C2xQ8:D4 - GroupNames

G = C₂×Q₈⋊D₄ order 128 = 2⁷

Direct product of C₂ and Q₈⋊D₄

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂×Q₈⋊D₄, C₂³⋊₇SD₁₆, C₂⁴.₁₇₅D₄, Q₈⋊₆(C₂×D₄), (C₂×Q₈)⋊₃₄D₄, C₄⋊C₄.₁C₂³, (Q₈×C₂³)⋊₅C₂, C₄.₃₅C₂²≀C₂, C₂²⋊₄(C₂×SD₁₆), C₄.₃₅(C₂²×D₄), C₂²⋊C₈⋊₆₆C₂², (C₂×C₄).₂₁₇C₂⁴, (C₂×C₈).₂₉₈C₂³, (C₂×D₄).₂₄C₂³, (C₂²×C₄).₄₁₇D₄, C₂³.₈₄₈(C₂×D₄), C₂.₆(C₂²×SD₁₆), Q₈⋊C₄⋊₇₃C₂², (C₂×SD₁₆)⋊₆₆C₂², (C₂²×SD₁₆)⋊₁₆C₂, (C₂×Q₈).₃₅₁C₂³, (C₂²×Q₈)⋊₅₃C₂², C₂².₁₁₄C₂²≀C₂, C₄⋊D₄.₁₄₄C₂², (C₂²×C₄).₉₅₅C₂³, (C₂²×C₈).₃₃₄C₂², (C₂³×C₄).₅₃₇C₂², C₂².₄₇₇(C₂²×D₄), (C₂²×D₄).₃₂₄C₂², C₂².₉₈(C₈.C₂²), (C₂×C₂²⋊C₈)⋊₃₄C₂, C₂.₇(C₂×C₈.C₂²), (C₂×Q₈⋊C₄)⋊₃₅C₂, C₂.₃₅(C₂×C₂²≀C₂), (C₂×C₄⋊D₄).₅₂C₂, (C₂×C₄).₁₀₉₁(C₂×D₄), (C₂×C₄⋊C₄).₅₇₈C₂², SmallGroup(128,1730)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — C₂×Q₈⋊D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — Q₈×C₂³ — C₂×Q₈⋊D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×Q₈⋊D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×Q₈⋊D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×Q₈⋊D₄

Generators and relations for C₂×Q₈⋊D₄
G = < a,b,c,d,e | a²=b⁴=d⁴=e²=1, c²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=dbd^-1=ebe=b^-1, dcd^-1=ece=b^-1c, ede=d^-1 >

Subgroups: 764 in 396 conjugacy classes, 124 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂²⋊C₈, Q₈⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄⋊D₄, C₂²×C₈, C₂×SD₁₆, C₂×SD₁₆, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₂×C₂²⋊C₈, C₂×Q₈⋊C₄, Q₈⋊D₄, C₂×C₄⋊D₄, C₂²×SD₁₆, Q₈×C₂³, C₂×Q₈⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₂⁴, C₂²≀C₂, C₂×SD₁₆, C₈.C₂², C₂²×D₄, Q₈⋊D₄, C₂×C₂²≀C₂, C₂²×SD₁₆, C₂×C₈.C₂², C₂×Q₈⋊D₄

Smallest permutation representation of C₂×Q₈⋊D₄
►On 64 points

Generators in S₆₄

(1 16)(2 13)(3 14)(4 15)(5 56)(6 53)(7 54)(8 55)(9 17)(10 18)(11 19)(12 20)(21 33)(22 34)(23 35)(24 36)(25 32)(26 29)(27 30)(28 31)(37 51)(38 52)(39 49)(40 50)(41 47)(42 48)(43 45)(44 46)(57 63)(58 64)(59 61)(60 62)

(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)

(1 56 3 54)(2 55 4 53)(5 14 7 16)(6 13 8 15)(9 58 11 60)(10 57 12 59)(17 64 19 62)(18 63 20 61)(21 41 23 43)(22 44 24 42)(25 40 27 38)(26 39 28 37)(29 49 31 51)(30 52 32 50)(33 47 35 45)(34 46 36 48)

(1 25 11 22)(2 28 12 21)(3 27 9 24)(4 26 10 23)(5 51 62 47)(6 50 63 46)(7 49 64 45)(8 52 61 48)(13 31 20 33)(14 30 17 36)(15 29 18 35)(16 32 19 34)(37 60 41 56)(38 59 42 55)(39 58 43 54)(40 57 44 53)

(1 17)(2 20)(3 19)(4 18)(5 59)(6 58)(7 57)(8 60)(9 16)(10 15)(11 14)(12 13)(21 33)(22 36)(23 35)(24 34)(25 30)(26 29)(27 32)(28 31)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(53 64)(54 63)(55 62)(56 61)

