# gps_char downloaded from the LMFDB on 23 July 2026.
# Search link: https://www.lmfdb.org/Groups/Abstract/ComplexCharacters?group=1584.401
# Query "{'group': '1584.401'}" returned 132 gps_chars, sorted by group.

# Each entry in the following data list has the form:
#    [Label, Degree, Type, Faithful, Field of Traces, $\Q$-character, Group, Image Order, Kernel Order]
# For more details, see the definitions at the bottom of the file.



"1584.401.1a"	1	1	false	[0, 1]	NULL	"1584.401"	1	1584
"1584.401.1b"	1	1	false	[0, 1]	NULL	"1584.401"	2	792
"1584.401.1c"	1	1	false	[0, 1]	NULL	"1584.401"	2	792
"1584.401.1d"	1	1	false	[0, 1]	NULL	"1584.401"	2	792
"1584.401.1e1"	1	0	false	[1, 0, 1]	NULL	"1584.401"	4	396
"1584.401.1e2"	1	0	false	[1, 0, 1]	NULL	"1584.401"	4	396
"1584.401.1f1"	1	0	false	[1, 0, 1]	NULL	"1584.401"	4	396
"1584.401.1f2"	1	0	false	[1, 0, 1]	NULL	"1584.401"	4	396
"1584.401.1g1"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g10"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g2"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g3"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g4"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g5"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g6"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g7"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g8"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1g9"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	11	144
"1584.401.1h1"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h10"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h2"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h3"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h4"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h5"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h6"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h7"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h8"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1h9"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i1"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i10"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i2"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i3"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i4"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i5"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i6"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i7"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i8"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1i9"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j1"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j10"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j2"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j3"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j4"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j5"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j6"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j7"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j8"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1j9"	1	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	22	72
"1584.401.1k1"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k10"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k11"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k12"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k13"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k14"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k15"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k16"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k17"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k18"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k19"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k2"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k20"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k3"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k4"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k5"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k6"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k7"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k8"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1k9"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l1"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l10"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l11"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l12"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l13"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l14"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l15"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l16"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l17"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l18"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l19"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l2"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l20"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l3"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l4"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l5"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l6"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l7"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l8"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.1l9"	1	0	false	[1, 0, 1]	NULL	"1584.401"	44	36
"1584.401.2a"	2	1	false	[0, 1]	NULL	"1584.401"	8	198
"1584.401.2b"	2	-1	false	[0, 1]	NULL	"1584.401"	8	198
"1584.401.2c1"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c10"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c2"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c3"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c4"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c5"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c6"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c7"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c8"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2c9"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d1"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d10"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d2"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d3"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d4"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d5"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d6"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d7"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d8"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.2d9"	2	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	88	18
"1584.401.8a"	8	1	false	[0, 1]	NULL	"1584.401"	72	22
"1584.401.8b"	8	1	false	[0, 1]	NULL	"1584.401"	144	11
"1584.401.8c1"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c10"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c2"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c3"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c4"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c5"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c6"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c7"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c8"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8c9"	8	0	false	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	792	2
"1584.401.8d1"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d10"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d2"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d3"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d4"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d5"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d6"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d7"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d8"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1
"1584.401.8d9"	8	0	true	[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]	NULL	"1584.401"	1584	1


# Label --
#    The **label** of an irreducible complex character of a group has the format $N.i.dcj$ where $N.i$ is the label for the group, $d$ is the dimension, $c$ is a lower-case letter code for the rational class of this character, and $j$ is an enumeration of characters within that rational class ($j$ is omitted if the rational class is a singleton).


#Degree (dim) --
#    The **degree** of a character $\chi$ of a group $G$ is the value given by $\chi(1)$ where $1$ is the trivial element of $G$.


#Type (indicator) --
#    Let $\rho:G\to\GL_n(\C)$ be an irreducible complex group representation.  Then $\rho$ is one of three **types**:

#    1. **Real** if $\rho$ is conjugate to a representation $G\to \GL_n(\R)$
#    1. **Complex** if some character value $\textrm{Tr}(\rho(g))$ is not contained in $\R$
#    1. **Quaternionic (or Symplectic)** if the character values are all real but the representation is not conjugate to a real representation.

