Abstract group 51840.cr: $C_2\times F_{81}:C_4$

• Label: 51840.cr
• Order: \( 2^{7} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $83$
• Trans deg.: $162$
• Rank: $3$

Group information

Description:	$C_2\times F_{81}:C_4$
Order:	\(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \)
Exponent:	\(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times C_3^4.C_{80}:C_2.C_2$, of order \(103680\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_3$ x 4, $C_5$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	5	6	8	10	12	16	20	40	80
Elements	1	343	80	6264	324	1520	16848	972	8640	7776	1296	2592	5184	51840
Conjugacy classes	1	5	1	8	1	3	10	3	4	8	4	8	16	72
Divisions	1	5	1	6	1	3	6	3	2	4	2	2	2	38
Autjugacy classes	1	5	1	6	1	3	8	3	2	4	4	8	8	54

Dimension	1	2	4	8	16	32	80	160
Irr. complex chars.	16	12	36	0	0	0	8	0	72
Irr. rational chars.	8	8	8	4	2	2	4	2	38

Minimal presentations

Permutation degree:	$83$
Transitive degree:	$162$
Rank:	$3$
Inequivalent generating triples:	$423198720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	80	80	80
Arbitrary	80	80	80

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{80}=c^{3}=d^{3}=e^{6}=f^{3}=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $83$ $\langle(2,12,3,17,57,16,48,50,30,29,34,31,67,51,76,62,9,38,66,59,61,4,22,39,69,24,8,35,44,45,40,42,33,23,15,55,77,73,36,78,6,27,13,49,63,19,56,47,72,60,70,79,81,14,52,11,26,20,7,32,64,18,58,71,75,28,68,65,54,53,21,46,25,5,10,41,80,37,74,43) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(F_{81}:C_4)$
Semidirect product:	$(C_2\times F_{81})$ $\,\rtimes\,$ $C_4$	$F_{81}$ $\,\rtimes\,$ $(C_2\times C_4)$ (2)	$(C_2\times C_3^4:C_{16})$ $\,\rtimes\,$ $F_5$	$(C_3^3:S_3)$ $\,\rtimes\,$ $(C_{80}:C_4)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^4:(C_8\times D_5))$ . $Q_8$	$(C_3^4:(C_8\times D_5))$ . $D_4$	$(C_3^4:C_4)$ . $(D_{10}.D_4)$	$(C_3^4:D_{10})$ . $(C_8:C_4)$	all 37
Aut. group:	$\Aut(C_3\wr C_5)$	$\Aut(C_3^4:C_{20})$	$\Aut(C_3^4:C_{20})$	$\Aut(C_2\times C_3^4:C_{10})$	all 13

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

Homology

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

Subgroups

There are 96258 subgroups in 450 conjugacy classes, 68 normal (50 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $F_{81}:C_4$
Commutator:	$G' \simeq$ $C_3^4:C_{40}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times F_{81}:C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_6$	$G/\operatorname{Fit} \simeq$ $C_{80}:C_4$
Radical:	$R \simeq$ $C_2\times F_{81}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_{80}:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 51840: $C_2\times F_{81}:C_4$

Order 25920: $F_{81}:C_4$ x 4, $C_2\times C_3^4.C_{80}.C_2$, $C_2\times C_3^4.(C_8.F_5)$, $C_3^4:D_{10}.\SD_{16}$

Order 12960: $C_3^4.(C_8.F_5)$ x 3, $C_3^4.C_{80}.C_2$ x 3, $C_3^4:C_{40}:C_4$ x 3, $C_2\times C_3^4.(C_8\times D_5)$, $C_2\times C_3^4.C_{40}.C_2$, $C_2\times C_3^4:C_5.\OD_{16}$, $C_2\times C_3^4.D_5.D_4$, $C_3^4.C_{80}:C_2$, $C_2\times F_{81}$

Order 10368: $C_3^4:\OD_{32}:C_4$

Order 6480: $C_3^4:(C_8\times D_5)$ x 4, $C_3^4:C_5.\OD_{16}$ x 3, $C_3^4:C_{20}:C_4$ x 3, $C_3^4.C_{40}.C_2$ x 2, $F_{81}$ x 2, $C_2\times C_3^4.D_{10}.C_2$, $C_2\times C_3^4:C_5.C_8$, $C_2\times C_3^4:C_5.C_8$, $C_2\times C_3^4:C_{40}$, $C_3^4:(C_2^2\times F_5)$

