C2xC2^2.D8 - GroupNames

G = C₂×C₂².D₈ order 128 = 2⁷

Direct product of C₂ and C₂².D₈

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

Aliases: C₂×C₂².D₈, C₂³.₅₂D₈, C₂⁴.₁₇₉D₄, C₂.₈(C₂²×D₈), C₄⋊C₄.₄₈C₂³, C₂².₂₁(C₂×D₈), C₂.D₈⋊₅₁C₂², C₂²⋊C₈⋊₅₂C₂², (C₂×C₈).₁₄₀C₂³, (C₂×C₄).₂₈₃C₂⁴, (C₂×D₄).₇₅C₂³, (C₂²×C₄).₄₃₄D₄, C₂³.₆₆₁(C₂×D₄), D₄⋊C₄⋊₆₅C₂², C₄⋊D₄.₁₅₁C₂², (C₂²×C₈).₁₄₅C₂², (C₂³×C₄).₅₅₃C₂², C₂².₅₄₃(C₂²×D₄), (C₂²×C₄).₁₀₀₂C₂³, C₄.₅₅(C₂².D₄), (C₂²×D₄).₃₅₆C₂², C₂².₁₀₉(C₈.C₂²), C₂².₁₀₆(C₂².D₄), (C₂×C₂.D₈)⋊₂₃C₂, C₄.₉₃(C₂×C₄○D₄), (C₂²×C₄⋊C₄)⋊₃₃C₂, (C₂×C₂²⋊C₈)⋊₂₁C₂, (C₂×C₄).₈₄₅(C₂×D₄), (C₂×D₄⋊C₄)⋊₂₃C₂, (C₂×C₄⋊C₄)⋊₁₁₆C₂², (C₂×C₄⋊D₄).₅₅C₂, C₂.₂₅(C₂×C₈.C₂²), (C₂×C₄).₈₄₁(C₄○D₄), C₂.₄₈(C₂×C₂².D₄), SmallGroup(128,1817)

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

Derived series

C₁ — C₂×C₄ — C₂×C₂².D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂²×C₄⋊C₄ — C₂×C₂².D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₂².D₈

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×C₂².D₈

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₂².D₈

Generators and relations for C₂×C₂².D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd^-1 >

Subgroups: 540 in 256 conjugacy classes, 108 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄⋊D₄, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, C₂×C₂.D₈, C₂².D₈, C₂²×C₄⋊C₄, C₂×C₄⋊D₄, C₂×C₂².D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂×D₈, C₈.C₂², C₂²×D₄, C₂×C₄○D₄, C₂².D₈, C₂×C₂².D₄, C₂²×D₈, C₂×C₈.C₂², C₂×C₂².D₈

Smallest permutation representation of C₂×C₂².D₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

C₄○D₄

D₈

C₈.C₂²

kernel

C₂×C₂².D₈

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

C₂×C₂.D₈

C₂².D₈

C₂²×C₄⋊C₄

C₂×C₄⋊D₄

C₂²×C₄

C₂⁴

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂×C₂².D₈ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	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

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	4
0	0	0	0	0	16

1	0	0	0	0	0
0	1	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

14	3	0	0	0	0
14	14	0	0	0	0
0	0	14	3	0	0
0	0	14	14	0	0
0	0	0	0	4	0
0	0	0	0	15	13

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	0
0	0	0	0	9	1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,4,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,4,15,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,9,0,0,0,0,0,1] >;

C₂×C₂².D₈ in GAP, Magma, Sage, TeX

C_2\times C_2^2.D_8

% in TeX

G:=Group("C2xC2^2.D8");

// GroupNames label

G:=SmallGroup(128,1817);

// by ID

G=gap.SmallGroup(128,1817);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;

// generators/relations

G = C2×C22.D8 order 128 = 27

Direct product of C2 and C22.D8

G = C₂×C₂².D₈ order 128 = 2⁷

Direct product of C₂ and C₂².D₈