C2xQ8:6D4 - GroupNames

G = C₂xQ₈:₆D₄ order 128 = 2⁷

Direct product of C₂ and Q₈:₆D₄

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂xQ₈:₆D₄, C₂³.₂₄C₂⁴, C₂².₅₆C₂⁵, C₄².₅₅₅C₂³, C₂⁴.₄₉₃C₂³, C₂².₁₁₀2₊¹⁺⁴, (C₂xQ₈):₄₃D₄, Q₈:₁₁(C₂xD₄), (C₄xQ₈):₉₂C₂², C₂.₂₃(D₄xC₂³), (C₄xD₄):₁₀₄C₂², C₄:₁D₄:₄₇C₂², C₄:D₄:₇₂C₂², C₄:C₄.₄₆₈C₂³, (C₂xC₄).₅₉₈C₂⁴, C₄.₁₁₂(C₂²xD₄), (C₂xD₄).₄₅₂C₂³, C₂²:C₄.₈₃C₂³, (C₂xQ₈).₄₈₅C₂³, (C₂xC₄²).₉₂₇C₂², (C₂³xC₄).₅₉₆C₂², C₂².₁₆₃(C₂²xD₄), C₂.₁₇(C₂x2₊¹⁺⁴), (C₂²xC₄).₁₁₉₃C₂³, (C₂²xD₄).₄₂₂C₂², (C₂²xQ₈).₅₁₄C₂², Q₈ o₂(C₂xC₄:C₄), C₄:C₄ o₃(C₂xQ₈), (C₂xC₄xD₄):₈₂C₂, C₄:₃(C₂xC₄oD₄), (C₂xC₄xQ₈):₅₁C₂, C₄:C₄ o(C₂²xQ₈), (C₂xC₄):₂₀(C₄oD₄), (C₂xC₄:₁D₄):₂₅C₂, (C₂xC₄:D₄):₆₂C₂, (C₂xC₄).₁₁₁₂(C₂xD₄), (C₂²xC₄oD₄):₁₉C₂, (C₂xC₄oD₄):₇₃C₂², C₂.₂₈(C₂²xC₄oD₄), (C₂xC₄:C₄).₉₅₆C₂², C₂².₁₅₈(C₂xC₄oD₄), (C₂xC₂²:C₄).₅₃₉C₂², (C₂xQ₈)o₂(C₂xC₄:C₄), (C₂xC₄:C₄)o(C₂²xQ₈), SmallGroup(128,2199)

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

Derived series

C₁ — C₂² — C₂xQ₈:₆D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄oD₄ — C₂xQ₈:₆D₄

Lower central

C₁ — C₂² — C₂xQ₈:₆D₄

Upper central

C₁ — C₂³ — C₂xQ₈:₆D₄

Jennings

C₁ — C₂² — C₂xQ₈:₆D₄

Generators and relations for C₂xQ₈:₆D₄
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=b^-1, be=eb, dcd^-1=ece=b²c, ede=d^-1 >

Subgroups: 1436 in 868 conjugacy classes, 444 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₂xC₄², C₂xC₂²:C₄, C₂xC₄:C₄, C₂xC₄:C₄, C₄xD₄, C₄xQ₈, C₄:D₄, C₄:₁D₄, C₂³xC₄, C₂²xD₄, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄oD₄, C₂xC₄xD₄, C₂xC₄xQ₈, C₂xC₄:D₄, C₂xC₄:₁D₄, Q₈:₆D₄, C₂²xC₄oD₄, C₂xQ₈:₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₄oD₄, C₂⁴, C₂²xD₄, C₂xC₄oD₄, 2₊¹⁺⁴, C₂⁵, Q₈:₆D₄, D₄xC₂³, C₂²xC₄oD₄, C₂x2₊¹⁺⁴, C₂xQ₈:₆D₄

Smallest permutation representation of C₂xQ₈:₆D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4X	4Y	···	4AD
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	2	···	2	4	···	4

50 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	C₄oD₄	2₊¹⁺⁴
kernel	C₂xQ₈:₆D₄	C₂xC₄xD₄	C₂xC₄xQ₈	C₂xC₄:D₄	C₂xC₄:₁D₄	Q₈:₆D₄	C₂²xC₄oD₄	C₂xQ₈	C₂xC₄	C₂²
# reps	1	3	1	6	3	16	2	8	8	2

Matrix representation of C₂xQ₈:₆D₄ ►in GL₅(F₅)

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	2	4
0	0	0	0	3

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	3	0
0	0	0	2	2

1	0	0	0	0
0	0	1	0	0
0	4	0	0	0
0	0	0	4	3
0	0	0	1	1

4	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	0	0	1	2
0	0	0	0	4

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,4,3],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,3,2,0,0,0,0,2],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,3,1],[4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,4] >;

C₂xQ₈:₆D₄ in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_6D_4

% in TeX

G:=Group("C2xQ8:6D4");

// GroupNames label

G:=SmallGroup(128,2199);

// by ID

G=gap.SmallGroup(128,2199);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,184,570,136]);

// 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=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;

// generators/relations

G = C2xQ8:6D4 order 128 = 27

Direct product of C2 and Q8:6D4

G = C₂xQ₈:₆D₄ order 128 = 2⁷

Direct product of C₂ and Q₈:₆D₄