(C2xD4).303D4 - GroupNames

G = (C₂×D₄).₃₀₃D₄ order 128 = 2⁷

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

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

Aliases: (C₂×D₄).₃₀₃D₄, C₂.₁₆(D₄○D₈), (C₂×Q₈).₂₃₈D₄, C₂.₁₆(Q₈○D₈), C₄⋊C₄.₃₉₆C₂³, (C₂×C₈).₁₄₅C₂³, (C₂×C₄).₂₉₆C₂⁴, (C₂×D₄).₈₃C₂³, C₂³.₂₄₇(C₂×D₄), (C₂×Q₈).₇₁C₂³, C₄.Q₈.₁₃C₂², C₂.D₈.₈₄C₂², C₄⋊D₄.₂₄C₂², C₂³.₂₄D₄⋊₉C₂, C₂².D₈⋊₁₄C₂, C₂²⋊C₈.₁₈C₂², M₄(2)⋊C₄⋊₂₈C₂, C₂²⋊Q₈.₂₄C₂², C₂³.₂₀D₄⋊₁₅C₂, C₂³.₄₈D₄⋊₁₄C₂, C₂³.₃₆D₄⋊₁₁C₂, C₂³.₁₉D₄⋊₁₅C₂, (C₂²×C₈).₁₄₈C₂², C₂².₅₅₆(C₂²×D₄), D₄⋊C₄.₁₅₉C₂², (C₂²×C₄).₁₀₁₂C₂³, Q₈⋊C₄.₁₅₁C₂², C₄.₆₁(C₂².D₄), (C₂×M₄(2)).₇₈C₂², C₂³.₃₃C₂³⋊₁₀C₂, C₄²⋊C₂.₁₂₅C₂², C₂².₃₁C₂⁴.₈C₂, C₂².₂₃(C₂².D₄), (C₂×C₂.D₈)⋊₂₇C₂, C₄.₁₀₆(C₂×C₄○D₄), (C₂×C₄).₄₉₁(C₂×D₄), (C₂²×C₈)⋊C₂⋊₁₀C₂, (C₂×C₄).₂₉₈(C₄○D₄), (C₂×C₄⋊C₄).₆₁₂C₂², (C₂×C₄○D₄).₁₄₁C₂², C₂.₆₁(C₂×C₂².D₄), SmallGroup(128,1830)

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

Derived series

C₁ — C₂×C₄ — (C₂×D₄).₃₀₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₄²⋊C₂ — C₂³.₃₃C₂³ — (C₂×D₄).₃₀₃D₄

Lower central

C₁ — C₂ — C₂×C₄ — (C₂×D₄).₃₀₃D₄

Upper central

C₁ — C₂² — C₂×C₄○D₄ — (C₂×D₄).₃₀₃D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — (C₂×D₄).₃₀₃D₄

Generators and relations for (C₂×D₄).₃₀₃D₄
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, dbd^-1=ab^-1, dcd^-1=ece=ab²c, ede=ad³ >

Subgroups: 380 in 199 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, 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×D₄, C₂×Q₈, C₄○D₄, C₂²⋊C₈, D₄⋊C₄, D₄⋊C₄, Q₈⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×C₄○D₄, (C₂²×C₈)⋊C₂, C₂³.₂₄D₄, C₂³.₃₆D₄, C₂×C₂.D₈, M₄(2)⋊C₄, C₂².D₈, C₂³.₁₉D₄, C₂³.₄₈D₄, C₂³.₂₀D₄, C₂³.₃₃C₂³, C₂².₃₁C₂⁴, (C₂×D₄).₃₀₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₂².D₄, D₄○D₈, Q₈○D₈, (C₂×D₄).₃₀₃D₄

Smallest permutation representation of (C₂×D₄).₃₀₃D₄
►On 64 points

Generators in S₆₄

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

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

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

(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)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○D₈

Q₈○D₈

kernel

(C₂×D₄).₃₀₃D₄

(C₂²×C₈)⋊C₂

C₂³.₂₄D₄

C₂³.₃₆D₄

C₂×C₂.D₈

M₄(2)⋊C₄

C₂².D₈

C₂³.₁₉D₄

C₂³.₄₈D₄

C₂³.₂₀D₄

C₂³.₃₃C₂³

C₂².₃₁C₂⁴

C₂×D₄

C₂×Q₈

C₂×C₄

C₂

# reps

Matrix representation of (C₂×D₄).₃₀₃D₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	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	1

16	2	0	0	0	0
0	1	0	0	0	0
0	0	11	0	3	0
0	0	0	11	0	3
0	0	16	0	6	0
0	0	0	16	0	6

1	15	0	0	0	0
0	16	0	0	0	0
0	0	0	9	0	15
0	0	8	0	2	0
0	0	0	7	0	8
0	0	10	0	9	0

13	0	0	0	0	0
13	4	0	0	0	0
0	0	3	14	6	11
0	0	3	3	6	6
0	0	14	3	14	3
0	0	14	14	14	14

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

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,11,0,16,0,0,0,0,11,0,16,0,0,3,0,6,0,0,0,0,3,0,6],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,8,0,10,0,0,9,0,7,0,0,0,0,2,0,9,0,0,15,0,8,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,3,3,14,14,0,0,14,3,3,14,0,0,6,6,14,14,0,0,11,6,3,14],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C₂×D₄).₃₀₃D₄ in GAP, Magma, Sage, TeX

(C_2\times D_4)._{303}D_4

% in TeX

G:=Group("(C2xD4).303D4");

// GroupNames label

G:=SmallGroup(128,1830);

// by ID

G=gap.SmallGroup(128,1830);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,248,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = (C2×D4).303D4 order 128 = 27

56th non-split extension by C2×D4 of D4 acting via D4/C22=C2

G = (C₂×D₄).₃₀₃D₄ order 128 = 2⁷

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