Abstract group 92880.a: $C_{215}\times D_{216}$

• Label: 92880.a
• Order: \( 2^{4} \cdot 3^{3} \cdot 5 \cdot 43 \)
• Exponent: \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \cdot 43 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \cdot 43 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{6} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{3} \cdot 7 \)
• Perm deg.: $83$
• Trans deg.: $46440$
• Rank: $2$

Group information

Description:	$C_{215}\times D_{216}$
Order:	\(92880\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 43 \)
Exponent:	\(46440\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \cdot 43 \)
Automorphism group:	Group of order \(2612736\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 3, $C_5$, $C_{43}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	18	20	24	27	30	36	40	43	45	54	60	72	86	90	108	120	129	135	172	180	215	216	258	270	344	360	387	430	516	540	645	774	860	1032	1080	1161	1290	1548	1720	1935	2322	2580	3096	3870	4644	5160	5805	7740	9288	11610	15480	23220	46440
Elements	1	217	2	2	4	2	4	6	868	4	8	6	8	8	18	8	12	16	42	24	18	16	24	9114	24	36	32	84	72	84	48	168	72	84	72	168	96	252	36456	168	144	336	252	336	336	288	756	336	504	672	1008	756	672	1008	1008	1512	1344	3024	2016	3024	3024	4032	6048	12096	92880
Conjugacy classes	1	3	1	1	4	1	2	3	12	2	4	3	4	4	9	4	6	8	42	12	9	8	12	126	12	18	16	42	36	42	24	168	36	42	36	84	48	126	504	84	72	168	126	168	168	144	378	168	252	336	504	378	336	504	504	756	672	1512	1008	1512	1512	2016	3024	6048	23865
Divisions	1	3	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	72
Autjugacy classes	1	2	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	68

Minimal presentations

Permutation degree:	$83$
Transitive degree:	$46440$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{46440}=1, b^{a}=b^{431} \rangle$
Permutation group:	Degree $83$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 421 & 30 \\ 189 & 421 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 430 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{431})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{7740}$ . $D_6$	$C_{5160}$ . $D_9$	$C_{860}$ . $D_{54}$	$C_{645}$ . $D_{72}$	all 60

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

Homology

Abelianization:	$C_{2} \times C_{430} \simeq C_{2}^{2} \times C_{5} \times C_{43}$
Schur multiplier:	$C_{2}$
Commutator length:	not computed

Subgroups

There are 2464 subgroups in 176 conjugacy classes, 76 normal (68 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{430}$	$G/Z \simeq$ $D_{108}$
Commutator:	$G' \simeq$ $C_{108}$	$G/G' \simeq$ $C_2\times C_{430}$
Frattini:	$\Phi \simeq$ $C_{36}$	$G/\Phi \simeq$ $S_3\times C_{430}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{46440}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{215}\times D_{216}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1290}$	$G/\operatorname{soc} \simeq$ $D_{36}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_{27}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 92880: $C_{215}\times D_{216}$