Order 5184: $C_3^4:C_{16}:C_4$ x 4, $C_2\times C_3^4.C_8:C_4$, $C_3^4:C_2^3.Q_8$, $C_2\times C_3^4:\OD_{32}$

Order 3240: $C_3^4:(C_4\times D_5)$ x 4, $C_3^4:(C_2\times F_5)$ x 3, $C_3^4:C_5:C_8$ x 2, $C_3^4:C_5:C_8$ x 2, $C_3^4:C_{40}$ x 2, $C_2\times C_3^4:(C_5:C_4)$, $C_2\times C_3^4:C_{20}$, $C_2\times C_3^4:D_{10}$, $C_2\times C_3^4:F_5$

Order 2592: $C_3^4:\OD_{32}$ x 4, $C_3^2\wr C_2.\SD_{16}$ x 3, $C_3^2:F_9.C_4$ x 3, $C_2\times C_3^4:C_{16}$ x 2, $C_2\times C_3^4.C_4:C_4$, $C_3^2:(C_2^2\times F_9)$, $C_2\times C_3^4:\OD_{16}$

Order 1620: $C_3^4:D_{10}$ x 3, $C_3^4:F_5$ x 2, $C_3^4:C_{20}$ x 2, $C_3^4:C_5:C_4$ x 2, $C_2\times C_3^4:C_{10}$, $C_3^4:D_{10}$

Order 1296: $C_3^2:(C_2\times F_9)$ x 4, $C_3^4:C_{16}$ x 4, $C_3^4:\OD_{16}$ x 3, $C_3^2\wr C_2.D_4$ x 3, $C_2\times C_3^2:F_9$ x 2, $C_3^4:(C_2^2\times C_4)$, $C_3^3:(C_4\times D_6)$, $C_2\times C_3^2:F_9$

Order 810: $C_3^4:C_{10}$ x 2, $C_3^4:D_5$ x 2, $C_3^4:C_{10}$

Order 648: $C_3^4:(C_2\times C_4)$ x 4, $C_3^2:F_9$ x 4, $C_3^3:(C_4\times S_3)$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_3^2:F_9$ x 2, $C_3^4:C_2^3$, $C_2\times C_3\wr C_4$

Order 640: $C_2\times C_{40}:C_4.C_2$

Order 576: $C_6^2.\SD_{16}$

Order 432: $C_3^2:C_4\times D_6$

Order 405: $C_3^4:C_5$

Order 324: $C_3^4:C_4$ x 4, $C_3^2:S_3^2$ x 3, $C_3\wr C_4$ x 2, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 320: $C_{80}:C_4$ x 4, $C_{16}:D_{10}$, $C_2\times C_{40}.C_4$, $D_{10}.\SD_{16}$

Order 288: $F_9:C_4$ x 4, $C_6^2.Q_8$, $C_6^2.D_4$, $C_2^2\times F_9$

Order 216: $C_3^2:C_4\times S_3$ x 4, $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$

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

Order 160: $C_{80}:C_2$ x 4, $C_{40}.C_4$ x 3, $C_{40}:C_4$ x 3, $C_2\times C_{80}$, $D_{10}.D_4$, $C_{10}:\OD_{16}$, $C_{10}:C_{16}$, $C_8\times D_{10}$

Order 144: $C_2\times F_9$ x 7, $C_2.\PSU(3,2)$ x 3, $C_2.\SOPlus(4,2)$ x 3, $C_6^2:C_4$ x 2, $C_2^2.S_3^2$

Order 128: $\OD_{32}:C_4$

Order 108: $C_3:S_3^2$ x 4, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$

Order 81: $C_3^4$

Order 80: $C_8\times D_5$ x 4, $C_{20}:C_4$ x 3, $C_5:\OD_{16}$ x 3, $C_{80}$ x 2, $C_5:C_{16}$ x 2, $C_{10}:C_8$, $C_2^2\times F_5$, $C_4\times D_{10}$, $C_{10}:C_8$, $C_2\times C_{40}$

Order 72: $C_2\times C_3^2:C_4$ x 11, $F_9$ x 6, $C_6.D_6$ x 3, $C_6:D_6$, $S_3\times D_6$, $C_6:C_{12}$

