: $C_2\times D_{14}$ x 19, $C_7:D_4$ x 14, $D_{28}$ x 9, $C_7\times D_4$ x 6, $C_4\times D_7$ x 6, $C_2\times C_{28}$ x 4, $C_{14}:C_4$ x 3, $C_2^2\times C_{14}$ x 3

Order 48: $C_6\times D_4$ x 28, $C_2^2:C_{12}$ x 8, $C_2\times S_4$ x 7, $C_2^3\times C_6$ x 4, $C_2^2\times A_4$ x 4, $C_2^2\times C_{12}$ x 4, $C_4\times A_4$ x 2, $C_4:C_{12}$ x 2, $S_3\times D_4$, $A_4:C_4$, $C_4\times C_{12}$

Order 42: $F_7$ x 8, $C_7:C_6$ x 6, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 36: $C_6\times S_3$ x 7, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 28: $D_{14}$ x 24, $C_2\times C_{14}$ x 9, $C_{28}$ x 4, $C_7:C_4$ x 4

Order 24: $C_3\times D_4$ x 31, $C_2^2\times C_6$ x 22, $C_2\times C_{12}$ x 13, $C_2\times A_4$ x 6, $S_4$ x 4, $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2\times D_4$ x 28, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

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

Order 12: $C_2\times C_6$ x 37, $C_{12}$ x 10, $D_6$ x 7, $A_4$ x 2, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $D_4$ x 29, $C_2^3$ x 22, $C_2\times C_4$ x 13

Order 7: $C_7$

Order 6: $C_6$ x 17, $S_3$ x 4

Order 4: $C_2^2$ x 33, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4032: $C_{28}:(C_6\times S_4)$

Order 2016: $C_4\times A_4:F_7$, $C_{28}:(C_3\times S_4)$, $A_4\times C_{28}:C_6$, $(C_{14}\times S_4):C_6$, $C_2\times S_4\times F_7$

Order 1344: $C_{28}:(C_6\times D_4)$, $D_{28}:S_4$

Order 1008: $C_2\times A_4\times F_7$, $C_2\times A_4:F_7$, $C_7:C_6\times S_4$, $S_4\times F_7$, $C_7:C_{12}\times A_4$, $D_{12}:F_7$, $(C_7\times A_4):C_{12}$

Order 672: $C_2\times D_4\times F_7$ x 2, $(D_4\times C_{14}):C_6$, $(C_2\times D_{28}):C_6$, $(D_4\times C_{14}):C_6$, $C_4\times D_{14}:C_6$, $(C_2\times D_{28}):C_6$, $D_{28}:C_{12}$, $(C_2\times D_{28}):C_6$, $(C_2\times D_{28}):C_6$, $S_4\times D_{14}$, $D_{28}:A_4$, $A_4\times D_{28}$, $C_2\times D_{28}:C_6$, $C_{28}:S_4$, $C_{28}:S_4$, $C_7:\GL(2,\mathbb{Z}/4)$

Order 576: $\GL(2,\mathbb{Z}/4):C_6$

Order 504: $C_{84}:C_6$, $C_{84}:C_6$, $C_{84}:C_6$, $D_6:F_7$, $C_7:C_6\times A_4$, $D_6\times F_7$, $A_4\times F_7$, $A_4:F_7$, $C_7:C_3\times S_4$

Order 448: $D_{28}:D_4$

Order 336: $D_4\times F_7$ x 5, $C_2^3:F_7$ x 4, $D_{28}:C_6$ x 4, $D_{14}:C_{12}$ x 3, $C_2^3\times F_7$ x 2, $C_2\times C_{28}:C_6$ x 2, $D_{14}:C_{12}$ x 2, $C_{28}:C_{12}$, $(C_2\times C_{14}):C_{12}$, $C_2^3.F_7$, $D_{14}:A_4$, $A_4\times D_{14}$, $C_{14}:S_4$, $C_{14}\times S_4$, $D_7\times S_4$, $C_{28}:A_4$, $A_4\times C_{28}$, $C_2\times C_{14}:C_{12}$, $C_{12}:D_{14}$, $C_{14}.(C_2\times C_{12})$, $C_{28}:C_{12}$, $C_{14}.S_4$

Order 288: $C_2^3.C_6^2$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_2\times C_6\times S_4$