Order 46440: $C_{215}\times D_{108}$ x 2, $C_{46440}$

Order 30960: $D_{72}\times C_{215}$

Order 23220: $D_{27}\times C_{430}$ x 2, $C_{23220}$

Order 18576: $C_{43}\times D_{216}$

Order 15480: $D_{36}\times C_{215}$ x 2, $C_{15480}$

Order 11610: $D_{27}\times C_{215}$ x 2, $C_{11610}$

Order 10320: $D_{24}\times C_{215}$

Order 9288: $C_{43}\times D_{108}$ x 2, $C_{9288}$

Order 7740: $D_9\times C_{430}$ x 2, $C_{7740}$

Order 6192: $C_{43}\times D_{72}$

Order 5805: $C_{5805}$

Order 5160: $D_{12}\times C_{215}$ x 2, $C_{5160}$

Order 4644: $C_{86}\times D_{27}$ x 2, $C_{4644}$

Order 3870: $D_9\times C_{215}$ x 2, $C_{3870}$

Order 3440: $D_8\times C_{215}$

Order 3096: $C_{43}\times D_{36}$ x 2, $C_{3096}$

Order 2580: $C_{430}\times S_3$ x 2, $C_{2580}$

Order 2322: $C_{43}\times D_{27}$ x 2, $C_{2322}$

Order 2160: $C_5\times D_{216}$

Order 2064: $C_{43}\times D_{24}$

Order 1935: $C_{1935}$

Order 1720: $D_4\times C_{215}$ x 2, $C_{1720}$

Order 1548: $D_9\times C_{86}$ x 2, $C_{1548}$

Order 1290: $S_3\times C_{215}$ x 2, $C_{1290}$

Order 1161: $C_{1161}$

Order 1080: $C_5\times D_{108}$ x 2, $C_{1080}$

Order 1032: $D_{12}\times C_{43}$ x 2, $C_{1032}$

Order 860: $C_2\times C_{430}$ x 2, $C_{860}$

Order 774: $D_9\times C_{43}$ x 2, $C_{774}$

Order 720: $C_5\times D_{72}$

Order 688: $D_8\times C_{43}$

Order 645: $C_{645}$

Order 540: $C_5\times D_{54}$ x 2, $C_{540}$

Order 516: $S_3\times C_{86}$ x 2, $C_{516}$

Order 432: $D_{216}$

Order 430: $C_{430}$ x 3

Order 387: $C_{387}$

Order 360: $C_5\times D_{36}$ x 2, $C_{360}$

Order 344: $D_4\times C_{43}$ x 2, $C_{344}$

Order 270: $C_5\times D_{27}$ x 2, $C_{270}$

Order 258: $S_3\times C_{43}$ x 2, $C_{258}$

Order 240: $C_5\times D_{24}$

Order 216: $D_{108}$ x 2, $C_{216}$

Order 215: $C_{215}$

Order 180: $C_5\times D_{18}$ x 2, $C_{180}$

Order 172: $C_2\times C_{86}$ x 2, $C_{172}$

Order 144: $D_{72}$

Order 135: $C_{135}$

Order 129: $C_{129}$

Order 120: $C_5\times D_{12}$ x 2, $C_{120}$

Order 108: $D_{54}$ x 2, $C_{108}$

Order 90: $C_5\times D_9$ x 2, $C_{90}$

Order 86: $C_{86}$ x 3

Order 80: $C_5\times D_8$

Order 72: $D_{36}$ x 2, $C_{72}$

Order 60: $S_3\times C_{10}$ x 2, $C_{60}$

Order 54: $D_{27}$ x 2, $C_{54}$

Order 48: $D_{24}$

Order 45: $C_{45}$

Order 43: $C_{43}$

Order 40: $C_5\times D_4$ x 2, $C_{40}$

Order 36: $D_{18}$ x 2, $C_{36}$

Order 30: $C_5\times S_3$ x 2, $C_{30}$

Order 27: $C_{27}$

Order 24: $D_{12}$ x 2, $C_{24}$

Order 20: $C_2\times C_{10}$ x 2, $C_{20}$

Order 18: $D_9$ x 2, $C_{18}$

Order 16: $D_8$

Order 15: $C_{15}$

Order 12: $D_6$ x 2, $C_{12}$

Order 10: $C_{10}$ x 3

Order 9: $C_9$

Order 8: $D_4$ x 2, $C_8$

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

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 92880: $C_{215}\times D_{216}$

