Abstract group 5184.ff: $C_3^4:C_4^2:C_2^2$

• Label: 5184.ff
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $16$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$C_3^4:C_4^2:C_2^2$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4.C_2^3.C_2^5.C_2^4$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	343	80	2088	1520	1152	5184
Conjugacy classes	1	13	7	20	27	16	84
Divisions	1	13	7	12	27	8	68
Autjugacy classes	1	5	3	3	6	1	19

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	0	2	48	0	2	84
Irr. rational chars.	16	8	2	32	8	2	68

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	$669614400$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{2}=c^{4}=d^{6}=e^{3}=f^{3}=g^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,3)(2,5)(4,9)(6,8)(7,11)(10,12)(13,14)(15,16), (1,4)(3,7)(13,15,14,16), (1,2)(3,6)(4,8)(5,7)(9,10)(11,12), (5,10)(7,9)(13,14)(15,16)\rangle$
Transitive group:	24T7644	24T7650	36T6055		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.S_3^3)$ $\,\rtimes\,$ $C_2^2$	$(C_3:S_3^3:C_4)$ $\,\rtimes\,$ $C_2$	$(C_3^4:C_4^2)$ $\,\rtimes\,$ $C_2^2$	$C_3^4$ $\,\rtimes\,$ $(C_4^2:C_2^2)$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:S_3^3)$ . $C_2^2$	$(C_3^3:D_6)$ . $C_2^4$	$(C_3^4:C_2^3)$ . $C_2^3$ (2)	$C_2$ . $(C_3:S_3^3:C_2^2)$	all 15
Aut. group:	$\Aut(C_8\times C_2^3:C_{48})$

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 52150 subgroups in 1550 conjugacy classes, 123 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3:S_3^3:C_2^2$
Commutator:	$G' \simeq$ $C_3^3:S_3$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3:S_3^3:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_6$	$G/\operatorname{Fit} \simeq$ $D_4:C_2^2$
Radical:	$R \simeq$ $C_3^4:C_4^2:C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $D_4:C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 5184: $C_3^4:C_4^2:C_2^2$

Order 2592: $C_3^4:(C_4\times D_4)$ x 4, $C_3:S_3^3:C_4$ x 4, $C_3^4:(C_4\times D_4)$ x 4, $C_3^4:C_4^2:C_2$ x 2, $C_3^4:(C_2^2\times D_4)$

Order 1296: $(C_3^3\times C_6).D_4$ x 12, $C_3^4:(C_2\times D_4)$ x 8, $C_3^4:(C_2^2\times C_4)$ x 5, $C_2\times C_3^4:D_4$ x 4, $C_6.S_3^3$ x 4, $C_3^4:C_4^2$ x 4, $C_6:S_3^3$ x 2, $(C_3^3\times C_6).Q_8$ x 2, $(C_3^3\times C_6).D_4$ x 2

Order 648: $C_3^4:D_4$ x 16, $C_3:S_3^3$ x 10, $C_3^4:(C_2\times C_4)$ x 8, $C_2\times C_3^4:C_4$ x 6, $C_3^4:C_2^3$ x 5, $C_3^3:(C_4\times S_3)$ x 4, $C_3\times C_6.S_3^2$ x 4, $C_3^4:C_2^3$ x 2

Order 432: $C_2.S_3^3$ x 4, $C_2\times S_3^3$ x 2

Order 324: $C_3^2:S_3^2$ x 15, $C_3\wr C_2^2$ x 8, $C_3^4:C_4$ x 8, $C_3^2:C_6^2$ x 5, $C_3^3:C_{12}$ x 4, $C_3^3:D_6$

Order 288: $C_6^2.D_4$ x 4, $C_4\times \SOPlus(4,2)$ x 4, $C_6^2:D_4$

Order 216: $S_3^3$ x 16, $C_6:S_3^2$ x 10, $C_6.S_3^2$ x 8, $C_6.S_3^2$ x 8, $C_6\times S_3^2$ x 6, $C_6.S_3^2$ x 4, $C_6.S_3^2$ x 4, $C_6:S_3^2$ x 2

Order 162: $C_3^2\wr C_2$ x 10, $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 144: $S_3^2:C_4$ x 24, $S_3^2:C_2^2$ x 16, $C_4\times S_3^2$ x 8, $D_6^2$ x 6, $C_6^2:C_4$ x 5, $C_2^2.S_3^2$ x 4, $C_3^2:C_4^2$ x 4, $C_2.\SOPlus(4,2)$ x 4, $C_2.\PSU(3,2)$ x 2

Order 108: $C_3:S_3^2$ x 40, $C_3\times S_3^2$ x 24, $C_3^2:D_6$ x 10, $C_3^2\times D_6$ x 10, $C_3:S_3^2$ x 8, $C_3^2:C_{12}$ x 8, $C_3^2:C_{12}$ x 8, $C_3^2:D_6$ x 7

