Abstract group 1400.32: $D_{175}:C_4$

• Label: 1400.32
• Order: \( 2^{3} \cdot 5^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 5^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 5^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 3 \cdot 5 \)
• Perm deg.: $32$
• Trans deg.: $175$
• Rank: $2$

Group information

Description:	$D_{175}:C_4$
Order:	\(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \)
Exponent:	\(700\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 7 \)
Automorphism group:	$F_7\times D_{25}.C_{10}$, of order \(21000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_5$ x 2, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	7	10	14	25	28	35	50	175
Elements	1	207	400	4	6	28	150	20	300	24	140	120	1400
Conjugacy classes	1	3	4	1	3	1	3	5	6	3	5	15	50
Divisions	1	3	2	1	1	1	1	1	1	1	1	1	15
Autjugacy classes	1	3	4	1	1	1	1	1	2	1	1	1	18

Dimension	1	2	4	6	8	12	20	24	120
Irr. complex chars.	8	12	12	0	18	0	0	0	0	50
Irr. rational chars.	4	2	2	2	0	1	2	1	1	15

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$175$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	120
Arbitrary	6	6	26

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{175}=[a,b]=1, c^{a}=c^{76}, c^{b}=c^{43} \rangle$
Permutation group:	Degree $32$ $\langle(27,28)(29,30)(31,32), (2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25) \!\cdots\! \rangle$
Direct product:	$D_7$ $\, \times\, $ $(C_{25}:C_4)$
Semidirect product:	$D_{175}$ $\,\rtimes\,$ $C_4$	$(D_7\times C_{25})$ $\,\rtimes\,$ $C_4$	$C_{25}$ $\,\rtimes\,$ $(C_4\times D_7)$	$C_7$ $\,\rtimes\,$ $(C_{50}:C_4)$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{25}$ . $D_{14}$	$(C_5\times D_7)$ . $F_5$	$C_5$ . $(D_7\times F_5)$	$(D_7\times D_{25})$ . $C_2$	all 6

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

Homology

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

Subgroups

There are 1146 subgroups in 48 conjugacy classes, 17 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{175}:C_4$
Commutator:	$G' \simeq$ $C_{175}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $D_7\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{175}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_4$
Radical:	$R \simeq$ $D_{175}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{35}$	$G/\operatorname{soc} \simeq$ $C_2\times F_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
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 1400: $D_{175}:C_4$

Order 700: $D_7\times D_{25}$, $C_{175}:C_4$, $C_{25}:C_{28}$

Order 350: $D_{175}$, $D_7\times C_{25}$, $C_7\times D_{25}$

Order 280: $D_7\times F_5$

Order 200: $C_{50}:C_4$

Order 175: $C_{175}$

Order 140: $D_5\times D_7$, $C_{35}:C_4$, $C_7\times F_5$

Order 100: $C_{25}:C_4$ x 2, $D_{50}$

Order 70: $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 56: $C_4\times D_7$

Order 50: $D_{25}$ x 2, $C_{50}$

Order 40: $C_2\times F_5$

Order 35: $C_{35}$

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

Order 25: $C_{25}$

Order 20: $F_5$ x 2, $D_{10}$

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

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2\times C_4$

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1400: $D_{175}:C_4$

Order 700: $D_7\times D_{25}$, $C_{175}:C_4$, $C_{25}:C_{28}$

Order 350: $D_{175}$, $D_7\times C_{25}$, $C_7\times D_{25}$

Order 280: $D_7\times F_5$

Order 200: $C_{50}:C_4$

Order 175: $C_{175}$

Order 140: $D_5\times D_7$, $C_{35}:C_4$, $C_7\times F_5$

Order 100: $C_{25}:C_4$ x 2, $D_{50}$

Order 70: $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 56: $C_4\times D_7$

Order 50: $D_{25}$ x 2, $C_{50}$

Order 40: $C_2\times F_5$

Order 35: $C_{35}$

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

Order 25: $C_{25}$

Order 20: $F_5$ x 2, $D_{10}$

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

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2\times C_4$

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1400: $D_{175}:C_4$ ($C_1$)

Order 700: $D_7\times D_{25}$ ($C_2$), $C_{175}:C_4$ ($C_2$), $C_{25}:C_{28}$ ($C_2$)

Order 350: $D_{175}$ ($C_4$), $D_7\times C_{25}$ ($C_4$), $C_7\times D_{25}$ ($C_2^2$)

Order 175: $C_{175}$ ($C_2\times C_4$)

