(C2xD4).302D4 - GroupNames

G = (C₂×D₄).₃₀₂D₄ order 128 = 2⁷

55^th non-split extension by C₂×D₄ of D₄ acting via D₄/C₂²=C₂

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

Aliases: (C₂×D₄).₃₀₂D₄, (C₂×Q₈).₂₃₇D₄, C₂.₁₅(Q₈○D₈), C₄⋊C₄.₃₉₅C₂³, (C₂×C₈).₃₁₂C₂³, (C₂×C₄).₂₉₅C₂⁴, C₂³.₂₄₆(C₂×D₄), (C₂×Q₈).₇₀C₂³, C₄.Q₈.₁₂C₂², C₂.D₈.₈₃C₂², C₂.₂₄(D₄○SD₁₆), C₂³.₄₇D₄⋊₃C₂, C₂³.₂₅D₄⋊₈C₂, C₂²⋊C₈.₁₇C₂², M₄(2)⋊C₄⋊₂₇C₂, C₂²⋊Q₈.₂₃C₂², C₂³.₃₈D₄⋊₁₁C₂, C₂³.₂₀D₄⋊₁₄C₂, C₂³.₄₈D₄⋊₁₃C₂, (C₂²×C₈).₁₈₅C₂², C₂².₅₅₅(C₂²×D₄), (C₂²×C₄).₁₀₁₁C₂³, Q₈⋊C₄.₁₅₀C₂², C₄.₈₃(C₂².D₄), (C₂×M₄(2)).₇₇C₂², (C₂²×Q₈).₂₉₃C₂², C₄²⋊C₂.₁₂₄C₂², C₂³.₃₃C₂³.₉C₂, C₂³.₃₈C₂³.₁₂C₂, C₂².₁₉(C₂².D₄), C₄.₁₀₅(C₂×C₄○D₄), (C₂×C₄).₄₉₀(C₂×D₄), (C₂×Q₈⋊C₄)⋊₃₀C₂, (C₂×C₄).₂₉₇(C₄○D₄), (C₂×C₄⋊C₄).₆₁₁C₂², (C₂²×C₈)⋊C₂.₃C₂, (C₂×C₄○D₄).₁₄₀C₂², C₂.₆₀(C₂×C₂².D₄), SmallGroup(128,1829)

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

Derived series

C₁ — C₂×C₄ — (C₂×D₄).₃₀₂D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₃₃C₂³ — (C₂×D₄).₃₀₂D₄

Lower central

C₁ — C₂ — C₂×C₄ — (C₂×D₄).₃₀₂D₄

Upper central

C₁ — C₂² — C₂×C₄○D₄ — (C₂×D₄).₃₀₂D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — (C₂×D₄).₃₀₂D₄

Generators and relations for (C₂×D₄).₃₀₂D₄
G = < a,b,c,d,e | a²=b⁴=c²=1, d⁴=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, dbd^-1=ebe^-1=ab^-1, dcd^-1=ece^-1=ab²c, ede^-1=ab²d³ >

Subgroups: 340 in 193 conjugacy classes, 92 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂²⋊C₈, Q₈⋊C₄, C₄.Q₈, C₄.Q₈, C₂.D₈, C₂.D₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂²×Q₈, C₂×C₄○D₄, (C₂²×C₈)⋊C₂, C₂×Q₈⋊C₄, C₂³.₃₈D₄, C₂³.₂₅D₄, M₄(2)⋊C₄, C₂³.₄₇D₄, C₂³.₄₈D₄, C₂³.₂₀D₄, C₂³.₃₃C₂³, C₂³.₃₈C₂³, (C₂×D₄).₃₀₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₂².D₄, D₄○SD₁₆, Q₈○D₈, (C₂×D₄).₃₀₂D₄

Smallest permutation representation of (C₂×D₄).₃₀₂D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	4R	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4	8	8	8	8	8	8
size	1	1	1	1	2	2	4	4	2	2	2	2	4	···	4	8	8	8	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

Q₈○D₈

kernel

(C₂×D₄).₃₀₂D₄

(C₂²×C₈)⋊C₂

C₂×Q₈⋊C₄

C₂³.₃₈D₄

C₂³.₂₅D₄

M₄(2)⋊C₄

C₂³.₄₇D₄

C₂³.₄₈D₄

C₂³.₂₀D₄

C₂³.₃₃C₂³

C₂³.₃₈C₂³

C₂×D₄

C₂×Q₈

C₂×C₄

C₂

# reps

Matrix representation of (C₂×D₄).₃₀₂D₄ ►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	16	0
0	0	0	0	0	16

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

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

0	9	0	0	0	0
2	0	0	0	0	0
0	0	5	12	0	0
0	0	5	5	0	0
0	0	0	0	5	12
0	0	0	0	5	5

0	15	0	0	0	0
8	0	0	0	0	0
0	0	0	0	12	5
0	0	0	0	5	5
0	0	12	5	0	0
0	0	5	5	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,5,0,0,0,0,5,5,0,0] >;

(C₂×D₄).₃₀₂D₄ in GAP, Magma, Sage, TeX

(C_2\times D_4)._{302}D_4

% in TeX

G:=Group("(C2xD4).302D4");

// GroupNames label

G:=SmallGroup(128,1829);

// by ID

G=gap.SmallGroup(128,1829);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,1018,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = (C2×D4).302D4 order 128 = 27

55th non-split extension by C2×D4 of D4 acting via D4/C22=C2

G = (C₂×D₄).₃₀₂D₄ order 128 = 2⁷

55^th non-split extension by C₂×D₄ of D₄ acting via D₄/C₂²=C₂