D112:C2 - GroupNames

G = D₁₁₂⋊C₂ order 448 = 2⁶·7

6^th semidirect product of D₁₁₂ and C₂ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₆⋊₃D₁₄, D₁₁₂⋊₆C₂, Q₁₆⋊₂D₁₄, SD₃₂⋊₁D₇, D₁₄.₇D₈, D₈.₃D₁₄, C₁₁₂⋊₃C₂², D₅₆⋊₆C₂², Dic₇.₉D₈, C₅₆.₁₇C₂³, C₇⋊C₈.₃D₄, (D₇×D₈)⋊₅C₂, C₄.₅(D₄×D₇), C₇⋊D₁₆⋊₃C₂, C₇⋊C₁₆⋊₂C₂², (C₄×D₇).₈D₄, C₂.₂₀(D₇×D₈), C₁₆⋊D₇⋊₁C₂, C₇⋊₃(C₁₆⋊C₂²), (C₇×SD₃₂)⋊₁C₂, C₁₄.₃₆(C₂×D₈), C₂₈.₁₁(C₂×D₄), C₇⋊SD₃₂⋊₂C₂, Q₈.D₁₄⋊₃C₂, (C₇×Q₁₆)⋊₅C₂², (C₇×D₈).₃C₂², (C₈×D₇).₄C₂², C₈.₂₃(C₂²×D₇), SmallGroup(448,448)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅₆ — D₁₁₂⋊C₂

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₅₆ — C₈×D₇ — D₇×D₈ — D₁₁₂⋊C₂

Lower central

C₇ — C₁₄ — C₂₈ — C₅₆ — D₁₁₂⋊C₂

Upper central

C₁ — C₂ — C₄ — C₈ — SD₃₂

Generators and relations for D₁₁₂⋊C₂
G = < a,b,c | a¹¹²=b²=c²=1, bab=a^-1, cac=a⁷¹, bc=cb >

Subgroups: 736 in 90 conjugacy classes, 31 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₇, C₈, C₈, C₂×C₄, D₄, Q₈, C₂³, D₇, C₁₄, C₁₄, C₁₆, C₁₆, C₂×C₈, D₈, D₈, SD₁₆, Q₁₆, C₂×D₄, C₄○D₄, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, M₅(2), D₁₆, SD₃₂, SD₃₂, C₂×D₈, C₄○D₈, C₇⋊C₈, C₅₆, C₄×D₇, C₄×D₇, D₂₈, C₇⋊D₄, C₇×D₄, C₇×Q₈, C₂²×D₇, C₁₆⋊C₂², C₇⋊C₁₆, C₁₁₂, C₈×D₇, D₅₆, D₄⋊D₇, Q₈⋊D₇, C₇×D₈, C₇×Q₁₆, D₄×D₇, Q₈⋊₂D₇, C₁₆⋊D₇, D₁₁₂, C₇⋊D₁₆, C₇⋊SD₃₂, C₇×SD₃₂, D₇×D₈, Q₈.D₁₄, D₁₁₂⋊C₂
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, D₈, C₂×D₄, D₁₄, C₂×D₈, C₂²×D₇, C₁₆⋊C₂², D₄×D₇, D₇×D₈, D₁₁₂⋊C₂

Smallest permutation representation of D₁₁₂⋊C₂
►On 112 points

Generators in S₁₁₂

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)

(1 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 84)(9 83)(10 82)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)

(1 43)(3 73)(4 32)(5 103)(6 62)(7 21)(8 92)(9 51)(11 81)(12 40)(13 111)(14 70)(15 29)(16 100)(17 59)(19 89)(20 48)(22 78)(23 37)(24 108)(25 67)(27 97)(28 56)(30 86)(31 45)(33 75)(35 105)(36 64)(38 94)(39 53)(41 83)(44 72)(46 102)(47 61)(49 91)(52 80)(54 110)(55 69)(57 99)(60 88)(63 77)(65 107)(68 96)(71 85)(76 104)(79 93)(84 112)(87 101)(95 109)

G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,43)(3,73)(4,32)(5,103)(6,62)(7,21)(8,92)(9,51)(11,81)(12,40)(13,111)(14,70)(15,29)(16,100)(17,59)(19,89)(20,48)(22,78)(23,37)(24,108)(25,67)(27,97)(28,56)(30,86)(31,45)(33,75)(35,105)(36,64)(38,94)(39,53)(41,83)(44,72)(46,102)(47,61)(49,91)(52,80)(54,110)(55,69)(57,99)(60,88)(63,77)(65,107)(68,96)(71,85)(76,104)(79,93)(84,112)(87,101)(95,109)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,43)(3,73)(4,32)(5,103)(6,62)(7,21)(8,92)(9,51)(11,81)(12,40)(13,111)(14,70)(15,29)(16,100)(17,59)(19,89)(20,48)(22,78)(23,37)(24,108)(25,67)(27,97)(28,56)(30,86)(31,45)(33,75)(35,105)(36,64)(38,94)(39,53)(41,83)(44,72)(46,102)(47,61)(49,91)(52,80)(54,110)(55,69)(57,99)(60,88)(63,77)(65,107)(68,96)(71,85)(76,104)(79,93)(84,112)(87,101)(95,109) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,84),(9,83),(10,82),(11,81),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103)], [(1,43),(3,73),(4,32),(5,103),(6,62),(7,21),(8,92),(9,51),(11,81),(12,40),(13,111),(14,70),(15,29),(16,100),(17,59),(19,89),(20,48),(22,78),(23,37),(24,108),(25,67),(27,97),(28,56),(30,86),(31,45),(33,75),(35,105),(36,64),(38,94),(39,53),(41,83),(44,72),(46,102),(47,61),(49,91),(52,80),(54,110),(55,69),(57,99),(60,88),(63,77),(65,107),(68,96),(71,85),(76,104),(79,93),(84,112),(87,101),(95,109)]])

