Abstract group 1824.1172: $C_{38}:C_6\times Q_8$

• Label: 1824.1172
• Order: \( 2^{5} \cdot 3 \cdot 19 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{3} \)
• Perm deg.: $31$
• Trans deg.: $608$
• Rank: $4$

Group information

Description:	$C_{38}:C_6\times Q_8$
Order:	\(1824\)\(\medspace = 2^{5} \cdot 3 \cdot 19 \)
Exponent:	\(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \)
Automorphism group:	$C_2^4.C_2^4.C_{57}.C_3.C_9.C_2^3$, of order \(3151872\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 19 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{19}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	12	19	38	76
Elements	1	7	38	24	266	912	18	126	432	1824
Conjugacy classes	1	7	2	12	14	24	6	42	72	180
Divisions	1	7	1	12	7	12	1	7	12	60
Autjugacy classes	1	2	2	1	4	2	1	2	1	16

Dimension	1	2	3	4	6	18	36
Irr. complex chars.	48	12	96	0	24	0	0	180
Irr. rational chars.	16	20	0	4	0	16	4	60

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$608$
Rank:	$4$
Inequivalent generating quadruples:	$1066800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=d^{76}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=1, c^{6}=d^{38}, d^{c}=d^{11} \rangle$
Permutation group:	Degree $31$ $\langle(20,21)(22,24)(23,25)(26,27)(28,29)(30,31), (20,21)(22,24)(23,25)(26,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 77 & 76 \\ 76 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 77 & 38 \\ 38 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 77 & 0 \\ 0 & 77 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 57 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/114\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $Q_8$ $\, \times\, $ $(C_{19}:C_3)$
Semidirect product:	$(Q_8\times C_{38})$ $\,\rtimes\,$ $C_6$	$C_{38}$ $\,\rtimes\,$ $(C_6\times Q_8)$	$(C_{76}.C_2^3)$ $\,\rtimes\,$ $C_3$	$C_{19}$ $\,\rtimes\,$ $(C_6.C_2^4)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times C_{76})$ . $C_6$	$(C_{38}:C_6)$ . $C_2^3$	$(C_{19}:C_6)$ . $C_2^4$	$C_{76}$ . $(C_2^2\times C_6)$	all 18

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

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 1716 subgroups in 312 conjugacy classes, 234 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_{38}:C_6$
Commutator:	$G' \simeq$ $C_{38}$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times C_{38}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{76}.C_2^3$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_{38}:C_6\times Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{38}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times Q_8$
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

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Classes of subgroups up to automorphism

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 1824: $C_{38}:C_6\times Q_8$

Order 912: $C_{19}:C_6\times Q_8$ x 12, $C_2\times C_{38}:C_{12}$ x 3

Order 608: $C_{76}.C_2^3$

Order 456: $C_{38}:C_{12}$ x 18, $C_{76}.C_6$ x 16, $C_2\times C_{38}:C_6$

Order 304: $Q_8\times C_{38}$ x 12, $C_2^2\times C_{76}$ x 3

Order 228: $C_{19}:C_{12}$ x 12, $C_{38}:C_6$ x 7

Order 152: $C_2\times C_{76}$ x 18, $Q_8\times C_{19}$ x 16, $C_2^2\times C_{38}$

Order 114: $C_{19}:C_6$ x 7

Order 96: $C_6.C_2^4$

Order 76: $C_{76}$ x 12, $C_2\times C_{38}$ x 7

Order 57: $C_{19}:C_3$

Order 48: $C_6\times Q_8$ x 12, $C_2^2\times C_{12}$ x 3

Order 38: $C_{38}$ x 7

Order 32: $C_2^2\times Q_8$

Order 24: $C_2\times C_{12}$ x 18, $C_3\times Q_8$ x 16, $C_2^2\times C_6$

Order 19: $C_{19}$

Order 16: $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 3

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 18, $Q_8$ x 16, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1824: $C_{38}:C_6\times Q_8$

Order 912: $C_{19}:C_6\times Q_8$, $C_2\times C_{38}:C_{12}$

Order 608: $C_{76}.C_2^3$

Order 456: $C_2\times C_{38}:C_6$, $C_{76}.C_6$, $C_{38}:C_{12}$

Order 304: $Q_8\times C_{38}$, $C_2^2\times C_{76}$

Order 228: $C_{38}:C_6$ x 2, $C_{19}:C_{12}$

Order 152: $C_2\times C_{76}$, $C_2^2\times C_{38}$, $Q_8\times C_{19}$

Order 114: $C_{19}:C_6$ x 2

Order 96: $C_6.C_2^4$

Order 76: $C_2\times C_{38}$ x 2, $C_{76}$

Order 57: $C_{19}:C_3$

Order 48: $C_6\times Q_8$, $C_2^2\times C_{12}$

Order 38: $C_{38}$ x 2

Order 32: $C_2^2\times Q_8$

Order 24: $C_2\times C_{12}$, $C_2^2\times C_6$, $C_3\times Q_8$

Order 19: $C_{19}$

Order 16: $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 2, $C_{12}$

Order 8: $C_2^3$, $Q_8$, $C_2\times C_4$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1824: $C_{38}:C_6\times Q_8$ ($C_1$)

Order 912: $C_{19}:C_6\times Q_8$ ($C_2$) x 12, $C_2\times C_{38}:C_{12}$ ($C_2$) x 3

Order 608: $C_{76}.C_2^3$ ($C_3$)