Order 252: $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$, $A_4\times C_7:C_3$, $S_3\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 224: $D_4\times D_{14}$ x 2, $C_4:D_{28}$, $D_{28}:C_4$, $D_{14}:D_4$, $C_2^2:D_{28}$, $C_2^2\times D_{28}$, $C_{28}:D_4$, $C_{28}:D_4$, $D_{14}:D_4$, $C_{28}:D_4$

Order 192: $D_4\times S_4$, $C_3\times D_4^2$

Order 168: $C_2^2\times F_7$ x 9, $C_{28}:C_6$ x 6, $D_{14}:C_6$ x 6, $C_{28}:C_6$ x 4, $C_4\times F_7$ x 3, $C_{14}:C_{12}$ x 3, $C_{14}:C_{12}$ x 3, $C_2\times C_{14}:C_6$ x 2, $C_{14}:A_4$, $A_4\times C_{14}$, $S_3\times D_{14}$, $D_7:A_4$, $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$, $C_4\times D_{21}$, $C_7\times D_{12}$, $C_3\times D_{28}$, $C_{21}:D_4$

Order 144: $C_6\times S_4$ x 3, $C_2^2:C_6^2$, $C_{12}:D_6$, $C_{12}\times A_4$, $A_4:C_{12}$

Order 126: $C_{21}:C_6$, $C_{21}:C_6$, $C_3\times F_7$, $C_{21}:C_6$

Order 112: $D_4\times D_7$ x 5, $C_{14}:D_4$ x 4, $C_2\times D_{28}$ x 4, $D_{14}:C_4$ x 3, $C_2^2\times D_{14}$ x 2, $D_4\times C_{14}$ x 2, $C_4\times D_{14}$ x 2, $C_2^2\times C_{28}$, $C_4:C_{28}$, $C_2^2:C_{28}$, $C_{14}.D_4$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $C_{12}:C_2^3$ x 3, $C_{12}:D_4$ x 3, $D_4\times A_4$ x 2, $C_2^4:C_6$ x 2, $D_4\times C_{12}$ x 2, $C_2^2\times S_4$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $C_4\times S_4$, $C_{12}:D_4$

Order 84: $C_2\times F_7$ x 12, $C_{14}:C_6$ x 6, $C_7:C_{12}$ x 4, $C_7:C_{12}$ x 3, $S_3\times D_7$, $C_{84}$, $C_{21}:C_4$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$, $C_7:A_4$, $C_7\times A_4$

Order 72: $C_6\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6\times D_6$, $D_4\times C_3^2$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $D_4^2$

Order 63: $C_{21}:C_3$

Order 56: $C_2\times D_{14}$ x 9, $C_7:D_4$ x 6, $D_{28}$ x 5, $C_7\times D_4$ x 4, $C_2\times C_{28}$ x 3, $C_{14}:C_4$ x 3, $C_4\times D_7$ x 3, $C_2^2\times C_{14}$ x 2

Order 48: $C_6\times D_4$ x 15, $C_2^2:C_{12}$ x 5, $C_2\times S_4$ x 3, $C_2^2\times C_{12}$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times A_4$ x 2, $C_4\times A_4$ x 2, $C_4:C_{12}$ x 2, $S_3\times D_4$, $A_4:C_4$, $C_4\times C_{12}$

Order 42: $C_7:C_6$ x 5, $F_7$ x 4, $C_{42}$, $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 36: $C_6\times S_3$ x 3, $C_3\times C_{12}$, $C_3:C_{12}$, $C_6^2$, $C_3\times A_4$

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

Order 28: $D_{14}$ x 11, $C_2\times C_{14}$ 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 11, $C_2\times C_{12}$ x 9, $C_2\times A_4$ x 4, $S_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_2\times D_6$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$ x 15, $C_2^2:C_4$ x 5, $C_2^2\times C_4$ x 3, $C_4:C_4$ x 2, $C_2^4$ x 2, $C_4^2$

Order 14: $C_{14}$ x 4, $D_7$ x 3

Order 12: $C_2\times C_6$ x 19, $C_{12}$ x 8, $D_6$ x 3, $A_4$ x 2, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $D_4$ x 15, $C_2^3$ x 11, $C_2\times C_4$ x 9