Order 64: $C_{16}:C_4$ x 4, $C_2\times \OD_{32}$, $C_2^3.Q_8$, $C_2^3.D_4$

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

Order 48: $C_4\times D_6$

Order 40: $C_4\times D_5$ x 4, $C_2\times F_5$ x 4, $C_5:C_8$ x 2, $C_{40}$ x 2, $C_5:C_8$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_2\times D_{10}$

Order 36: $C_3^2:C_4$ x 8, $C_6:S_3$ x 8, $S_3^2$ x 4, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 32: $\OD_{32}$ x 4, $C_8.C_4$ x 3, $C_8:C_4$ x 3, $C_2\times C_{16}$ x 2, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_2^2.D_4$

Order 27: $C_3^3$

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

Order 20: $D_{10}$ x 4, $F_5$ x 2, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 18: $C_3:S_3$ x 8, $C_3\times C_6$ x 5, $C_3\times S_3$ x 4

Order 16: $C_2\times C_8$ x 7, $C_{16}$ x 4, $\OD_{16}$ x 3, $C_4:C_4$ x 3, $C_2^2\times C_4$ x 2

Order 12: $D_6$ x 6, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 10, $C_8$ x 6, $C_2^3$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 51840: $C_2\times F_{81}:C_4$

Order 25920: $C_2\times C_3^4.C_{80}.C_2$, $C_2\times C_3^4.(C_8.F_5)$, $F_{81}:C_4$, $C_3^4:D_{10}.\SD_{16}$

Order 12960: $C_3^4.(C_8.F_5)$ x 2, $C_3^4.C_{80}.C_2$ x 2, $C_3^4:C_{40}:C_4$ x 2, $C_2\times C_3^4.(C_8\times D_5)$, $C_2\times C_3^4.C_{40}.C_2$, $C_2\times C_3^4:C_5.\OD_{16}$, $C_2\times C_3^4.D_5.D_4$, $C_2\times F_{81}$

Order 10368: $C_3^4:\OD_{32}:C_4$

Order 6480: $C_3^4:(C_8\times D_5)$ x 4, $C_3^4:C_5.\OD_{16}$ x 2, $C_3^4:C_{20}:C_4$ x 2, $C_2\times C_3^4.D_{10}.C_2$, $C_3^4.C_{40}.C_2$, $C_2\times C_3^4:C_5.C_8$, $C_2\times C_3^4:C_5.C_8$, $C_2\times C_3^4:C_{40}$, $C_3^4:(C_2^2\times F_5)$, $F_{81}$

Order 5184: $C_2\times C_3^4.C_8:C_4$, $C_3^4:C_2^3.Q_8$, $C_2\times C_3^4:\OD_{32}$, $C_3^4:C_{16}:C_4$

Order 3240: $C_3^4:(C_4\times D_5)$ x 4, $C_3^4:C_5:C_8$ x 2, $C_3^4:(C_2\times F_5)$ x 2, $C_3^4:C_{40}$ x 2, $C_2\times C_3^4:(C_5:C_4)$, $C_2\times C_3^4:C_{20}$, $C_3^4:C_5:C_8$, $C_2\times C_3^4:D_{10}$, $C_2\times C_3^4:F_5$

Order 2592: $C_3^2\wr C_2.\SD_{16}$ x 2, $C_2\times C_3^4:C_{16}$ x 2, $C_3^2:F_9.C_4$ x 2, $C_3^4:\OD_{32}$ x 2, $C_2\times C_3^4.C_4:C_4$, $C_3^2:(C_2^2\times F_9)$, $C_2\times C_3^4:\OD_{16}$

Order 1620: $C_3^4:D_{10}$ x 3, $C_3^4:C_{20}$ x 2, $C_3^4:C_5:C_4$ x 2, $C_2\times C_3^4:C_{10}$, $C_3^4:D_{10}$, $C_3^4:F_5$

Order 1296: $C_3^2:(C_2\times F_9)$ x 4, $C_2\times C_3^2:F_9$ x 2, $C_3^4:\OD_{16}$ x 2, $C_3^2\wr C_2.D_4$ x 2, $C_3^4:C_{16}$ x 2, $C_3^4:(C_2^2\times C_4)$, $C_3^3:(C_4\times D_6)$, $C_2\times C_3^2:F_9$