Order 456: $C_{38}:C_{12}$ ($C_2^2$) x 18, $C_{76}.C_6$ ($C_2^2$) x 16, $C_2\times C_{38}:C_6$ ($C_2^2$)

Order 304: $Q_8\times C_{38}$ ($C_6$) x 12, $C_2^2\times C_{76}$ ($C_6$) x 3

Order 228: $C_{19}:C_{12}$ ($C_2^3$) x 12, $C_{38}:C_6$ ($Q_8$) x 4, $C_{38}:C_6$ ($C_2^3$) x 3

Order 152: $C_2\times C_{76}$ ($C_2\times C_6$) x 18, $Q_8\times C_{19}$ ($C_2\times C_6$) x 16, $C_2^2\times C_{38}$ ($C_2\times C_6$)

Order 114: $C_{19}:C_6$ ($C_2\times Q_8$) x 6, $C_{19}:C_6$ ($C_2^4$)

Order 76: $C_{76}$ ($C_2^2\times C_6$) x 12, $C_2\times C_{38}$ ($C_3\times Q_8$) x 4, $C_2\times C_{38}$ ($C_2^2\times C_6$) x 3

Order 57: $C_{19}:C_3$ ($C_2^2\times Q_8$)

Order 38: $C_{38}$ ($C_6\times Q_8$) x 6, $C_{38}$ ($C_2^3\times C_6$)

Order 32: $C_2^2\times Q_8$ ($C_{19}:C_3$)

Order 19: $C_{19}$ ($C_6.C_2^4$)

Order 16: $C_2\times Q_8$ ($C_{19}:C_6$) x 12, $C_2^2\times C_4$ ($C_{19}:C_6$) x 3

Order 8: $C_2\times C_4$ ($C_{38}:C_6$) x 18, $Q_8$ ($C_{38}:C_6$) x 16, $C_2^3$ ($C_{38}:C_6$)

Order 4: $C_4$ ($C_2\times C_{38}:C_6$) x 12, $C_2^2$ ($C_{76}.C_6$) x 4, $C_2^2$ ($C_2\times C_{38}:C_6$) x 3

Order 2: $C_2$ ($C_{19}:C_6\times Q_8$) x 6, $C_2$ ($C_2^2\times C_{38}:C_6$)

Order 1: $C_1$ ($C_{38}:C_6\times Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1824: $C_{38}:C_6\times Q_8$ ($C_1$)

Order 912: $C_{19}:C_6\times Q_8$ ($C_2$), $C_2\times C_{38}:C_{12}$ ($C_2$)

Order 608: $C_{76}.C_2^3$ ($C_3$)

Order 456: $C_2\times C_{38}:C_6$ ($C_2^2$), $C_{76}.C_6$ ($C_2^2$), $C_{38}:C_{12}$ ($C_2^2$)

Order 304: $Q_8\times C_{38}$ ($C_6$), $C_2^2\times C_{76}$ ($C_6$)

Order 228: $C_{38}:C_6$ ($C_2^3$), $C_{38}:C_6$ ($Q_8$), $C_{19}:C_{12}$ ($C_2^3$)

Order 152: $C_2\times C_{76}$ ($C_2\times C_6$), $C_2^2\times C_{38}$ ($C_2\times C_6$), $Q_8\times C_{19}$ ($C_2\times C_6$)

Order 114: $C_{19}:C_6$ ($C_2^4$), $C_{19}:C_6$ ($C_2\times Q_8$)

Order 76: $C_2\times C_{38}$ ($C_2^2\times C_6$), $C_2\times C_{38}$ ($C_3\times Q_8$), $C_{76}$ ($C_2^2\times C_6$)

Order 57: $C_{19}:C_3$ ($C_2^2\times Q_8$)

Order 38: $C_{38}$ ($C_2^3\times C_6$), $C_{38}$ ($C_6\times Q_8$)

Order 32: $C_2^2\times Q_8$ ($C_{19}:C_3$)

Order 19: $C_{19}$ ($C_6.C_2^4$)

Order 16: $C_2\times Q_8$ ($C_{19}:C_6$), $C_2^2\times C_4$ ($C_{19}:C_6$)

Order 8: $C_2^3$ ($C_{38}:C_6$), $Q_8$ ($C_{38}:C_6$), $C_2\times C_4$ ($C_{38}:C_6$)

Order 4: $C_2^2$ ($C_2\times C_{38}:C_6$), $C_2^2$ ($C_{76}.C_6$), $C_4$ ($C_2\times C_{38}:C_6$)

Order 2: $C_2$ ($C_2^2\times C_{38}:C_6$), $C_2$ ($C_{19}:C_6\times Q_8$)

Order 1: $C_1$ ($C_{38}:C_6\times Q_8$)

Series

Derived series

$C_{38}:C_6\times Q_8$

$\rhd$

$C_{38}$

$\rhd$

$C_1$

Chief series

$C_{38}:C_6\times Q_8$

$\rhd$

$C_2\times C_{38}:C_{12}$

$\rhd$

$C_{38}:C_{12}$

$\rhd$

$C_{19}:C_{12}$

$\rhd$

$C_{76}$

$\rhd$

$C_{38}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Lower central series

$C_{38}:C_6\times Q_8$

$\rhd$

$C_{38}$

$\rhd$

$C_{19}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times Q_8$

Character theory

Complex character table

See the $180 \times 180$ 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 $60 \times 60$ rational character table.