data:
- - character_diagonal:
    - 1
    - 2
    - 4
    - 7
    - 12
    - 12
    - 14
    - 39
    - 40
    - 29
    character_matrix:
    - - 1
      - 0
      - 0
      - 1
      - 0
      - 2
      - 0
      - 1
      - 0
      - 0
    - - 0
      - 2
      - 0
      - 0
      - 4
      - 0
      - 4
      - 0
      - 6
      - 0
    - - 0
      - 0
      - 4
      - 2
      - 0
      - 1
      - 0
      - 9
      - 0
      - 8
    - - 1
      - 0
      - 2
      - 7
      - 0
      - 7
      - 0
      - 12
      - 0
      - 6
    - - 0
      - 4
      - 0
      - 0
      - 12
      - 0
      - 12
      - 0
      - 20
      - 0
    - - 2
      - 0
      - 1
      - 7
      - 0
      - 12
      - 0
      - 12
      - 0
      - 5
    - - 0
      - 4
      - 0
      - 0
      - 12
      - 0
      - 14
      - 0
      - 22
      - 0
    - - 1
      - 0
      - 9
      - 12
      - 0
      - 12
      - 0
      - 39
      - 0
      - 29
    - - 0
      - 6
      - 0
      - 0
      - 20
      - 0
      - 22
      - 0
      - 40
      - 0
    - - 0
      - 0
      - 8
      - 6
      - 0
      - 5
      - 0
      - 29
      - 0
      - 29
    component_group: '12.4'
    component_group_number: 4
    components: 12
    counts:
    - - a_1
      - - - 0
          - 7
    degree: 4
    first_a2_moment: 1
    fourth_trace_moment: 18
    gens:
    - - - \zeta_{12}
        - '0'
        - '0'
        - '0'
      - - '0'
        - \zeta_{12}^{11}
        - '0'
        - '0'
      - - '0'
        - '0'
        - \zeta_{12}^{11}
        - '0'
      - - '0'
        - '0'
        - '0'
        - \zeta_{12}
    - - - '0'
        - '1'
        - '0'
        - '0'
      - - '-1'
        - '0'
        - '0'
        - '0'
      - - '0'
        - '0'
        - '0'
        - '1'
      - - '0'
        - '0'
        - '-1'
        - '0'
    id: 15
    identity_component: U(1)_2
    label: 1.4.F.12.4c
    label_components:
    - 1
    - 4
    - 5
    - 12
    - 4
    - 2
    maximal: false
    moments:
    - - a_1
      - 1
      - 0
      - 2
      - 0
      - 18
      - 0
      - 200
      - 0
      - 2450
      - 0
      - 31752
      - 0
      - 427812
    - - a_2
      - 1
      - 1
      - 5
      - 16
      - 77
      - 356
      - 1803
      - 9262
      - 48933
      - 262372
      - 1427255
      - 7850822
      - 43608271
    name: D_6
    old_label: 1.4.1.12.4a
    pretty: D_6
    rational: true
    real_dimension: 1
    second_trace_moment: 2
    simplex:
    - 1
    - 2
    - 5
    - 8
    - 18
    - 16
    - 36
    - 84
    - 200
    - 77
    - 172
    - 412
    - 1000
    - 2450
    - 356
    - 856
    - 2088
    - 5140
    - 12740
    - 31752
    - 1803
    - 4386
    - 10842
    - 26980
    - 67494
    - 169596
    - 427812
    st0_label: 1.4.F
    subgroup_multiplicities:
    - 1
    - 2
    - 1
    subgroups:
    - 1.4.F.4.2a
    - 1.4.F.6.1a
    - 1.4.F.6.2c
    supgroups:
    - 1.4.F.24.14a
    trace_histogram: data:image/png;base64,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
    trace_zero_density: 7/12
    weight: 1
    zvector:
    - 7
    - 0
    - 0
    - 0
    - 0
    - 0
    - 0
    - 0
    - 0
    - 0
    - 0
- - degree: 4
    description: '\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}:
      \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}'
    hodge_circle: u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)
    id: 8
    label: 1.4.F
    label_components:
    - 1
    - 4
    - 5
    name: U(1)_2
    pretty: \mathrm{U}(1)_2
    real_dimension: 1
    symplectic_form: \begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}
    weight: 1
- - Agroup: true
    Zgroup: false
    abelian: false
    abelian_quotient: '4.2'
    all_subgroups_known: true