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 67, $C_2\times C_3^2:C_4$ x 36, $\SOPlus(4,2)$ x 32, $C_6.D_6$ x 24, $S_3\times C_{12}$ x 8, $C_6.D_6$ x 8, $C_6\times D_6$ x 6, $C_6:D_6$ x 5, $C_{12}:S_3$ x 4, $C_6:C_{12}$ x 4

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2:C_6$ x 20, $S_3\times C_3^2$ x 20, $C_3^2:S_3$ x 14, $C_3^2\times C_6$ x 7

Order 48: $C_4\times D_6$ x 4, $C_2^2\times D_6$ x 2

Order 36: $S_3^2$ x 104, $C_6\times S_3$ x 54, $C_6:S_3$ x 47, $C_3^2:C_4$ x 32, $C_3:C_{12}$ x 24, $C_6^2$ x 5, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4

Order 32: $C_4\times D_4$ x 8, $C_2^3:C_4$ x 4, $C_4^2:C_2$ x 2, $C_2^2\times D_4$

Order 27: $C_3^3$ x 7

Order 24: $C_2\times D_6$ x 32, $C_4\times S_3$ x 16, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4, $C_2^2\times C_6$ x 2

Order 18: $C_3\times S_3$ x 60, $C_3:S_3$ x 54, $C_3\times C_6$ x 32

Order 16: $C_2^2:C_4$ x 12, $C_2\times D_4$ x 12, $C_2^2\times C_4$ x 9, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 12: $D_6$ x 81, $C_2\times C_6$ x 18, $C_{12}$ x 8, $C_3:C_4$ x 8

Order 9: $C_3^2$ x 22

Order 8: $C_2\times C_4$ x 22, $C_2^3$ x 17, $D_4$ x 16

Order 6: $S_3$ x 34, $C_6$ x 27

Order 4: $C_2^2$ x 29, $C_4$ x 12

Order 3: $C_3$ x 7

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $C_3^4:C_4^2:C_2^2$

Order 2592: $C_3^4:(C_4\times D_4)$, $C_3^4:C_4^2:C_2$, $C_3:S_3^3:C_4$, $C_3^4:(C_4\times D_4)$, $C_3^4:(C_2^2\times D_4)$

Order 1296: $C_3^4:(C_2^2\times C_4)$ x 2, $(C_3^3\times C_6).D_4$ x 2, $C_6:S_3^3$, $C_2\times C_3^4:D_4$, $C_3^4:(C_2\times D_4)$, $C_6.S_3^3$, $(C_3^3\times C_6).Q_8$, $(C_3^3\times C_6).D_4$, $C_3^4:C_4^2$

Order 648: $C_3^4:C_2^3$ x 2, $C_2\times C_3^4:C_4$ x 2, $C_3:S_3^3$ x 2, $C_3^4:(C_2\times C_4)$ x 2, $C_3^4:C_2^3$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3\times C_6.S_3^2$

Order 432: $C_2\times S_3^3$, $C_2.S_3^3$

Order 324: $C_3^2:S_3^2$ x 4, $C_3^2:C_6^2$ x 2, $C_3^4:C_4$ x 2, $C_3^3:D_6$, $C_3\wr C_2^2$, $C_3^3:C_{12}$

Order 288: $C_6^2.D_4$, $C_4\times \SOPlus(4,2)$, $C_6^2:D_4$

Order 216: $C_6:S_3^2$ x 3, $C_6\times S_3^2$ x 2, $S_3^3$ x 2, $C_6:S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 162: $C_3^3:S_3$ x 2, $C_3^2\wr C_2$ x 2, $C_3^3\times C_6$

Order 144: $S_3^2:C_2^2$ x 3, $S_3^2:C_4$ x 3, $D_6^2$ x 2, $C_6^2:C_4$ x 2, $C_2^2.S_3^2$, $C_4\times S_3^2$, $C_3^2:C_4^2$, $C_2.\PSU(3,2)$, $C_2.\SOPlus(4,2)$

Order 108: $C_3:S_3^2$ x 6, $C_3^2:D_6$ x 3, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3\times S_3^2$ x 2, $C_3:S_3^2$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 12, $C_2\times C_3^2:C_4$ x 8, $C_6.D_6$ x 4, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $\SOPlus(4,2)$ x 2, $C_{12}:S_3$, $C_6:C_{12}$, $S_3\times C_{12}$, $C_6.D_6$

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2:S_3$ x 6, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3

Order 48: $C_2^2\times D_6$, $C_4\times D_6$

Order 36: $C_6:S_3$ x 14, $S_3^2$ x 14, $C_6\times S_3$ x 9, $C_3^2:C_4$ x 6, $C_3:C_{12}$ x 3, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

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

Order 27: $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 7, $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_2^2\times C_6$

Order 18: $C_3:S_3$ x 18, $C_3\times C_6$ x 10, $C_3\times S_3$ x 7

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

Order 12: $D_6$ x 14, $C_2\times C_6$ x 4, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 8