Order 810: $C_3^4:C_{10}$ x 2, $C_3^4:D_5$ x 2, $C_3^4:C_{10}$

Order 648: $C_3^4:(C_2\times C_4)$ x 4, $C_3^2:F_9$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^2:F_9$ x 2, $C_3^4:C_2^3$, $C_2\times C_3\wr C_4$

Order 640: $C_2\times C_{40}:C_4.C_2$

Order 576: $C_6^2.\SD_{16}$

Order 432: $C_3^2:C_4\times D_6$

Order 405: $C_3^4:C_5$

Order 324: $C_3^4:C_4$ x 4, $C_3^2:S_3^2$ x 3, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3\wr C_4$

Order 320: $C_{16}:D_{10}$, $C_{80}:C_4$, $C_2\times C_{40}.C_4$, $D_{10}.\SD_{16}$

Order 288: $F_9:C_4$ x 2, $C_6^2.Q_8$, $C_6^2.D_4$, $C_2^2\times F_9$

Order 216: $C_3^2:C_4\times S_3$ x 2, $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$

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

Order 160: $C_{40}.C_4$ x 2, $C_{40}:C_4$ x 2, $C_{80}:C_2$ x 2, $C_2\times C_{80}$, $D_{10}.D_4$, $C_{10}:\OD_{16}$, $C_{10}:C_{16}$, $C_8\times D_{10}$

Order 144: $C_2\times F_9$ x 7, $C_6^2:C_4$ x 2, $C_2.\PSU(3,2)$ x 2, $C_2.\SOPlus(4,2)$ x 2, $C_2^2.S_3^2$

Order 128: $\OD_{32}:C_4$

Order 108: $C_3:S_3^2$ x 4, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 81: $C_3^4$

Order 80: $C_8\times D_5$ x 4, $C_{20}:C_4$ x 2, $C_5:\OD_{16}$ x 2, $C_{10}:C_8$, $C_2^2\times F_5$, $C_4\times D_{10}$, $C_{10}:C_8$, $C_2\times C_{40}$, $C_{80}$, $C_5:C_{16}$

Order 72: $C_2\times C_3^2:C_4$ x 10, $F_9$ x 5, $C_6.D_6$ x 2, $C_6:D_6$, $S_3\times D_6$, $C_6:C_{12}$

Order 64: $C_{16}:C_4$, $C_2\times \OD_{32}$, $C_2^3.Q_8$, $C_2^3.D_4$

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

Order 48: $C_4\times D_6$

Order 40: $C_4\times D_5$ x 4, $C_2\times F_5$ x 3, $C_{40}$ x 2, $C_5:C_8$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_5:C_8$, $C_2\times D_{10}$

Order 36: $C_6:S_3$ x 8, $C_3^2:C_4$ x 7, $S_3^2$ x 4, $C_6\times S_3$ x 2, $C_3:C_{12}$, $C_6^2$

Order 32: $\OD_{32}$ x 2, $C_2\times C_{16}$ x 2, $C_8.C_4$ x 2, $C_8:C_4$ x 2, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_2^2.D_4$

Order 27: $C_3^3$

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

Order 20: $D_{10}$ x 4, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$, $F_5$

Order 18: $C_3:S_3$ x 8, $C_3\times C_6$ x 5, $C_3\times S_3$ x 4

Order 16: $C_2\times C_8$ x 7, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_{16}$ x 2

Order 12: $D_6$ x 6, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $C_2^3$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 51840: $C_2\times F_{81}:C_4$ ($C_1$)

Order 25920: $F_{81}:C_4$ ($C_2$) x 4, $C_2\times C_3^4.C_{80}.C_2$ ($C_2$), $C_2\times C_3^4.(C_8.F_5)$ ($C_2$), $C_3^4:D_{10}.\SD_{16}$ ($C_2$)

Order 12960: $C_3^4.(C_8.F_5)$ ($C_2^2$) x 2, $C_3^4.C_{80}.C_2$ ($C_2^2$) x 2, $C_3^4:C_{40}:C_4$ ($C_2^2$) x 2, $C_2\times C_3^4.(C_8\times D_5)$ ($C_2^2$), $C_2\times C_3^4.C_{40}.C_2$ ($C_4$), $C_3^4.C_{80}:C_2$ ($C_4$), $C_3^4.C_{80}.C_2$ ($C_4$), $C_2\times F_{81}$ ($C_4$)

