C2xD4:D4 - GroupNames

G = C₂×D₄⋊D₄ order 128 = 2⁷

Direct product of C₂ and D₄⋊D₄

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

Aliases: C₂×D₄⋊D₄, C₂⁴.₁₀₁D₄, D₄⋊₇(C₂×D₄), Q₈⋊₇(C₂×D₄), (C₂×D₄)⋊₄₈D₄, (C₂×Q₈)⋊₃₅D₄, (C₂²×D₈)⋊₆C₂, C₄⋊C₄.₃C₂³, C₄.₃₇C₂²≀C₂, (C₂×D₈)⋊₃₆C₂², C₄.₃₇(C₂²×D₄), C₄⋊D₄⋊₄₈C₂², C₂²⋊C₈⋊₅₆C₂², (C₂×C₄).₂₁₉C₂⁴, (C₂×C₈).₁₂₇C₂³, (C₂×D₄).₂₅C₂³, C₂³.₈₅₀(C₂×D₄), (C₂²×C₄).₇₁₂D₄, D₄⋊C₄⋊₇₀C₂², Q₈⋊C₄⋊₆₄C₂², (C₂×SD₁₆)⋊₆₇C₂², (C₂²×SD₁₆)⋊₁₇C₂, C₂².₈₆(C₄○D₈), (C₂×Q₈).₃₅₂C₂³, C₂².₁₁₆C₂²≀C₂, (C₂²×C₈).₁₃₄C₂², (C₂³×C₄).₅₃₉C₂², (C₂²×C₄).₉₅₇C₂³, C₂².₄₇₉(C₂²×D₄), C₂².₁₁₁(C₈⋊C₂²), (C₂²×D₄).₃₂₅C₂², (C₂²×Q₈).₄₆₅C₂², C₂.₇(C₂×C₄○D₈), C₂.₉(C₂×C₈⋊C₂²), (C₂×C₄⋊D₄)⋊₄₄C₂, (C₂×C₂²⋊C₈)⋊₂₃C₂, (C₂²×C₄○D₄)⋊₇C₂, (C₂×D₄⋊C₄)⋊₃₆C₂, (C₂×Q₈⋊C₄)⋊₂₂C₂, C₂.₃₇(C₂×C₂²≀C₂), (C₂×C₄).₁₀₉₃(C₂×D₄), (C₂×C₄○D₄)⋊₆₃C₂², (C₂×C₄⋊C₄).₅₈₀C₂², SmallGroup(128,1732)

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

Derived series

C₁ — C₂×C₄ — C₂×D₄⋊D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂×D₄⋊D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×D₄⋊D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×D₄⋊D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×D₄⋊D₄

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

Subgroups: 844 in 406 conjugacy classes, 116 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄⋊D₄, C₂²×C₈, C₂×D₈, C₂×D₈, C₂×SD₁₆, C₂×SD₁₆, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, C₂×Q₈⋊C₄, D₄⋊D₄, C₂×C₄⋊D₄, C₂²×D₈, C₂²×SD₁₆, C₂²×C₄○D₄, C₂×D₄⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₄○D₈, C₈⋊C₂², C₂²×D₄, D₄⋊D₄, C₂×C₂²≀C₂, C₂×C₄○D₈, C₂×C₈⋊C₂², C₂×D₄⋊D₄

Smallest permutation representation of C₂×D₄⋊D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	···	2M	2N	2O	4A	···	4H	4I	4J	4K	4L	4M	4N	8A	···	8H
order	1	2	···	2	2	···	2	2	2	4	···	4	4	4	4	4	4	4	8	···	8
size	1	1	···	1	4	···	4	8	8	2	···	2	4	4	4	4	8	8	4	···	4

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₈

C₈⋊C₂²

kernel

C₂×D₄⋊D₄

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

C₂×Q₈⋊C₄

D₄⋊D₄

C₂×C₄⋊D₄

C₂²×D₈

C₂²×SD₁₆

C₂²×C₄○D₄

C₂²×C₄

C₂×D₄

C₂×Q₈

C₂⁴

C₂²

# reps

Matrix representation of C₂×D₄⋊D₄ ►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	16	0
0	0	0	0	0	16

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	0	1
0	0	0	0	16	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	16	2	0	0
0	0	0	1	0	0
0	0	0	0	3	14
0	0	0	0	14	14

0	16	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	1	16	0	0
0	0	0	0	0	13
0	0	0	0	13	0

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

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,3,14,0,0,0,0,14,14],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C₂×D₄⋊D₄ in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_4

% in TeX

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

// GroupNames label

G:=SmallGroup(128,1732);

// by ID

G=gap.SmallGroup(128,1732);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2804,1411,172]);

// Polycyclic

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

// generators/relations

G = C2×D4⋊D4 order 128 = 27

Direct product of C2 and D4⋊D4

G = C₂×D₄⋊D₄ order 128 = 2⁷

Direct product of C₂ and D₄⋊D₄