Order 7: $C_7$

Order 6: $C_6$ x 11, $S_3$ x 2

Order 4: $C_2^2$ x 17, $C_4$ x 6

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4032: $C_{28}:(C_6\times S_4)$ ($C_1$)

Order 2016: $(C_{14}\times S_4):C_6$ ($C_2$) x 2, $C_2\times S_4\times F_7$ ($C_2$) x 2, $C_4\times A_4:F_7$ ($C_2$), $C_{28}:(C_3\times S_4)$ ($C_2$), $A_4\times C_{28}:C_6$ ($C_2$)

Order 1344: $D_{28}:S_4$ ($C_3$)

Order 1008: $C_2\times A_4\times F_7$ ($C_2^2$) x 2, $C_7:C_6\times S_4$ ($C_2^2$) x 2, $C_2\times A_4:F_7$ ($C_2^2$), $C_7:C_{12}\times A_4$ ($C_2^2$), $(C_7\times A_4):C_{12}$ ($C_2^2$)

Order 672: $S_4\times D_{14}$ ($C_6$) x 2, $C_7:\GL(2,\mathbb{Z}/4)$ ($C_6$) x 2, $A_4\times D_{28}$ ($C_6$), $C_2\times D_{28}:C_6$ ($S_3$), $C_{28}:S_4$ ($C_6$), $C_{28}:S_4$ ($C_6$)

Order 504: $A_4:F_7$ ($D_4$) x 2, $C_7:C_6\times A_4$ ($C_2^3$)

Order 336: $A_4\times D_{14}$ ($C_2\times C_6$) x 2, $C_2^3\times F_7$ ($D_6$) x 2, $C_{14}\times S_4$ ($C_2\times C_6$) x 2, $C_{14}:S_4$ ($C_2\times C_6$), $A_4\times C_{28}$ ($C_2\times C_6$), $C_2\times C_{14}:C_{12}$ ($D_6$), $C_{14}.S_4$ ($C_2\times C_6$)

Order 252: $A_4\times C_7:C_3$ ($C_2\times D_4$)

Order 224: $C_2^2\times D_{28}$ ($C_3\times S_3$)

Order 168: $C_7:S_4$ ($C_3\times D_4$) x 2, $C_{28}:C_6$ ($S_4$), $A_4\times C_{14}$ ($C_2^2\times C_6$), $C_2\times C_{14}:C_6$ ($C_2\times D_6$)

Order 112: $C_2^2\times D_{14}$ ($C_6\times S_3$) x 2, $C_2^2\times C_{28}$ ($C_6\times S_3$)

Order 96: $C_4:S_4$ ($F_7$)

Order 84: $C_2\times F_7$ ($C_2\times S_4$) x 2, $C_{14}:C_6$ ($S_3\times D_4$), $C_7:C_{12}$ ($C_2\times S_4$), $C_7\times A_4$ ($C_6\times D_4$)

Order 56: $D_{28}$ ($C_3\times S_4$), $C_2^2\times C_{14}$ ($C_6\times D_6$)

Order 48: $C_2\times S_4$ ($C_2\times F_7$) x 2, $C_4\times A_4$ ($C_2\times F_7$)

Order 42: $C_7:C_6$ ($C_2^2\times S_4$)

Order 28: $D_{14}$ ($C_6\times S_4$) x 2, $C_2\times C_{14}$ ($C_{12}:D_6$), $C_{28}$ ($C_6\times S_4$)

Order 24: $C_2\times A_4$ ($C_2^2\times F_7$)

Order 21: $C_7:C_3$ ($D_4\times S_4$)

Order 16: $C_2^2\times C_4$ ($S_3\times F_7$)

Order 14: $C_{14}$ ($C_2\times C_6\times S_4$)

Order 12: $A_4$ ($D_4\times F_7$)

Order 8: $C_2^3$ ($D_6\times F_7$)

Order 7: $C_7$ ($\GL(2,\mathbb{Z}/4):C_6$)

Order 4: $C_2^2$ ($D_{12}:F_7$), $C_4$ ($S_4\times F_7$)

Order 2: $C_2$ ($C_2\times S_4\times F_7$)

Order 1: $C_1$ ($C_{28}:(C_6\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4032: $C_{28}:(C_6\times S_4)$ ($C_1$)

