Abstract group 5160.d: $C_{645}:Q_8$

• Label: 5160.d
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 43 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \cdot 43 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \cdot 43 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 7 \)
• Perm deg.: $59$
• Trans deg.: $5160$
• Rank: $2$

Group information

Description:	$C_{645}:Q_8$
Order:	\(5160\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 43 \)
Exponent:	\(2580\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 43 \)
Automorphism group:	$D_5\times C_5^2:D_{20}$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$, $C_{43}$
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	10	12	15	20	30	43	60	86	129	172	215	258	430	516	645	860	1290	2580
Elements	1	1	2	14	4	2	4	4	8	56	8	42	16	42	84	588	168	84	168	168	336	2352	336	672	5160
Conjugacy classes	1	1	1	3	4	1	4	2	4	12	4	42	8	42	42	126	168	42	168	84	168	504	168	336	1935
Divisions	1	1	1	3	1	1	1	1	1	3	1	1	1	1	1	3	1	1	1	1	1	3	1	1	32
Autjugacy classes	1	1	1	2	1	1	1	1	1	2	1	1	1	1	1	2	1	1	1	1	1	2	1	1	28

Minimal presentations

Permutation degree:	$59$
Transitive degree:	$5160$
Rank:	$2$
Inequivalent generating pairs:	$792$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{2580}=1, a^{2}=b^{1962}, b^{a}=b^{431} \rangle$
Permutation group:	Degree $59$ $\langle(1,2,4,7)(3,6,8,5)(9,10,11)(12,13,15,14,16)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 327 & 144 \\ 390 & 327 \end{array}\right), \left(\begin{array}{rr} 257 & 17 \\ 367 & 174 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{431})$
Direct product:	$C_5$ $\, \times\, $ $C_{43}$ $\, \times\, $ $(C_3:Q_8)$
Semidirect product:	$C_{645}$ $\,\rtimes\,$ $Q_8$	$C_{129}$ $\,\rtimes\,$ $(C_5\times Q_8)$	$C_{15}$ $\,\rtimes\,$ $(Q_8\times C_{43})$	$C_3$ $\,\rtimes\,$ $(Q_8\times C_{215})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{860}$ . $S_3$	$C_{430}$ . $D_6$	$C_{60}$ . $C_{86}$	$C_{2580}$ . $C_2$	all 20

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:	$C_1$
Commutator length:	$1$

Subgroups

There are 72 subgroups in 48 conjugacy classes, 36 normal (28 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{430}$	$G/Z \simeq$ $D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2\times C_{430}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{645}:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2580}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{645}:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1290}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 5160: $C_{645}:Q_8$

Order 2580: $C_{215}\times C_3:C_4$ x 2, $C_{2580}$

Order 1720: $Q_8\times C_{215}$

Order 1290: $C_{1290}$

Order 1032: $C_{129}:Q_8$

Order 860: $C_{860}$ x 3

Order 645: $C_{645}$

Order 516: $C_3:C_{172}$ x 2, $C_{516}$

Order 430: $C_{430}$

Order 344: $Q_8\times C_{43}$

Order 258: $C_{258}$

Order 215: $C_{215}$

Order 172: $C_{172}$ x 3

Order 129: $C_{129}$

Order 120: $C_{15}:Q_8$

Order 86: $C_{86}$

Order 60: $C_3:C_{20}$ x 2, $C_{60}$

Order 43: $C_{43}$

Order 40: $C_5\times Q_8$

Order 30: $C_{30}$

Order 24: $C_3:Q_8$

Order 20: $C_{20}$ x 3

Order 15: $C_{15}$

Order 12: $C_3:C_4$ x 2, $C_{12}$

Order 10: $C_{10}$

Order 8: $Q_8$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5160: $C_{645}:Q_8$

Order 2580: $C_{215}\times C_3:C_4$, $C_{2580}$

Order 1720: $Q_8\times C_{215}$

Order 1290: $C_{1290}$

Order 1032: $C_{129}:Q_8$

Order 860: $C_{860}$ x 2

Order 645: $C_{645}$

Order 516: $C_{516}$, $C_3:C_{172}$

Order 430: $C_{430}$

Order 344: $Q_8\times C_{43}$

Order 258: $C_{258}$

Order 215: $C_{215}$

Order 172: $C_{172}$ x 2

Order 129: $C_{129}$

Order 120: $C_{15}:Q_8$

Order 86: $C_{86}$

Order 60: $C_{60}$, $C_3:C_{20}$

Order 43: $C_{43}$

Order 40: $C_5\times Q_8$

Order 30: $C_{30}$

Order 24: $C_3:Q_8$

Order 20: $C_{20}$ x 2

Order 15: $C_{15}$

Order 12: $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$

Order 8: $Q_8$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5160: $C_{645}:Q_8$ ($C_1$)