49 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	7A	7B	7C	8A	8B	8C	14A	14B	14C	14D	14E	14F	16A	16B	16C	16D	28A	28B	28C	28D	28E	28F	56A	···	56F	112A	···	112L
order	1	2	2	2	2	2	4	4	4	7	7	7	8	8	8	14	14	14	14	14	14	16	16	16	16	28	28	28	28	28	28	56	···	56	112	···	112
size	1	1	8	14	56	56	2	8	14	2	2	2	2	2	28	2	2	2	16	16	16	4	4	28	28	4	4	4	16	16	16	4	···	4	4	···	4

49 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₈

D₁₄

C₁₆⋊C₂²

D₄×D₇

D₇×D₈

D₁₁₂⋊C₂

kernel

D₁₁₂⋊C₂

C₁₆⋊D₇

D₁₁₂

C₇⋊D₁₆

C₇⋊SD₃₂

C₇×SD₃₂

D₇×D₈

Q₈.D₁₄

C₇⋊C₈

C₄×D₇

SD₃₂

Dic₇

D₁₄

C₁₆

D₈

Q₁₆

C₇

C₄

C₂

C₁

# reps

Matrix representation of D₁₁₂⋊C₂ ►in GL₈(𝔽₁₁₃)

76	84	37	29	0	0	0	0
76	0	37	0	0	0	0	0
76	84	76	84	0	0	0	0
76	0	76	0	0	0	0	0
0	0	0	0	25	0	0	111
0	0	0	0	100	102	1	1
0	0	0	0	8	74	11	10
0	0	0	0	30	82	0	88

0	0	25	89	0	0	0	0
0	0	26	88	0	0	0	0
25	89	0	0	0	0	0	0
26	88	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	73	27	0	1
0	0	0	0	40	86	1	0

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	98	27	0	112
0	0	0	0	41	108	112	0

G:=sub<GL(8,GF(113))| [76,76,76,76,0,0,0,0,84,0,84,0,0,0,0,0,37,37,76,76,0,0,0,0,29,0,84,0,0,0,0,0,0,0,0,0,25,100,8,30,0,0,0,0,0,102,74,82,0,0,0,0,0,1,11,0,0,0,0,0,111,1,10,88],[0,0,25,26,0,0,0,0,0,0,89,88,0,0,0,0,25,26,0,0,0,0,0,0,89,88,0,0,0,0,0,0,0,0,0,0,62,82,73,40,0,0,0,0,62,51,27,86,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,62,82,98,41,0,0,0,0,62,51,27,108,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0] >;

D₁₁₂⋊C₂ in GAP, Magma, Sage, TeX

D_{112}\rtimes C_2

% in TeX

G:=Group("D112:C2");

// GroupNames label

G:=SmallGroup(448,448);

// by ID

G=gap.SmallGroup(448,448);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,135,184,346,185,192,851,438,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^112=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^71,b*c=c*b>;

// generators/relations

76	84	37	29	0	0	0	0
76	0	37	0	0	0	0	0
76	84	76	84	0	0	0	0
76	0	76	0	0	0	0	0
0	0	0	0	25	0	0	111
0	0	0	0	100	102	1	1
0	0	0	0	8	74	11	10
0	0	0	0	30	82	0	88

0	0	25	89	0	0	0	0
0	0	26	88	0	0	0	0
25	89	0	0	0	0	0	0
26	88	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	73	27	0	1
0	0	0	0	40	86	1	0

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	98	27	0	112
0	0	0	0	41	108	112	0

76	84	37	29	0	0	0	0
76	0	37	0	0	0	0	0
76	84	76	84	0	0	0	0
76	0	76	0	0	0	0	0
0	0	0	0	25	0	0	111
0	0	0	0	100	102	1	1
0	0	0	0	8	74	11	10
0	0	0	0	30	82	0	88

0	0	25	89	0	0	0	0
0	0	26	88	0	0	0	0
25	89	0	0	0	0	0	0
26	88	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	73	27	0	1
0	0	0	0	40	86	1	0

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	98	27	0	112
0	0	0	0	41	108	112	0

G = D112⋊C2 order 448 = 26·7

6th semidirect product of D112 and C2 acting faithfully

G = D₁₁₂⋊C₂ order 448 = 2⁶·7

6^th semidirect product of D₁₁₂ and C₂ acting faithfully

76	84	37	29	0	0	0	0
76	0	37	0	0	0	0	0
76	84	76	84	0	0	0	0
76	0	76	0	0	0	0	0
0	0	0	0	25	0	0	111
0	0	0	0	100	102	1	1
0	0	0	0	8	74	11	10
0	0	0	0	30	82	0	88

0	0	25	89	0	0	0	0
0	0	26	88	0	0	0	0
25	89	0	0	0	0	0	0
26	88	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	73	27	0	1
0	0	0	0	40	86	1	0

0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	0	0	62	62	0	0
0	0	0	0	82	51	0	0
0	0	0	0	98	27	0	112
0	0	0	0	41	108	112	0