Abstract group 224.104: $C_4.D_{28}$

• Label: 224.104
• Order: \( 2^{5} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $23$
• Trans deg.: $112$
• Rank: $3$

Group information

Description:	$C_4.D_{28}$
Order:	\(224\)\(\medspace = 2^{5} \cdot 7 \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$F_7\times D_4^2$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	8	14	28	56
Elements	1	31	88	6	8	18	24	48	224
Conjugacy classes	1	3	5	3	2	6	9	12	41
Divisions	1	3	5	1	2	2	2	2	18
Autjugacy classes	1	3	4	1	1	2	2	1	15

Dimension	1	2	4	6	12	24
Irr. complex chars.	8	26	7	0	0	0	41
Irr. rational chars.	8	2	1	4	2	1	18

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$112$
Rank:	$3$
Inequivalent generating triples:	$1344$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	8	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{4}=[a,c]=1, b^{28}=c^{2}, b^{a}=b^{27}, c^{b}=c^{3} \rangle$
Permutation group:	Degree $23$ $\langle(1,2)(3,7)(4,11)(5,14)(6,9)(8,12)(10,16)(13,15)(18,19)(20,21)(22,23), (1,3,6,5) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\OD_{16}$ $\,\rtimes\,$ $D_7$	$(C_{14}:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_7:Q_{16})$ $\,\rtimes\,$ $C_2$ (2)	$C_7$ $\,\rtimes\,$ $(Q_{16}:C_2)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{28}$ . $D_4$	$C_8$ . $D_{14}$ (2)	$C_4$ . $D_{28}$	$D_{28}$ . $C_2^2$	all 15

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 270 subgroups in 60 conjugacy classes, 29 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_{28}$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_7\times \OD_{16}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_4.D_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 224: $C_4.D_{28}$

Order 112: $C_7:Q_{16}$ x 2, $C_{56}:C_2$ x 2, $D_{28}:C_2$, $C_{14}:Q_8$, $C_7\times \OD_{16}$

Order 56: $C_7:Q_8$ x 4, $C_{56}$ x 2, $C_2\times C_{28}$, $C_7:D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$

Order 32: $Q_{16}:C_2$

Order 28: $C_7:C_4$ x 3, $C_{28}$ x 2, $C_2\times C_{14}$, $D_{14}$

Order 16: $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 14: $C_{14}$ x 2, $D_7$

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$ x 2

Order 7: $C_7$

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 224: $C_4.D_{28}$

Order 112: $C_7:Q_{16}$, $C_{56}:C_2$, $D_{28}:C_2$, $C_{14}:Q_8$, $C_7\times \OD_{16}$

Order 56: $C_7:Q_8$ x 3, $C_2\times C_{28}$, $C_7:D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_{56}$

Order 32: $Q_{16}:C_2$

Order 28: $C_{28}$ x 2, $C_7:C_4$ x 2, $C_2\times C_{14}$, $D_{14}$

Order 16: $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 14: $C_{14}$ x 2, $D_7$

Order 8: $Q_8$ x 3, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 224: $C_4.D_{28}$ ($C_1$)

Order 112: $C_7:Q_{16}$ ($C_2$) x 2, $C_{56}:C_2$ ($C_2$) x 2, $D_{28}:C_2$ ($C_2$), $C_{14}:Q_8$ ($C_2$), $C_7\times \OD_{16}$ ($C_2$)

Order 56: $C_7:Q_8$ ($C_2^2$) x 3, $C_{56}$ ($C_2^2$) x 2, $C_2\times C_{28}$ ($C_2^2$), $D_{28}$ ($C_2^2$)

Order 28: $C_2\times C_{14}$ ($D_4$), $C_{28}$ ($C_2^3$), $C_{28}$ ($D_4$)

Order 16: $\OD_{16}$ ($D_7$)

Order 14: $C_{14}$ ($C_2\times D_4$)

Order 8: $C_8$ ($D_{14}$) x 2, $C_2\times C_4$ ($D_{14}$)

Order 7: $C_7$ ($Q_{16}:C_2$)

Order 4: $C_2^2$ ($D_{28}$), $C_4$ ($D_{28}$), $C_4$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($C_2\times D_{28}$)

Order 1: $C_1$ ($C_4.D_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 224: $C_4.D_{28}$ ($C_1$)

Order 112: $C_7:Q_{16}$ ($C_2$), $C_{56}:C_2$ ($C_2$), $D_{28}:C_2$ ($C_2$), $C_{14}:Q_8$ ($C_2$), $C_7\times \OD_{16}$ ($C_2$)

Order 56: $C_7:Q_8$ ($C_2^2$) x 2, $C_2\times C_{28}$ ($C_2^2$), $D_{28}$ ($C_2^2$), $C_{56}$ ($C_2^2$)

Order 28: $C_2\times C_{14}$ ($D_4$), $C_{28}$ ($C_2^3$), $C_{28}$ ($D_4$)

Order 16: $\OD_{16}$ ($D_7$)

Order 14: $C_{14}$ ($C_2\times D_4$)

Order 8: $C_2\times C_4$ ($D_{14}$), $C_8$ ($D_{14}$)

Order 7: $C_7$ ($Q_{16}:C_2$)

Order 4: $C_2^2$ ($D_{28}$), $C_4$ ($D_{28}$), $C_4$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($C_2\times D_{28}$)

Order 1: $C_1$ ($C_4.D_{28}$)

Series

Derived series

$C_4.D_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_4.D_{28}$

$\rhd$

$D_{28}:C_2$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4.D_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$\OD_{16}$

Supergroups

This group is a maximal subgroup of 36 larger groups in the database.

This group is a maximal quotient of 43 larger groups in the database.

Character theory

Complex character table

See the $41 \times 41$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.