Order 2580: $C_{215}\times C_3:C_4$ ($C_2$) x 2, $C_{2580}$ ($C_2$)

Order 1290: $C_{1290}$ ($C_2^2$)

Order 1032: $C_{129}:Q_8$ ($C_5$)

Order 860: $C_{860}$ ($S_3$)

Order 645: $C_{645}$ ($Q_8$)

Order 516: $C_3:C_{172}$ ($C_{10}$) x 2, $C_{516}$ ($C_{10}$)

Order 430: $C_{430}$ ($D_6$)

Order 258: $C_{258}$ ($C_2\times C_{10}$)

Order 215: $C_{215}$ ($C_3:Q_8$)

Order 172: $C_{172}$ ($C_5\times S_3$)

Order 129: $C_{129}$ ($C_5\times Q_8$)

Order 120: $C_{15}:Q_8$ ($C_{43}$)

Order 86: $C_{86}$ ($S_3\times C_{10}$)

Order 60: $C_3:C_{20}$ ($C_{86}$) x 2, $C_{60}$ ($C_{86}$)

Order 43: $C_{43}$ ($C_{15}:Q_8$)

Order 30: $C_{30}$ ($C_2\times C_{86}$)

Order 24: $C_3:Q_8$ ($C_{215}$)

Order 20: $C_{20}$ ($S_3\times C_{43}$)

Order 15: $C_{15}$ ($Q_8\times C_{43}$)

Order 12: $C_3:C_4$ ($C_{430}$) x 2, $C_{12}$ ($C_{430}$)

Order 10: $C_{10}$ ($S_3\times C_{86}$)

Order 6: $C_6$ ($C_2\times C_{430}$)

Order 5: $C_5$ ($C_{129}:Q_8$)

Order 4: $C_4$ ($S_3\times C_{215}$)

Order 3: $C_3$ ($Q_8\times C_{215}$)

Order 2: $C_2$ ($C_{645}:C_2^2$)

Order 1: $C_1$ ($C_{645}:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5160: $C_{645}:Q_8$ ($C_1$)

Order 2580: $C_{215}\times C_3:C_4$ ($C_2$), $C_{2580}$ ($C_2$)

Order 1290: $C_{1290}$ ($C_2^2$)

Order 1032: $C_{129}:Q_8$ ($C_5$)

Order 860: $C_{860}$ ($S_3$)

Order 645: $C_{645}$ ($Q_8$)

Order 516: $C_{516}$ ($C_{10}$), $C_3:C_{172}$ ($C_{10}$)

Order 430: $C_{430}$ ($D_6$)

Order 258: $C_{258}$ ($C_2\times C_{10}$)

Order 215: $C_{215}$ ($C_3:Q_8$)

Order 172: $C_{172}$ ($C_5\times S_3$)

Order 129: $C_{129}$ ($C_5\times Q_8$)

Order 120: $C_{15}:Q_8$ ($C_{43}$)

Order 86: $C_{86}$ ($S_3\times C_{10}$)

Order 60: $C_{60}$ ($C_{86}$), $C_3:C_{20}$ ($C_{86}$)

Order 43: $C_{43}$ ($C_{15}:Q_8$)

Order 30: $C_{30}$ ($C_2\times C_{86}$)

Order 24: $C_3:Q_8$ ($C_{215}$)

Order 20: $C_{20}$ ($S_3\times C_{43}$)

Order 15: $C_{15}$ ($Q_8\times C_{43}$)

Order 12: $C_{12}$ ($C_{430}$), $C_3:C_4$ ($C_{430}$)

Order 10: $C_{10}$ ($S_3\times C_{86}$)

Order 6: $C_6$ ($C_2\times C_{430}$)

Order 5: $C_5$ ($C_{129}:Q_8$)

Order 4: $C_4$ ($S_3\times C_{215}$)

Order 3: $C_3$ ($Q_8\times C_{215}$)

Order 2: $C_2$ ($C_{645}:C_2^2$)

Order 1: $C_1$ ($C_{645}:Q_8$)

Series

Derived series

$C_{645}:Q_8$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_{645}:Q_8$

$\rhd$

$C_{215}\times C_3:C_4$

$\rhd$

$C_{1290}$

$\rhd$

$C_{645}$

$\rhd$

$C_{215}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$C_{645}:Q_8$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{430}$

$\lhd$

$C_{860}$

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

The $1935 \times 1935$ character table is not available for this group.

Rational character table

The $32 \times 32$ rational character table is not available for this group.