Order 100: $C_{25}:C_4$ ($D_7$)

Order 70: $C_5\times D_7$ ($F_5$)

Order 50: $D_{25}$ ($D_{14}$)

Order 35: $C_{35}$ ($C_2\times F_5$)

Order 25: $C_{25}$ ($C_4\times D_7$)

Order 14: $D_7$ ($C_{25}:C_4$)

Order 7: $C_7$ ($C_{50}:C_4$)

Order 5: $C_5$ ($D_7\times F_5$)

Order 1: $C_1$ ($D_{175}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1400: $D_{175}:C_4$ ($C_1$)

Order 700: $D_7\times D_{25}$ ($C_2$), $C_{175}:C_4$ ($C_2$), $C_{25}:C_{28}$ ($C_2$)

Order 350: $D_{175}$ ($C_4$), $D_7\times C_{25}$ ($C_4$), $C_7\times D_{25}$ ($C_2^2$)

Order 175: $C_{175}$ ($C_2\times C_4$)

Order 100: $C_{25}:C_4$ ($D_7$)

Order 70: $C_5\times D_7$ ($F_5$)

Order 50: $D_{25}$ ($D_{14}$)

Order 35: $C_{35}$ ($C_2\times F_5$)

Order 25: $C_{25}$ ($C_4\times D_7$)

Order 14: $D_7$ ($C_{25}:C_4$)

Order 7: $C_7$ ($C_{50}:C_4$)

Order 5: $C_5$ ($D_7\times F_5$)

Order 1: $C_1$ ($D_{175}:C_4$)

Series

Derived series

$D_{175}:C_4$

$\rhd$

$C_{175}$

$\rhd$

$C_1$

Chief series

$D_{175}:C_4$

$\rhd$

$D_7\times D_{25}$

$\rhd$

$C_7\times D_{25}$

$\rhd$

$C_{175}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{175}:C_4$

$\rhd$

$C_{175}$

Upper central series

$C_1$

Character theory

Complex character table

See the $50 \times 50$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	5A	7A	10A	14A	25A	28A	35A	50A	175A
	Size	1	7	25	175	50	350	4	6	28	150	20	300	24	140	120
	2 P	1A	1A	1A	1A	2B	2B	5A	7A	5A	7A	25A	14A	35A	25A	175A
	5 P	1A	2A	2B	2C	4A	4B	5A	7A	10A	14A	25A	28A	35A	50A	175A
	7 P	1A	2A	2B	2C	4A	4B	1A	7A	2A	14A	5A	28A	7A	10A	35A
1400.32.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1400.32.1b		$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$
1400.32.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$
1400.32.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$
1400.32.1e		$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$2$	$-2$	$-2$	$2$	$0$	$2$	$-2$	$2$
1400.32.1f		$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$2$	$2$	$-2$	$2$	$0$	$2$	$2$	$2$
1400.32.2a		$6$	$0$	$6$	$0$	$6$	$0$	$6$	$-1$	$0$	$-1$	$6$	$-1$	$-1$	$0$	$-1$
1400.32.2b		$6$	$0$	$6$	$0$	$-6$	$0$	$6$	$-1$	$0$	$-1$	$6$	$1$	$-1$	$0$	$-1$
1400.32.2c		$12$	$0$	$-12$	$0$	$0$	$0$	$12$	$-2$	$0$	$2$	$12$	$0$	$-2$	$0$	$-2$
1400.32.4a		$4$	$4$	$0$	$0$	$0$	$0$	$4$	$4$	$4$	$0$	$-1$	$0$	$4$	$-1$	$-1$
1400.32.4b		$4$	$-4$	$0$	$0$	$0$	$0$	$4$	$4$	$-4$	$0$	$-1$	$0$	$4$	$1$	$-1$
1400.32.4c		$20$	$20$	$0$	$0$	$0$	$0$	$-5$	$20$	$-5$	$0$	$0$	$0$	$-5$	$0$	$0$
1400.32.4d		$20$	$-20$	$0$	$0$	$0$	$0$	$-5$	$20$	$5$	$0$	$0$	$0$	$-5$	$0$	$0$
1400.32.8a		$24$	$0$	$0$	$0$	$0$	$0$	$24$	$-4$	$0$	$0$	$-6$	$0$	$-4$	$0$	$1$
1400.32.8b		$120$	$0$	$0$	$0$	$0$	$0$	$-30$	$-20$	$0$	$0$	$0$	$0$	$5$	$0$	$0$