(C2^2xD8).C2 - GroupNames

G = (C₂²×D₈).C₂ order 128 = 2⁷

3^rd non-split extension by C₂²×D₈ of C₂ acting faithfully

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

Aliases: C₄.₈C₂²≀C₂, (C₂×D₄).₉₁D₄, (C₂×C₈).₁₅₃D₄, (C₂×Q₈).₈₂D₄, (C₂²×D₈).₃C₂, C₄.₄₄(C₄⋊D₄), C₂.₁₄(C₈⋊₃D₄), C₂³.₉₀₀(C₂×D₄), (C₂²×C₄).₁₄₁D₄, C₂.₂₉(D₄⋊D₄), (C₂²×SD₁₆)⋊₁₂C₂, C₂.₁₆(D₄.₂D₄), C₂².₂₀₂C₂²≀C₂, C₂.₁₆(C₈.₁₂D₄), C₂².₁₀₃(C₄○D₈), C₂².₇₃(C₄⋊₁D₄), C₂.₁₇(C₂³⋊₂D₄), C₂⁴.₃C₂²⋊₈C₂, (C₂×C₄²).₃₄₈C₂², (C₂²×C₈).₁₀₈C₂², (C₂²×D₄).₆₄C₂², (C₂²×Q₈).₅₃C₂², C₂².₂₂₁(C₄⋊D₄), C₂².₁₃₀(C₈⋊C₂²), (C₂²×C₄).₁₄₃₄C₂³, C₂².₇C₄²⋊₂₀C₂, (C₂×C₄.₄D₄)⋊₃C₂, (C₂×D₄⋊C₄)⋊₃₃C₂, (C₂×Q₈⋊C₄)⋊₂₀C₂, (C₂×C₄).₁₀₂₇(C₂×D₄), (C₂×C₄).₈₇₃(C₄○D₄), (C₂×C₄⋊C₄).₁₀₇C₂², SmallGroup(128,744)

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

Derived series

C₁ — C₂²×C₄ — (C₂²×D₈).C₂

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂×C₄.₄D₄ — (C₂²×D₈).C₂

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂²×D₈).C₂

Upper central

C₁ — C₂³ — C₂×C₄² — (C₂²×D₈).C₂

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂²×D₈).C₂

Generators and relations for (C₂²×D₈).C₂
G = < a,b,c,d,e | a²=b²=c⁸=d²=1, e²=c⁴, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c^-1, ece^-1=ac³, ede^-1=abc²d >

Subgroups: 536 in 213 conjugacy classes, 54 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, D₄⋊C₄, Q₈⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄.₄D₄, C₂²×C₈, C₂×D₈, C₂×SD₁₆, C₂²×D₄, C₂²×Q₈, C₂².₇C₄², C₂⁴.₃C₂², C₂×D₄⋊C₄, C₂×Q₈⋊C₄, C₂×C₄.₄D₄, C₂²×D₈, C₂²×SD₁₆, (C₂²×D₈).C₂
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₄○D₈, C₈⋊C₂², C₂³⋊₂D₄, D₄⋊D₄, D₄.₂D₄, C₈.₁₂D₄, C₈⋊₃D₄, (C₂²×D₈).C₂

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

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	8A	···	8H
order	1	2	···	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	8	···	8
size	1	1	···	1	8	8	8	8	2	2	2	2	4	4	4	4	8	8	8	8	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

C₄○D₈

C₈⋊C₂²

kernel

(C₂²×D₈).C₂

C₂².₇C₄²

C₂⁴.₃C₂²

C₂×D₄⋊C₄

C₂×Q₈⋊C₄

C₂×C₄.₄D₄

C₂²×D₈

C₂²×SD₁₆

C₂×C₈

C₂²×C₄

C₂×D₄

C₂×Q₈

C₂×C₄

C₂²

# reps

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

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	0	1

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

3	14	0	0	0	0
3	3	0	0	0	0
0	0	7	13	0	0
0	0	4	10	0	0
0	0	0	0	16	0
0	0	0	0	0	16

3	14	0	0	0	0
14	14	0	0	0	0
0	0	10	4	0	0
0	0	5	7	0	0
0	0	0	0	1	0
0	0	0	0	16	16

4	0	0	0	0	0
0	13	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	15
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,7,4,0,0,0,0,13,10,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,10,5,0,0,0,0,4,7,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;

(C₂²×D₈).C₂ in GAP, Magma, Sage, TeX

(C_2^2\times D_8).C_2

% in TeX

G:=Group("(C2^2xD8).C2");

// GroupNames label

G:=SmallGroup(128,744);

// by ID

G=gap.SmallGroup(128,744);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,2019,1018,521,248,2804,1411,718,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^2=1,e^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,e*c*e^-1=a*c^3,e*d*e^-1=a*b*c^2*d>;

// generators/relations

G = (C22×D8).C2 order 128 = 27

3rd non-split extension by C22×D8 of C2 acting faithfully

G = (C₂²×D₈).C₂ order 128 = 2⁷

3^rd non-split extension by C₂²×D₈ of C₂ acting faithfully