Order 8: $C_2\times C_4$ x 6, $C_2^3$ x 5, $D_4$

Order 6: $S_3$ x 9, $C_6$ x 6

Order 4: $C_2^2$ x 8, $C_4$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $C_3^4:C_4^2:C_2^2$ ($C_1$)

Order 2592: $C_3^4:(C_4\times D_4)$ ($C_2$) x 4, $C_3:S_3^3:C_4$ ($C_2$) x 4, $C_3^4:(C_4\times D_4)$ ($C_2$) x 4, $C_3^4:C_4^2:C_2$ ($C_2$) x 2, $C_3^4:(C_2^2\times D_4)$ ($C_2$)

Order 1296: $(C_3^3\times C_6).D_4$ ($C_2^2$) x 12, $C_3^4:(C_2\times D_4)$ ($C_4$) x 8, $C_3^4:(C_2^2\times C_4)$ ($C_2^2$) x 5, $C_2\times C_3^4:D_4$ ($C_2^2$) x 4, $C_6.S_3^3$ ($C_2^2$) x 4, $C_3^4:C_4^2$ ($C_2^2$) x 4, $C_6:S_3^3$ ($C_2^2$) x 2, $(C_3^3\times C_6).Q_8$ ($C_2^2$) x 2, $(C_3^3\times C_6).D_4$ ($C_2^2$) x 2

Order 648: $C_3^4:D_4$ ($C_2\times C_4$) x 16, $C_3:S_3^3$ ($C_2\times C_4$) x 8, $C_2\times C_3^4:C_4$ ($C_2^3$) x 6, $C_3^4:C_2^3$ ($C_2^3$) x 5, $C_3^4:(C_2\times C_4)$ ($C_2\times C_4$) x 4, $C_3^3:(C_4\times S_3)$ ($C_2^3$) x 4

Order 324: $C_3^2:S_3^2$ ($C_2^2\times C_4$) x 10, $C_3^4:C_4$ ($C_2^2\times C_4$) x 4, $C_3^3:D_6$ ($C_2^4$)

Order 162: $C_3^3\times C_6$ ($D_4:C_2^2$), $C_3^3:S_3$ ($D_4:C_2^2$), $C_3^3:S_3$ ($C_2^3\times C_4$)

Order 81: $C_3^4$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($C_3:S_3^3:C_2^2$)

Order 1: $C_1$ ($C_3^4:C_4^2:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $C_3^4:C_4^2:C_2^2$ ($C_1$)

Order 2592: $C_3^4:(C_4\times D_4)$ ($C_2$), $C_3^4:C_4^2:C_2$ ($C_2$), $C_3:S_3^3:C_4$ ($C_2$), $C_3^4:(C_4\times D_4)$ ($C_2$), $C_3^4:(C_2^2\times D_4)$ ($C_2$)

Order 1296: $C_3^4:(C_2^2\times C_4)$ ($C_2^2$) x 2, $(C_3^3\times C_6).D_4$ ($C_2^2$) x 2, $C_6:S_3^3$ ($C_2^2$), $C_2\times C_3^4:D_4$ ($C_2^2$), $C_3^4:(C_2\times D_4)$ ($C_4$), $C_6.S_3^3$ ($C_2^2$), $(C_3^3\times C_6).Q_8$ ($C_2^2$), $(C_3^3\times C_6).D_4$ ($C_2^2$), $C_3^4:C_4^2$ ($C_2^2$)

Order 648: $C_3^4:C_2^3$ ($C_2^3$) x 2, $C_2\times C_3^4:C_4$ ($C_2^3$) x 2, $C_3:S_3^3$ ($C_2\times C_4$), $C_3^4:D_4$ ($C_2\times C_4$), $C_3^4:(C_2\times C_4)$ ($C_2\times C_4$), $C_3^3:(C_4\times S_3)$ ($C_2^3$)

Order 324: $C_3^2:S_3^2$ ($C_2^2\times C_4$) x 2, $C_3^3:D_6$ ($C_2^4$), $C_3^4:C_4$ ($C_2^2\times C_4$)

Order 162: $C_3^3\times C_6$ ($D_4:C_2^2$), $C_3^3:S_3$ ($D_4:C_2^2$), $C_3^3:S_3$ ($C_2^3\times C_4$)

Order 81: $C_3^4$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($C_3:S_3^3:C_2^2$)

Order 1: $C_1$ ($C_3^4:C_4^2:C_2^2$)

Series

Derived series

$C_3^4:C_4^2:C_2^2$

$\rhd$

$C_3^4:C_4^2:C_2^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^4:C_4^2:C_2^2$

$\rhd$

$C_3^4:C_4^2:C_2^2$

$\rhd$

$C_3^4:(C_4\times D_4)$

$\rhd$

$C_3^4:(C_4\times D_4)$

$\rhd$

$C_6.S_3^3$

$\rhd$

$C_6.S_3^3$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^4:C_4^2:C_2^2$

$\rhd$

$C_3^4:C_4^2:C_2^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 4 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 $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.