Order 46440: $C_{46440}$

Order 30960: $D_{72}\times C_{215}$

Order 18576: $C_{43}\times D_{216}$

Order 10320: $D_{24}\times C_{215}$

Order 6192: $C_{43}\times D_{72}$

Order 5805: $C_{5805}$

Order 3440: $D_8\times C_{215}$

Order 2160: $C_5\times D_{216}$

Order 2064: $C_{43}\times D_{24}$

Order 1935: $C_{1935}$

Order 1161: $C_{1161}$

Order 720: $C_5\times D_{72}$

Order 688: $D_8\times C_{43}$

Order 645: $C_{645}$

Order 432: $D_{216}$

Order 387: $C_{387}$

Order 240: $C_5\times D_{24}$

Order 215: $C_{215}$

Order 144: $D_{72}$

Order 135: $C_{135}$

Order 129: $C_{129}$

Order 80: $C_5\times D_8$

Order 48: $D_{24}$

Order 45: $C_{45}$

Order 43: $C_{43}$

Order 27: $C_{27}$

Order 16: $D_8$

Order 15: $C_{15}$

Order 9: $C_9$

Order 5: $C_5$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 92880: $C_{215}\times D_{216}$ ($C_1$)

Order 46440: $C_{215}\times D_{108}$ ($C_2$) x 2, $C_{46440}$ ($C_2$)

Order 23220: $C_{23220}$ ($C_2^2$)

Order 18576: $C_{43}\times D_{216}$ ($C_5$)

Order 15480: $C_{15480}$ ($S_3$)

Order 11610: $C_{11610}$ ($D_4$)

Order 9288: $C_{43}\times D_{108}$ ($C_{10}$) x 2, $C_{9288}$ ($C_{10}$)

Order 7740: $C_{7740}$ ($D_6$)

Order 5805: $C_{5805}$ ($D_8$)

Order 5160: $C_{5160}$ ($D_9$)

Order 4644: $C_{4644}$ ($C_2\times C_{10}$)

Order 3870: $C_{3870}$ ($D_{12}$)

Order 3096: $C_{3096}$ ($C_5\times S_3$)

Order 2580: $C_{2580}$ ($D_{18}$)

Order 2322: $C_{2322}$ ($C_5\times D_4$)

Order 2160: $C_5\times D_{216}$ ($C_{43}$)

Order 1935: $C_{1935}$ ($D_{24}$)

Order 1720: $C_{1720}$ ($D_{27}$)

Order 1548: $C_{1548}$ ($S_3\times C_{10}$)

Order 1290: $C_{1290}$ ($D_{36}$)

Order 1161: $C_{1161}$ ($C_5\times D_8$)

Order 1080: $C_5\times D_{108}$ ($C_{86}$) x 2, $C_{1080}$ ($C_{86}$)

Order 1032: $C_{1032}$ ($C_5\times D_9$)

Order 860: $C_{860}$ ($D_{54}$)

Order 774: $C_{774}$ ($C_5\times D_{12}$)

Order 645: $C_{645}$ ($D_{72}$)

Order 540: $C_{540}$ ($C_2\times C_{86}$)

Order 516: $C_{516}$ ($C_5\times D_{18}$)

Order 432: $D_{216}$ ($C_{215}$)

Order 430: $C_{430}$ ($D_{108}$)

Order 387: $C_{387}$ ($C_5\times D_{24}$)

Order 360: $C_{360}$ ($S_3\times C_{43}$)

Order 344: $C_{344}$ ($C_5\times D_{27}$)

Order 270: $C_{270}$ ($D_4\times C_{43}$)

Order 258: $C_{258}$ ($C_5\times D_{36}$)

Order 216: $D_{108}$ ($C_{430}$) x 2, $C_{216}$ ($C_{430}$)

Order 215: $C_{215}$ ($D_{216}$)

Order 180: $C_{180}$ ($S_3\times C_{86}$)

Order 172: $C_{172}$ ($C_5\times D_{54}$)

Order 135: $C_{135}$ ($D_8\times C_{43}$)

Order 129: $C_{129}$ ($C_5\times D_{72}$)

Order 120: $C_{120}$ ($D_9\times C_{43}$)

Order 108: $C_{108}$ ($C_2\times C_{430}$)

Order 90: $C_{90}$ ($D_{12}\times C_{43}$)

Order 86: $C_{86}$ ($C_5\times D_{108}$)

Order 72: $C_{72}$ ($S_3\times C_{215}$)