Order 6480: $C_3^4.C_{40}.C_2$ ($C_2\times C_4$) x 2, $F_{81}$ ($C_2\times C_4$) x 2, $C_2\times C_3^4.D_{10}.C_2$ ($D_4$), $C_2\times C_3^4:C_5.C_8$ ($Q_8$), $C_2\times C_3^4:C_{40}$ ($C_2\times C_4$), $C_3^4:(C_8\times D_5)$ ($C_2^3$), $C_3^4:(C_8\times D_5)$ ($Q_8$), $C_3^4:(C_8\times D_5)$ ($D_4$), $C_3^4:(C_8\times D_5)$ ($C_2\times C_4$)

Order 3240: $C_3^4:(C_4\times D_5)$ ($\SD_{16}$) x 2, $C_2\times C_3^4:(C_5:C_4)$ ($\SD_{16}$), $C_2\times C_3^4:C_{20}$ ($C_4:C_4$), $C_3^4:C_5:C_8$ ($C_4:C_4$), $C_3^4:C_5:C_8$ ($C_2\times Q_8$), $C_3^4:(C_4\times D_5)$ ($C_4:C_4$), $C_3^4:(C_4\times D_5)$ ($C_2\times D_4$), $C_2\times C_3^4:D_{10}$ ($\SD_{16}$), $C_3^4:C_{40}$ ($C_4:C_4$), $C_3^4:C_{40}$ ($C_2^2\times C_4$)

Order 2592: $C_2\times C_3^4:C_{16}$ ($F_5$)

Order 1620: $C_2\times C_3^4:C_{10}$ ($C_8:C_4$), $C_3^4:D_{10}$ ($C_2\times \SD_{16}$), $C_3^4:D_{10}$ ($C_8:C_4$), $C_3^4:C_{20}$ ($C_2^2.D_4$), $C_3^4:C_{20}$ ($C_8:C_4$), $C_3^4:C_5:C_4$ ($C_2\times \SD_{16}$), $C_3^4:C_5:C_4$ ($C_8:C_4$)

Order 1296: $C_3^4:C_{16}$ ($C_2\times F_5$) x 2, $C_2\times C_3^2:F_9$ ($C_2\times F_5$)

Order 810: $C_3^4:C_{10}$ ($C_{16}:C_4$), $C_3^4:C_{10}$ ($C_{16}:C_4$), $C_3^4:C_{10}$ ($C_2^3.D_4$)

Order 648: $C_2\times C_3^4:C_4$ ($C_{20}:C_4$), $C_3^2:F_9$ ($C_2^2\times F_5$), $C_3^2:F_9$ ($C_{20}:C_4$)

Order 405: $C_3^4:C_5$ ($\OD_{32}:C_4$)

Order 324: $C_3^3:D_6$ ($C_{40}:C_4$), $C_3^4:C_4$ ($C_{40}:C_4$), $C_3^4:C_4$ ($D_{10}.D_4$)

Order 162: $C_3^3\times C_6$ ($C_{80}:C_4$), $C_3^3:S_3$ ($C_{80}:C_4$), $C_3^3:S_3$ ($D_{10}.\SD_{16}$)

Order 81: $C_3^4$ ($C_2\times C_{40}:C_4.C_2$)

Order 2: $C_2$ ($F_{81}:C_4$)

Order 1: $C_1$ ($C_2\times F_{81}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 51840: $C_2\times F_{81}:C_4$ ($C_1$)

Order 25920: $C_2\times C_3^4.C_{80}.C_2$ ($C_2$), $C_2\times C_3^4.(C_8.F_5)$ ($C_2$), $F_{81}:C_4$ ($C_2$), $C_3^4:D_{10}.\SD_{16}$ ($C_2$)

Order 12960: $C_3^4.(C_8.F_5)$ ($C_2^2$), $C_2\times C_3^4.(C_8\times D_5)$ ($C_2^2$), $C_2\times C_3^4.C_{40}.C_2$ ($C_4$), $C_3^4.C_{80}.C_2$ ($C_2^2$), $C_3^4.C_{80}.C_2$ ($C_4$), $C_3^4:C_{40}:C_4$ ($C_2^2$), $C_2\times F_{81}$ ($C_4$)

