Abstract group 608.190: $C_{76}.C_2^3$

• Label: 608.190
• Order: \( 2^{5} \cdot 19 \)
• Exponent: \( 2^{2} \cdot 19 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 19 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 19 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{4} \)
• Perm deg.: $31$
• Trans deg.: $608$
• Rank: $4$

Group information

Description:	$C_{76}.C_2^3$
Order:	\(608\)\(\medspace = 2^{5} \cdot 19 \)
Exponent:	\(76\)\(\medspace = 2^{2} \cdot 19 \)
Automorphism group:	$C_{18}\times C_2^6.(S_3\times S_4)$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \)
Composition factors:	$C_2$ x 5, $C_{19}$
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4	19	38	76
Elements	1	7	24	18	126	432	608
Conjugacy classes	1	7	12	18	126	216	380
Divisions	1	7	12	1	7	12	40
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	18	36
Irr. complex chars.	304	76	0	0	380
Irr. rational chars.	16	4	16	4	40

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$608$
Rank:	$4$
Inequivalent generating quadruples:	$253400$

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:	$\langle a, b, c, d \mid a^{2}=c^{2}=d^{76}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{2}=d^{38}, d^{b}=d^{39} \rangle$
Permutation group:	Degree $31$ $\langle(1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7)(9,10)(11,12), (1,4)(2,6)(3,8)(5,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 77 & 76 \\ 76 & 1 \end{array}\right), \left(\begin{array}{rr} 77 & 38 \\ 38 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 77 & 0 \\ 0 & 77 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 57 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/114\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_{19}$ $\, \times\, $ $Q_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{76}$ . $C_2^3$	$C_{38}$ . $C_2^4$	$(C_2^2\times C_{76})$ . $C_2$	$(C_2\times C_{76})$ . $C_2^2$	all 12

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

Homology

Abelianization:	$C_{2}^{3} \times C_{38} \simeq C_{2}^{4} \times C_{19}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 156 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

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

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

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

Order 38: $C_{38}$ x 7

Order 32: $C_2^2\times Q_8$

Order 19: $C_{19}$

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

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

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

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

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

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

Order 38: $C_{38}$ x 2

Order 32: $C_2^2\times Q_8$

Order 19: $C_{19}$

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

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 608: $C_{76}.C_2^3$ ($C_1$)

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

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

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

Order 38: $C_{38}$ ($C_2\times Q_8$) x 6, $C_{38}$ ($C_2^4$)

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

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

Order 16: $C_2\times Q_8$ ($C_{38}$) x 12, $C_2^2\times C_4$ ($C_{38}$) x 3

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

Order 4: $C_4$ ($C_2^2\times C_{38}$) x 12, $C_2^2$ ($Q_8\times C_{19}$) x 4, $C_2^2$ ($C_2^2\times C_{38}$) x 3

Order 2: $C_2$ ($Q_8\times C_{38}$) x 6, $C_2$ ($C_2^3\times C_{38}$)

Order 1: $C_1$ ($C_{76}.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 608: $C_{76}.C_2^3$ ($C_1$)

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

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

Order 76: $C_2\times C_{38}$ ($C_2^3$), $C_2\times C_{38}$ ($Q_8$), $C_{76}$ ($C_2^3$)

Order 38: $C_{38}$ ($C_2^4$), $C_{38}$ ($C_2\times Q_8$)

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

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

Order 16: $C_2\times Q_8$ ($C_{38}$), $C_2^2\times C_4$ ($C_{38}$)

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

Order 4: $C_2^2$ ($C_2^2\times C_{38}$), $C_2^2$ ($Q_8\times C_{19}$), $C_4$ ($C_2^2\times C_{38}$)

Order 2: $C_2$ ($C_2^3\times C_{38}$), $C_2$ ($Q_8\times C_{38}$)

Order 1: $C_1$ ($C_{76}.C_2^3$)

Series

Derived series

$C_{76}.C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{76}.C_2^3$

$\rhd$

$C_2^2\times C_{76}$

$\rhd$

$C_2\times C_{76}$

$\rhd$

$C_2\times C_{38}$

$\rhd$

$C_{38}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Lower central series

$C_{76}.C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{38}$

$\lhd$

$C_{76}.C_2^3$

Supergroups

This group is a maximal subgroup of 54 larger groups in the database.

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

Character theory

Complex character table

See the $380 \times 380$ 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 $40 \times 40$ rational character table.