C2xC2^3:Q8 - GroupNames

G = C₂×C₂³⋊Q₈ order 128 = 2⁷

Direct product of C₂ and C₂³⋊Q₈

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

Aliases: C₂×C₂³⋊Q₈, C₂⁴⋊₃Q₈, C₂⁵.₂₇C₂², C₂⁴.₆₄₇C₂³, C₂³.₂₈₅C₂⁴, C₂³⋊₃(C₂×Q₈), (Q₈×C₂³)⋊₁C₂, C₂³.₈₃₀(C₂×D₄), (C₂²×C₄).₃₆₃D₄, (C₂²×Q₈)⋊₅₁C₂², C₂².₁₀₉C₂²≀C₂, C₂³.₃₆₅(C₄○D₄), C₂².₅₄(C₂²×Q₈), (C₂³×C₄).₃₁₇C₂², (C₂²×C₄).₇₇₆C₂³, C₂².₁₆₈(C₂²×D₄), C₂².₉₀(C₂²⋊Q₈), C₂.C₄²⋊₆₂C₂², C₂².₇₆(C₄.₄D₄), C₂.₇(C₂×C₂²⋊Q₈), C₂.₆(C₂×C₂²≀C₂), (C₂×C₄).₂₈₇(C₂×D₄), C₂.₆(C₂×C₄.₄D₄), C₂².₁₆₅(C₂×C₄○D₄), (C₂×C₂.C₄²)⋊₂₄C₂, (C₂²×C₂²⋊C₄).₁₈C₂, (C₂×C₂²⋊C₄).₄₈₃C₂², SmallGroup(128,1117)

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

Derived series

C₁ — C₂³ — C₂×C₂³⋊Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂×C₂³⋊Q₈

Lower central

C₁ — C₂³ — C₂×C₂³⋊Q₈

Upper central

C₁ — C₂⁴ — C₂×C₂³⋊Q₈

Jennings

C₁ — C₂³ — C₂×C₂³⋊Q₈

Generators and relations for C₂×C₂³⋊Q₈
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=e², ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 996 in 518 conjugacy classes, 164 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂⁵, C₂×C₂.C₄², C₂³⋊Q₈, C₂²×C₂²⋊C₄, Q₈×C₂³, C₂×C₂³⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂³⋊Q₈, C₂×C₂²≀C₂, C₂×C₂²⋊Q₈, C₂×C₄.₄D₄, C₂×C₂³⋊Q₈

Smallest permutation representation of C₂×C₂³⋊Q₈
►On 64 points

Generators in S₆₄

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

(2 48)(4 46)(5 62)(6 35)(7 64)(8 33)(10 22)(12 24)(14 52)(16 50)(18 44)(20 42)(25 31)(26 53)(27 29)(28 55)(30 59)(32 57)(34 38)(36 40)(37 61)(39 63)(54 58)(56 60)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	···	2O	2P	2Q	2R	2S	4A	···	4X
order	1	2	···	2	2	2	2	2	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄
kernel	C₂×C₂³⋊Q₈	C₂×C₂.C₄²	C₂³⋊Q₈	C₂²×C₂²⋊C₄	Q₈×C₂³	C₂²×C₄	C₂⁴	C₂³
# reps	1	3	8	3	1	12	4	12

Matrix representation of C₂×C₂³⋊Q₈ ►in GL₇(𝔽₅)

4	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	4	0	0	0
0	0	0	0	4	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

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

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	4	0
0	0	0	0	0	0	4

1	0	0	0	0	0	0
0	4	0	0	0	0	0
0	0	4	0	0	0	0
0	0	0	4	0	0	0
0	0	0	0	4	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

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

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

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,3,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0] >;

C₂×C₂³⋊Q₈ in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes Q_8

% in TeX

G:=Group("C2xC2^3:Q8");

// GroupNames label

G:=SmallGroup(128,1117);

// by ID

G=gap.SmallGroup(128,1117);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723]);

// Polycyclic

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

// generators/relations

G = C2×C23⋊Q8 order 128 = 27

Direct product of C2 and C23⋊Q8

G = C₂×C₂³⋊Q₈ order 128 = 2⁷

Direct product of C₂ and C₂³⋊Q₈