C2xD4xQ8 - GroupNames

G = C₂×D₄×Q₈ order 128 = 2⁷

Direct product of C₂, D₄ and Q₈

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C₂×D₄×Q₈, C₂².₅₅C₂⁵, C₄².₅₅₄C₂³, C₂³.₁₂₅C₂⁴, C₂⁴.₆₁₅C₂³, C₂².₈₁2_-¹⁺⁴, C₄⋊₂(C₂²×Q₈), C₂³⋊₆(C₂×Q₈), C₄⋊Q₈⋊₈₂C₂², (C₂×C₄).₅₇C₂⁴, (Q₈×C₂³)⋊₁₁C₂, (C₄×Q₈)⋊₉₁C₂², C₂.₂₂(D₄×C₂³), C₂.₁₀(Q₈×C₂³), C₄⋊C₄.₄₆₇C₂³, C₂²⋊₂(C₂²×Q₈), C₄.₁₁₁(C₂²×D₄), C₂²⋊Q₈⋊₈₅C₂², (C₄×D₄).₃₅₀C₂², (C₂×D₄).₅₀₁C₂³, C₂²⋊C₄.₈₂C₂³, (C₂×Q₈).₄₃₀C₂³, (C₂²×Q₈)⋊₆₃C₂², (C₂×C₄²).₉₂₆C₂², (C₂³×C₄).₅₉₅C₂², C₂².₁₆₂(C₂²×D₄), C₂.₁₃(C₂×2_-¹⁺⁴), (C₂²×C₄).₁₁₉₂C₂³, (C₂²×D₄).₆₁₅C₂², (C₂×C₄×Q₈)⋊₅₀C₂, (C₂×C₄⋊Q₈)⋊₅₁C₂, (C₂×C₄)⋊₁₁(C₂×Q₈), (C₂×C₄×D₄).₈₆C₂, (C₂×Q₈)○(C₂²×D₄), (C₂×D₄)○(C₂²×Q₈), (C₂×C₂²⋊Q₈)⋊₇₀C₂, (C₂×C₄).₁₁₁₁(C₂×D₄), (C₂²×D₄)○(C₂²×Q₈), (C₂×C₄⋊C₄).₉₅₅C₂², (C₂×C₂²⋊C₄).₅₃₈C₂², SmallGroup(128,2198)

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

Derived series

C₁ — C₂² — C₂×D₄×Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — Q₈×C₂³ — C₂×D₄×Q₈

Lower central

C₁ — C₂² — C₂×D₄×Q₈

Upper central

C₁ — C₂³ — C₂×D₄×Q₈

Jennings

C₁ — C₂² — C₂×D₄×Q₈

Generators and relations for C₂×D₄×Q₈
G = < a,b,c,d,e | a²=b⁴=c²=d⁴=1, e²=d², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 1084 in 784 conjugacy classes, 484 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄⋊Q₈, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₂²×Q₈, C₂×C₄×D₄, C₂×C₄×Q₈, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, D₄×Q₈, Q₈×C₂³, C₂×D₄×Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₂⁴, C₂²×D₄, C₂²×Q₈, 2_-¹⁺⁴, C₂⁵, D₄×Q₈, D₄×C₂³, Q₈×C₂³, C₂×2_-¹⁺⁴, C₂×D₄×Q₈

Smallest permutation representation of C₂×D₄×Q₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4P	4Q	···	4AH
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	2	···	2	4	···	4

50 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	2_-¹⁺⁴
kernel	C₂×D₄×Q₈	C₂×C₄×D₄	C₂×C₄×Q₈	C₂×C₂²⋊Q₈	C₂×C₄⋊Q₈	D₄×Q₈	Q₈×C₂³	C₂×D₄	C₂×Q₈	C₂²
# reps	1	3	1	6	3	16	2	8	8	2

Matrix representation of C₂×D₄×Q₈ ►in GL₅(𝔽₅)

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	1	0
0	0	0	0	1

4	0	0	0	0
0	0	4	0	0
0	1	0	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	4	0	0
0	0	0	1	0
0	0	0	0	1

4	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	4
0	0	0	1	0

1	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	0	3
0	0	0	3	0

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,4,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,3,0] >;

C₂×D₄×Q₈ in GAP, Magma, Sage, TeX

C_2\times D_4\times Q_8

% in TeX

G:=Group("C2xD4xQ8");

// GroupNames label

G:=SmallGroup(128,2198);

// by ID

G=gap.SmallGroup(128,2198);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,570,136]);

// Polycyclic

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

// generators/relations

G = C2×D4×Q8 order 128 = 27

Direct product of C2, D4 and Q8

G = C₂×D₄×Q₈ order 128 = 2⁷

Direct product of C₂, D₄ and Q₈