G:=sub<Sym(64)| (1,16)(2,13)(3,14)(4,15)(5,56)(6,53)(7,54)(8,55)(9,17)(10,18)(11,19)(12,20)(21,33)(22,34)(23,35)(24,36)(25,32)(26,29)(27,30)(28,31)(37,51)(38,52)(39,49)(40,50)(41,47)(42,48)(43,45)(44,46)(57,63)(58,64)(59,61)(60,62), (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), (1,56,3,54)(2,55,4,53)(5,14,7,16)(6,13,8,15)(9,58,11,60)(10,57,12,59)(17,64,19,62)(18,63,20,61)(21,41,23,43)(22,44,24,42)(25,40,27,38)(26,39,28,37)(29,49,31,51)(30,52,32,50)(33,47,35,45)(34,46,36,48), (1,25,11,22)(2,28,12,21)(3,27,9,24)(4,26,10,23)(5,51,62,47)(6,50,63,46)(7,49,64,45)(8,52,61,48)(13,31,20,33)(14,30,17,36)(15,29,18,35)(16,32,19,34)(37,60,41,56)(38,59,42,55)(39,58,43,54)(40,57,44,53), (1,17)(2,20)(3,19)(4,18)(5,59)(6,58)(7,57)(8,60)(9,16)(10,15)(11,14)(12,13)(21,33)(22,36)(23,35)(24,34)(25,30)(26,29)(27,32)(28,31)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(53,64)(54,63)(55,62)(56,61)>;

G:=Group( (1,16)(2,13)(3,14)(4,15)(5,56)(6,53)(7,54)(8,55)(9,17)(10,18)(11,19)(12,20)(21,33)(22,34)(23,35)(24,36)(25,32)(26,29)(27,30)(28,31)(37,51)(38,52)(39,49)(40,50)(41,47)(42,48)(43,45)(44,46)(57,63)(58,64)(59,61)(60,62), (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), (1,56,3,54)(2,55,4,53)(5,14,7,16)(6,13,8,15)(9,58,11,60)(10,57,12,59)(17,64,19,62)(18,63,20,61)(21,41,23,43)(22,44,24,42)(25,40,27,38)(26,39,28,37)(29,49,31,51)(30,52,32,50)(33,47,35,45)(34,46,36,48), (1,25,11,22)(2,28,12,21)(3,27,9,24)(4,26,10,23)(5,51,62,47)(6,50,63,46)(7,49,64,45)(8,52,61,48)(13,31,20,33)(14,30,17,36)(15,29,18,35)(16,32,19,34)(37,60,41,56)(38,59,42,55)(39,58,43,54)(40,57,44,53), (1,17)(2,20)(3,19)(4,18)(5,59)(6,58)(7,57)(8,60)(9,16)(10,15)(11,14)(12,13)(21,33)(22,36)(23,35)(24,34)(25,30)(26,29)(27,32)(28,31)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(53,64)(54,63)(55,62)(56,61) );

G=PermutationGroup([[(1,16),(2,13),(3,14),(4,15),(5,56),(6,53),(7,54),(8,55),(9,17),(10,18),(11,19),(12,20),(21,33),(22,34),(23,35),(24,36),(25,32),(26,29),(27,30),(28,31),(37,51),(38,52),(39,49),(40,50),(41,47),(42,48),(43,45),(44,46),(57,63),(58,64),(59,61),(60,62)], [(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)], [(1,56,3,54),(2,55,4,53),(5,14,7,16),(6,13,8,15),(9,58,11,60),(10,57,12,59),(17,64,19,62),(18,63,20,61),(21,41,23,43),(22,44,24,42),(25,40,27,38),(26,39,28,37),(29,49,31,51),(30,52,32,50),(33,47,35,45),(34,46,36,48)], [(1,25,11,22),(2,28,12,21),(3,27,9,24),(4,26,10,23),(5,51,62,47),(6,50,63,46),(7,49,64,45),(8,52,61,48),(13,31,20,33),(14,30,17,36),(15,29,18,35),(16,32,19,34),(37,60,41,56),(38,59,42,55),(39,58,43,54),(40,57,44,53)], [(1,17),(2,20),(3,19),(4,18),(5,59),(6,58),(7,57),(8,60),(9,16),(10,15),(11,14),(12,13),(21,33),(22,36),(23,35),(24,34),(25,30),(26,29),(27,32),(28,31),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(53,64),(54,63),(55,62),(56,61)]])

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4N	4O	4P	8A	···	8H
order	1	2	···	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	8	···	8
size	1	1	···	1	2	2	2	2	8	8	2	2	2	2	4	···	4	8	8	4	···	4

38 irreducible representations

dim

type

image

C₁

C₂

D₄

SD₁₆

C₈.C₂²

kernel

C₂×Q₈⋊D₄

C₂×C₂²⋊C₈

C₂×Q₈⋊C₄

Q₈⋊D₄

C₂×C₄⋊D₄

C₂²×SD₁₆

Q₈×C₂³

C₂²×C₄

C₂×Q₈

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂×Q₈⋊D₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	1	15
0	0	0	0	1	16

0	1	0	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	10	7
0	0	0	0	5	7

0	16	0	0	0	0
1	0	0	0	0	0
0	0	0	16	0	0
0	0	1	0	0	0
0	0	0	0	16	0
0	0	0	0	16	1

1	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	1	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

C₂×Q₈⋊D₄ in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_4

% in TeX

G:=Group("C2xQ8:D4");

// GroupNames label

G:=SmallGroup(128,1730);

// by ID

G=gap.SmallGroup(128,1730);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=b^-1*c,e*d*e=d^-1>;

// generators/relations

G = C2×Q8⋊D4 order 128 = 27

Direct product of C2 and Q8⋊D4

G = C₂×Q₈⋊D₄ order 128 = 2⁷

Direct product of C₂ and Q₈⋊D₄