    almost_simple: false
    aut_gens:
    - - 1
      - 2
    - - 7
      - 10
    - - 1
      - 10
    - - 5
      - 2
    aut_group: '12.4'
    aut_order: 12
    aut_stats:
    - - 1
      - 1
      - 1
      - 1
    - - 2
      - 1
      - 1
      - 1
    - - 2
      - 3
      - 2
      - 1
    - - 3
      - 2
      - 1
      - 1
    - - 6
      - 2
      - 1
      - 1
    cc_stats:
    - - 1
      - 1
      - 1
    - - 2
      - 1
      - 1
    - - 2
      - 3
      - 2
    - - 3
      - 2
      - 1
    - - 6
      - 2
      - 1
    center_label: '2.1'
    central_product: true
    central_quotient: '6.1'
    commutator_count: 1
    commutator_label: '3.1'
    complements_known: true
    complete: false
    complex_characters_known: true
    composition_factors:
    - '2.1'
    - '2.1'
    - '3.1'
    composition_length: 3
    counter: 4
    cyclic: false
    derived_length: 2
    direct_factorization:
    - - '2.1'
      - 1
    - - '6.1'
      - 1
    direct_product: true
    div_stats:
    - - 1
      - 1
      - 1
      - 1
    - - 2
      - 1
      - 1
      - 1
    - - 2
      - 3
      - 1
      - 2
    - - 3
      - 2
      - 1
      - 1
    - - 6
      - 2
      - 1
      - 1
    element_repr_type: PC
    elementary: 1
    eulerian_function: 3
    exponent: 6
    exponents_of_order:
    - 2
    - 1
    factors_of_aut_order:
    - 2
    - 3
    factors_of_order:
    - 2
    - 3
    faithful_reps:
    - - 2
      - 1
      - 1
    frattini_label: '1.1'
    frattini_quotient: '12.4'
    hash: 4
    hyperelementary: 2
    id: 22
    irrC_degree: 2
    irrQ_degree: 2
    irrep_stats:
    - - 1
      - 4
    - - 2
      - 2
    label: '12.4'
    linC_degree: null
    linFp_degree: null
    linFq_degree: null
    linQ_degree: 2
    maximal_subgroups_known: true
    metabelian: true
    metacyclic: true
    monomial: true
    name: D6
    ngens: 3
    nilpotency_class: -1
    nilpotent: false
    normal_counts: null
    normal_index_bound: null
    normal_order_bound: null
    normal_subgroups_known: true
    number_autjugacy_classes: 5
    number_characteristic_subgroups: 5
    number_conjugacy_classes: 6
    number_divisions: 6
    number_normal_subgroups: 7
    number_subgroup_autclasses: 8
    number_subgroup_classes: 10
    number_subgroups: 16
    old_label: null
    order: 12
    order_factorization_type: 22
    order_stats:
    - - 1
      - 1
    - - 2
      - 7
    - - 3
      - 2
    - - 6
      - 2
    outer_equivalence: false
    outer_group: '2.1'
    outer_order: 2
    pc_rank: 2
    perfect: false
    permutation_degree: 5
    pgroup: 0
    primary_abelian_invariants:
    - 2
    - 2
    quasisimple: false
    rank: 2
    rational: true
    rational_characters_known: true
    ratrep_stats:
    - - 1
      - 4
    - - 2
      - 2
    representations:
      GLFp:
        d: 2
        gens:
        - 31
        - 55
        - 56
        p: 3
      GLZ:
        b: 3
        d: 2
        gens:
        - 17
        - 35
      PC:
        code: 43377
        gens:
        - 1
        - 2
        pres:
        - 3
        - -2
        - -2
        - -3
        - 61
        - 16
        - 74
      Perm:
        d: 5
        gens:
        - 6
        - 1
        - 30
    schur_multiplier:
    - 2
    semidirect_product: true
    simple: false
    smith_abelian_invariants:
    - 2
    - 2
    solvability_type: 6
    solvable: true
    subgroup_inclusions_known: true
    subgroup_index_bound: 0
    supersolvable: true
    sylow_subgroups_known: true
    tex_name: D_6
    transitive_degree: 6
    wreath_data: null
    wreath_product: false
label_cols:
- label
- name
- label
labels:
- 1.4.F.12.4c
- U(1)_2
- '12.4'
tables:
- gps_st
- gps_st0
- gps_groups
timestamp: '2024-05-16T09:09:42.187908'