Order 60: $C_{60}$ ($D_9\times C_{86}$)

Order 54: $C_{54}$ ($D_4\times C_{215}$)

Order 45: $C_{45}$ ($C_{43}\times D_{24}$)

Order 43: $C_{43}$ ($C_5\times D_{216}$)

Order 40: $C_{40}$ ($C_{43}\times D_{27}$)

Order 36: $C_{36}$ ($S_3\times C_{430}$)

Order 30: $C_{30}$ ($C_{43}\times D_{36}$)

Order 27: $C_{27}$ ($D_8\times C_{215}$)

Order 24: $C_{24}$ ($C_{1935}:C_2$)

Order 20: $C_{20}$ ($C_{43}\times D_{54}$)

Order 18: $C_{18}$ ($D_{12}\times C_{215}$)

Order 15: $C_{15}$ ($C_{43}\times D_{72}$)

Order 12: $C_{12}$ ($D_9\times C_{430}$)

Order 10: $C_{10}$ ($C_{43}\times D_{108}$)

Order 9: $C_9$ ($D_{24}\times C_{215}$)

Order 8: $C_8$ ($D_{27}\times C_{215}$)

Order 6: $C_6$ ($D_{36}\times C_{215}$)

Order 5: $C_5$ ($C_{43}\times D_{216}$)

Order 4: $C_4$ ($D_{27}\times C_{430}$)

Order 3: $C_3$ ($D_{72}\times C_{215}$)

Order 2: $C_2$ ($C_{215}\times D_{108}$)

Order 1: $C_1$ ($C_{215}\times D_{216}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 92880: $C_{215}\times D_{216}$ ($C_1$)

Order 46440: $C_{46440}$ ($C_2$)

Order 18576: $C_{43}\times D_{216}$ ($C_5$)

Order 5805: $C_{5805}$ ($D_8$)

Order 2160: $C_5\times D_{216}$ ($C_{43}$)

Order 1935: $C_{1935}$ ($D_{24}$)

Order 1161: $C_{1161}$ ($C_5\times D_8$)

Order 645: $C_{645}$ ($D_{72}$)

Order 432: $D_{216}$ ($C_{215}$)

Order 387: $C_{387}$ ($C_5\times D_{24}$)

Order 215: $C_{215}$ ($D_{216}$)

Order 135: $C_{135}$ ($D_8\times C_{43}$)

Order 129: $C_{129}$ ($C_5\times D_{72}$)

Order 45: $C_{45}$ ($C_{43}\times D_{24}$)

Order 43: $C_{43}$ ($C_5\times D_{216}$)

Order 27: $C_{27}$ ($D_8\times C_{215}$)

Order 15: $C_{15}$ ($C_{43}\times D_{72}$)

Order 9: $C_9$ ($D_{24}\times C_{215}$)

Order 5: $C_5$ ($C_{43}\times D_{216}$)

Order 3: $C_3$ ($D_{72}\times C_{215}$)

Order 1: $C_1$ ($C_{215}\times D_{216}$)

Series

Derived series

$C_{215}\times D_{216}$

$\rhd$

$C_{108}$

$\rhd$

$C_1$

Chief series

$C_{215}\times D_{216}$

$\rhd$

$C_{215}\times D_{108}$

$\rhd$

$C_{23220}$

$\rhd$

$C_{11610}$

$\rhd$

$C_{5805}$

$\rhd$

$C_{1935}$

$\rhd$

$C_{645}$

$\rhd$

$C_{215}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$C_{215}\times D_{216}$

$\rhd$

$C_{108}$

$\rhd$

$C_{54}$

$\rhd$

$C_{27}$

Upper central series

$C_1$

$\lhd$

$C_{430}$

$\lhd$

$C_{860}$

$\lhd$

$C_{1720}$

Supergroups

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

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

Character theory

Complex character table

The $23865 \times 23865$ character table is not available for this group.

Rational character table

The $72 \times 72$ rational character table is not available for this group.