Abstract group 912.72: $C_{57}:Q_{16}$

• Label: 912.72
• Order: \( 2^{4} \cdot 3 \cdot 19 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $38$
• Trans deg.: $912$
• Rank: $2$

Group information

Description:	$C_{57}:Q_{16}$
Order:	\(912\)\(\medspace = 2^{4} \cdot 3 \cdot 19 \)
Exponent:	\(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \)
Automorphism group:	$C_{38}.C_{18}.C_2^4$, of order \(10944\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 19 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{19}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	19	24	38	57	76	114	228
Elements	1	1	2	82	2	76	164	18	152	18	36	108	36	216	912
Conjugacy classes	1	1	2	3	2	2	6	9	4	9	18	27	18	54	156
Divisions	1	1	1	3	1	1	3	1	1	1	1	2	1	2	20
Autjugacy classes	1	1	1	3	1	1	3	1	1	1	1	2	1	2	20

Dimension	1	2	4	8	18	36	72
Irr. complex chars.	12	117	27	0	0	0	0	156
Irr. rational chars.	4	5	2	1	2	4	2	20

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$912$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	144
Arbitrary	4	6	26

Constructions

Presentation:	$\langle a, b, c \mid b^{8}=c^{57}=[a,c]=1, a^{2}=b^{4}, b^{a}=b^{7}, c^{b}=c^{37} \rangle$
Permutation group:	Degree $38$ $\langle(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(21,22)(23,24)(25,26)(27,28) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_{19}:Q_{16})$
Semidirect product:	$C_{57}$ $\,\rtimes\,$ $Q_{16}$	$C_{19}$ $\,\rtimes\,$ $(C_3\times Q_{16})$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{114}$ . $D_4$	$C_{12}$ . $D_{38}$	$C_{228}$ . $C_2^2$	$(C_3\times Q_8)$ . $D_{19}$	all 16

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 224 subgroups in 36 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_{19}:D_4$
Commutator:	$G' \simeq$ $C_{76}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3\times D_{38}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{57}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{57}:Q_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{114}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

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 912: $C_{57}:Q_{16}$

Order 456: $Q_8\times C_{57}$, $C_{19}:C_{24}$, $C_{57}:Q_8$

Order 304: $C_{19}:Q_{16}$

Order 228: $C_{228}$ x 2, $C_{19}:C_{12}$

Order 152: $C_{19}:Q_8$, $Q_8\times C_{19}$, $C_{19}:C_8$

Order 114: $C_{114}$

Order 76: $C_{76}$ x 2, $C_{19}:C_4$

Order 57: $C_{57}$

Order 48: $C_3\times Q_{16}$

Order 38: $C_{38}$

Order 24: $C_3\times Q_8$ x 2, $C_{24}$

Order 19: $C_{19}$

Order 16: $Q_{16}$

Order 12: $C_{12}$ x 3

Order 8: $Q_8$ x 2, $C_8$

Order 6: $C_6$

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 912: $C_{57}:Q_{16}$

Order 456: $Q_8\times C_{57}$, $C_{19}:C_{24}$, $C_{57}:Q_8$

Order 304: $C_{19}:Q_{16}$

Order 228: $C_{228}$ x 2, $C_{19}:C_{12}$

Order 152: $C_{19}:Q_8$, $Q_8\times C_{19}$, $C_{19}:C_8$

Order 114: $C_{114}$

Order 76: $C_{76}$ x 2, $C_{19}:C_4$

Order 57: $C_{57}$

Order 48: $C_3\times Q_{16}$

Order 38: $C_{38}$

Order 24: $C_3\times Q_8$ x 2, $C_{24}$

Order 19: $C_{19}$

Order 16: $Q_{16}$

Order 12: $C_{12}$ x 3

Order 8: $Q_8$ x 2, $C_8$

Order 6: $C_6$

Order 4: $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 912: $C_{57}:Q_{16}$ ($C_1$)

Order 456: $Q_8\times C_{57}$ ($C_2$), $C_{19}:C_{24}$ ($C_2$), $C_{57}:Q_8$ ($C_2$)

Order 304: $C_{19}:Q_{16}$ ($C_3$)

Order 228: $C_{228}$ ($C_2^2$)

Order 152: $C_{19}:Q_8$ ($C_6$), $Q_8\times C_{19}$ ($C_6$), $C_{19}:C_8$ ($C_6$)

Order 114: $C_{114}$ ($D_4$)

Order 76: $C_{76}$ ($C_2\times C_6$)

Order 57: $C_{57}$ ($Q_{16}$)

Order 38: $C_{38}$ ($C_3\times D_4$)

Order 24: $C_3\times Q_8$ ($D_{19}$)

Order 19: $C_{19}$ ($C_3\times Q_{16}$)

Order 12: $C_{12}$ ($D_{38}$)

Order 8: $Q_8$ ($C_3\times D_{19}$)

Order 6: $C_6$ ($C_{19}:D_4$)

Order 4: $C_4$ ($C_3\times D_{38}$)

Order 3: $C_3$ ($C_{19}:Q_{16}$)

Order 2: $C_2$ ($C_{57}:D_4$)

Order 1: $C_1$ ($C_{57}:Q_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 912: $C_{57}:Q_{16}$ ($C_1$)

Order 456: $Q_8\times C_{57}$ ($C_2$), $C_{19}:C_{24}$ ($C_2$), $C_{57}:Q_8$ ($C_2$)

Order 304: $C_{19}:Q_{16}$ ($C_3$)

Order 228: $C_{228}$ ($C_2^2$)

Order 152: $C_{19}:Q_8$ ($C_6$), $Q_8\times C_{19}$ ($C_6$), $C_{19}:C_8$ ($C_6$)

Order 114: $C_{114}$ ($D_4$)

Order 76: $C_{76}$ ($C_2\times C_6$)

Order 57: $C_{57}$ ($Q_{16}$)

Order 38: $C_{38}$ ($C_3\times D_4$)

Order 24: $C_3\times Q_8$ ($D_{19}$)

Order 19: $C_{19}$ ($C_3\times Q_{16}$)

Order 12: $C_{12}$ ($D_{38}$)

Order 8: $Q_8$ ($C_3\times D_{19}$)

Order 6: $C_6$ ($C_{19}:D_4$)

Order 4: $C_4$ ($C_3\times D_{38}$)

Order 3: $C_3$ ($C_{19}:Q_{16}$)

Order 2: $C_2$ ($C_{57}:D_4$)

Order 1: $C_1$ ($C_{57}:Q_{16}$)

Series

Derived series

$C_{57}:Q_{16}$

$\rhd$

$C_{76}$

$\rhd$

$C_1$

Chief series

$C_{57}:Q_{16}$

$\rhd$

$Q_8\times C_{57}$

$\rhd$

$C_{228}$

$\rhd$

$C_{114}$

$\rhd$

$C_{57}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Lower central series

$C_{57}:Q_{16}$

$\rhd$

$C_{76}$

$\rhd$

$C_{38}$

$\rhd$

$C_{19}$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_3\times Q_8$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $156 \times 156$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.