Order 2016: $C_4\times A_4:F_7$ ($C_2$), $C_{28}:(C_3\times S_4)$ ($C_2$), $A_4\times C_{28}:C_6$ ($C_2$), $(C_{14}\times S_4):C_6$ ($C_2$), $C_2\times S_4\times F_7$ ($C_2$)

Order 1344: $D_{28}:S_4$ ($C_3$)

Order 1008: $C_2\times A_4\times F_7$ ($C_2^2$), $C_2\times A_4:F_7$ ($C_2^2$), $C_7:C_6\times S_4$ ($C_2^2$), $C_7:C_{12}\times A_4$ ($C_2^2$), $(C_7\times A_4):C_{12}$ ($C_2^2$)

Order 672: $S_4\times D_{14}$ ($C_6$), $A_4\times D_{28}$ ($C_6$), $C_2\times D_{28}:C_6$ ($S_3$), $C_{28}:S_4$ ($C_6$), $C_{28}:S_4$ ($C_6$), $C_7:\GL(2,\mathbb{Z}/4)$ ($C_6$)

Order 504: $C_7:C_6\times A_4$ ($C_2^3$), $A_4:F_7$ ($D_4$)

Order 336: $A_4\times D_{14}$ ($C_2\times C_6$), $C_2^3\times F_7$ ($D_6$), $C_{14}:S_4$ ($C_2\times C_6$), $C_{14}\times S_4$ ($C_2\times C_6$), $A_4\times C_{28}$ ($C_2\times C_6$), $C_2\times C_{14}:C_{12}$ ($D_6$), $C_{14}.S_4$ ($C_2\times C_6$)

Order 252: $A_4\times C_7:C_3$ ($C_2\times D_4$)

Order 224: $C_2^2\times D_{28}$ ($C_3\times S_3$)

Order 168: $C_{28}:C_6$ ($S_4$), $A_4\times C_{14}$ ($C_2^2\times C_6$), $C_2\times C_{14}:C_6$ ($C_2\times D_6$), $C_7:S_4$ ($C_3\times D_4$)

Order 112: $C_2^2\times D_{14}$ ($C_6\times S_3$), $C_2^2\times C_{28}$ ($C_6\times S_3$)

Order 96: $C_4:S_4$ ($F_7$)

Order 84: $C_{14}:C_6$ ($S_3\times D_4$), $C_2\times F_7$ ($C_2\times S_4$), $C_7:C_{12}$ ($C_2\times S_4$), $C_7\times A_4$ ($C_6\times D_4$)

Order 56: $D_{28}$ ($C_3\times S_4$), $C_2^2\times C_{14}$ ($C_6\times D_6$)

Order 48: $C_2\times S_4$ ($C_2\times F_7$), $C_4\times A_4$ ($C_2\times F_7$)

Order 42: $C_7:C_6$ ($C_2^2\times S_4$)

Order 28: $C_2\times C_{14}$ ($C_{12}:D_6$), $D_{14}$ ($C_6\times S_4$), $C_{28}$ ($C_6\times S_4$)

Order 24: $C_2\times A_4$ ($C_2^2\times F_7$)

Order 21: $C_7:C_3$ ($D_4\times S_4$)

Order 16: $C_2^2\times C_4$ ($S_3\times F_7$)

Order 14: $C_{14}$ ($C_2\times C_6\times S_4$)

Order 12: $A_4$ ($D_4\times F_7$)

Order 8: $C_2^3$ ($D_6\times F_7$)

Order 7: $C_7$ ($\GL(2,\mathbb{Z}/4):C_6$)

Order 4: $C_2^2$ ($D_{12}:F_7$), $C_4$ ($S_4\times F_7$)

Order 2: $C_2$ ($C_2\times S_4\times F_7$)

Order 1: $C_1$ ($C_{28}:(C_6\times S_4)$)

Series

Derived series

$C_{28}:(C_6\times S_4)$

$\rhd$

$A_4\times C_{14}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{28}:(C_6\times S_4)$

$\rhd$

$C_{28}:(C_3\times S_4)$

$\rhd$

$C_7:C_{12}\times A_4$

$\rhd$

$A_4\times C_{28}$

$\rhd$

$A_4\times C_{14}$

$\rhd$

$C_7\times A_4$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_{28}:(C_6\times S_4)$

$\rhd$

$A_4\times C_{14}$

$\rhd$

$C_7\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $89 \times 89$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $63 \times 63$ rational character table.