Order 6480: $C_2\times C_3^4.D_{10}.C_2$ ($D_4$), $C_3^4.C_{40}.C_2$ ($C_2\times C_4$), $C_2\times C_3^4:C_5.C_8$ ($Q_8$), $C_2\times C_3^4:C_{40}$ ($C_2\times C_4$), $C_3^4:(C_8\times D_5)$ ($C_2^3$), $C_3^4:(C_8\times D_5)$ ($Q_8$), $C_3^4:(C_8\times D_5)$ ($D_4$), $C_3^4:(C_8\times D_5)$ ($C_2\times C_4$), $F_{81}$ ($C_2\times C_4$)

Order 3240: $C_3^4:(C_4\times D_5)$ ($\SD_{16}$) x 2, $C_2\times C_3^4:(C_5:C_4)$ ($\SD_{16}$), $C_2\times C_3^4:C_{20}$ ($C_4:C_4$), $C_3^4:C_5:C_8$ ($C_4:C_4$), $C_3^4:C_5:C_8$ ($C_2\times Q_8$), $C_3^4:(C_4\times D_5)$ ($C_4:C_4$), $C_3^4:(C_4\times D_5)$ ($C_2\times D_4$), $C_2\times C_3^4:D_{10}$ ($\SD_{16}$), $C_3^4:C_{40}$ ($C_4:C_4$), $C_3^4:C_{40}$ ($C_2^2\times C_4$)

Order 2592: $C_2\times C_3^4:C_{16}$ ($F_5$)

Order 1620: $C_2\times C_3^4:C_{10}$ ($C_8:C_4$), $C_3^4:D_{10}$ ($C_2\times \SD_{16}$), $C_3^4:D_{10}$ ($C_8:C_4$), $C_3^4:C_{20}$ ($C_2^2.D_4$), $C_3^4:C_{20}$ ($C_8:C_4$), $C_3^4:C_5:C_4$ ($C_2\times \SD_{16}$), $C_3^4:C_5:C_4$ ($C_8:C_4$)

Order 1296: $C_2\times C_3^2:F_9$ ($C_2\times F_5$), $C_3^4:C_{16}$ ($C_2\times F_5$)

Order 810: $C_3^4:C_{10}$ ($C_{16}:C_4$), $C_3^4:C_{10}$ ($C_{16}:C_4$), $C_3^4:C_{10}$ ($C_2^3.D_4$)

Order 648: $C_2\times C_3^4:C_4$ ($C_{20}:C_4$), $C_3^2:F_9$ ($C_2^2\times F_5$), $C_3^2:F_9$ ($C_{20}:C_4$)

Order 405: $C_3^4:C_5$ ($\OD_{32}:C_4$)

Order 324: $C_3^3:D_6$ ($C_{40}:C_4$), $C_3^4:C_4$ ($C_{40}:C_4$), $C_3^4:C_4$ ($D_{10}.D_4$)

Order 162: $C_3^3\times C_6$ ($C_{80}:C_4$), $C_3^3:S_3$ ($C_{80}:C_4$), $C_3^3:S_3$ ($D_{10}.\SD_{16}$)

Order 81: $C_3^4$ ($C_2\times C_{40}:C_4.C_2$)

Order 2: $C_2$ ($F_{81}:C_4$)

Order 1: $C_1$ ($C_2\times F_{81}:C_4$)

Series

Derived series

$C_2\times F_{81}:C_4$

$\rhd$

$C_3^4:C_{40}$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_2\times F_{81}:C_4$

$\rhd$

$F_{81}:C_4$

$\rhd$

$C_3^4:C_{40}:C_4$

$\rhd$

$C_3^4:(C_8\times D_5)$

$\rhd$

$C_3^4:(C_4\times D_5)$

$\rhd$

$C_3^4:C_{20}$

$\rhd$

$C_3^4:C_{10}$

$\rhd$

$C_3^4:C_5$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Lower central series

$C_2\times F_{81}:C_4$

$\rhd$

$C_3^4:C_{40}$

$\rhd$

$C_3^4:C_{20}$

$\rhd$

$C_3^4:C_{10}$

$\rhd$

$C_3^4:C_5$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.