Abstract group 224.50: $C_{28}.D_4$

• Label: 224.50
• Order: \( 2^{5} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $23$
• Trans deg.: $112$
• Rank: $2$

Group information

Description:	$C_{28}.D_4$
Order:	\(224\)\(\medspace = 2^{5} \cdot 7 \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$C_2\wr D_4\times C_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_7$
Nilpotency class:	$3$
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	7	8	14	28	56
Elements	1	3	12	6	16	18	72	96	224
Conjugacy classes	1	2	4	6	4	12	24	24	77
Divisions	1	2	4	1	2	2	4	2	18
Autjugacy classes	1	2	2	1	1	2	2	1	12

Dimension	1	2	4	6	12	24
Irr. complex chars.	56	14	7	0	0	0	77
Irr. rational chars.	4	4	1	4	4	1	18

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$112$
Rank:	$2$
Inequivalent generating pairs:	$24$

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

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

Homology

Abelianization:	$C_{2} \times C_{28} \simeq C_{2} \times C_{4} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 52 subgroups in 38 conjugacy classes, 24 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{14}$	$G/Z \simeq$ $C_2^2:C_4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times C_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{28}.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{28}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4$
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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Classes of subgroups up to conjugation

Order 224: $C_{28}.D_4$

Order 112: $C_7\times \OD_{16}$ x 2, $Q_8\times C_{14}$

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

Order 32: $C_4.D_4$

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$

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

Order 14: $C_{14}$ x 2

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

Order 7: $C_7$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 224: $C_{28}.D_4$

Order 112: $Q_8\times C_{14}$, $C_7\times \OD_{16}$

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

Order 32: $C_4.D_4$

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

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

Order 14: $C_{14}$ x 2

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

Order 7: $C_7$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 224: $C_{28}.D_4$ ($C_1$)

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

Order 56: $C_2\times C_{28}$ ($C_4$) x 2, $C_2\times C_{28}$ ($C_2^2$)

Order 32: $C_4.D_4$ ($C_7$)

Order 28: $C_{28}$ ($D_4$) x 2, $C_2\times C_{14}$ ($C_2\times C_4$)

Order 16: $\OD_{16}$ ($C_{14}$) x 2, $C_2\times Q_8$ ($C_{14}$)

Order 14: $C_{14}$ ($C_2^2:C_4$)

Order 8: $C_2\times C_4$ ($C_{28}$) x 2, $C_2\times C_4$ ($C_2\times C_{14}$)

Order 7: $C_7$ ($C_4.D_4$)

Order 4: $C_4$ ($C_7\times D_4$) x 2, $C_2^2$ ($C_2\times C_{28}$)

Order 2: $C_2$ ($C_2^2:C_{28}$)

Order 1: $C_1$ ($C_{28}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 224: $C_{28}.D_4$ ($C_1$)

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

Order 56: $C_2\times C_{28}$ ($C_2^2$), $C_2\times C_{28}$ ($C_4$)

Order 32: $C_4.D_4$ ($C_7$)

Order 28: $C_2\times C_{14}$ ($C_2\times C_4$), $C_{28}$ ($D_4$)

Order 16: $\OD_{16}$ ($C_{14}$), $C_2\times Q_8$ ($C_{14}$)

Order 14: $C_{14}$ ($C_2^2:C_4$)

Order 8: $C_2\times C_4$ ($C_2\times C_{14}$), $C_2\times C_4$ ($C_{28}$)

Order 7: $C_7$ ($C_4.D_4$)

Order 4: $C_2^2$ ($C_2\times C_{28}$), $C_4$ ($C_7\times D_4$)

Order 2: $C_2$ ($C_2^2:C_{28}$)

Order 1: $C_1$ ($C_{28}.D_4$)

Series

Derived series

$C_{28}.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{28}.D_4$

$\rhd$

$C_7\times \OD_{16}$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{28}.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{14}$

$\lhd$

$C_2\times C_{28}$

$\lhd$

$C_{28}.D_4$

Supergroups

This group is a maximal subgroup of 28 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 $77 \times 77$ 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.