#    The type of the representation can be computed via its Frobenius-Schur indicator.


# Faithful --
#    A representation $\rho$ is **faithful** if it is injective as a homomorphism to $\GL(V)$.  In terms of the character $\chi$ of $\rho$, for representations of finite groups in characteristic $0$, this will hold precisely when $\chi(g) = \chi(1)$ implies $g = 1$.


#Field of Traces (field) --
#    Given a complex character $\chi$ of a group $G$, define $\Q(\chi)$ to be the smallest field containing $\Q$ and the character values of $\chi$.  This field is abelian and hence has a **conductor**, a positive integer giving the minimum $n$ so that $\Q(\chi) \subseteq  \Q(\zeta_n)$.


#$\Q$-character (qchar) --
#    A **rational character** of a finite group $G$ is complex character taking values in $\Q$. Such a character can be obtained by taking the sum of Galois conjugates of an irreducible complex character, and we will refer to these as **irreducible rational characters** even though they are not irreducible as complex characters.  The irreducible rational characters form a $\Q$-basis for the $\Q$-vector space of rational characters.

#    Note that the representation corresponding to a rational character may not take values in $\GL_n(\Q)$ even though the character takes values in $\Q$ (the difference between the field of character values and the field in which matrix entries lie is measured by the Schur index).

#    The value taken by a rational character is constant on each rational conjugacy classes.


# Group --
#    We describe abstract groups using standard building blocks:
#    <ul>
#     <li> $C_n$ denotes the cyclic group of order $n$.
#     <li> $S_n$ denotes the symmetric group on $n$ letters.
#     <li> $A_n$ denotes the alternating group on $n$ letters.
#     <li> $D_n$ denotes the dihedral group of order $2n$.
#     <li> $Q_n$ denotes the (generalized) quaternion group of order $n$.
#     <li> $F_q$ denotes the Frobenius group for a prime power $q$. It is the group of affine linear transformations of the finite field $\mathbb{F}_q$. In other words, $F_q$ is a semidirect product  $\mathbb{F}_q : \mathbb{F}_q^{\times}$.
#     <li> Sporadic Groups (refer to this <a href="https://en.wikipedia.org/wiki/Sporadic_group">wikipage</a>):
#    <ul>
#     <li> $M_n$ for $n=11,12, 22,23, 24$ denotes the Mathieu groups
#     <li> $\Ru$ denotes the Rudvalis group
#     <li> $\McL$ denotes the McLaughlin group
#     <li> $\He$ denotes the Held group
#     <li> $\HS$ denotes the Higman–Sims group
#     <li> $J_1$, $J_2$, $J_3$ denote the Janko groups
#     <li> $\Co_2$ and $\Co_3$ denote the Conway groups
#    </ul>
#     <li> Here is a list of finite groups of Lie type.
#     <li> $\SD_n$ denotes the semidihedral group or quasidihedral group of order $n=2^k$.
#     <li> $\OD_n$ denotes the other-dihedral group (or modular maximal-cyclic group) of order $n=2^k$. It is the non-trivial semidirect product $C_{2^{k-1}} : C_2$ which is not isomorphic to either $\SD_n$ or $D_{2^{k-1}}$.
#     <li> $\He_p$ denotes the Heisenberg group, the unique non-abelian group of order $p^3$ and exponent $p$ for an odd prime $p$.
#    </ul>

#    Groups $A$ and $B$ may be used to construct a larger group:

#    - $A\times B$ for the direct product of $A$ and $B$.
#    - $A:B$ for the semidirect product of $A$ and $B$ (with normal subgroup $A$).
#    - $A.B$ an extension with normal subgroup $A$ and quotient isomorphic to $B$.
#    - $A\wr B$ for the wreath product of $A$ and $B$.


#Image Order (image_order) --
#    The **image** of a representation of a finite group $G$ is the image of the defining homomorphism $\rho : G \to \GL(V)$ for some vector  space $V$.


#Kernel Order (kernel_order) --
#    The **kernel** of the character $\chi$ of a group $G$ is the subgroup of $G$ given by the set: $$\ker(\phi)=\{g \in G : \chi(g)=\chi(1)\}.$$