(C2xD4):Q8 - GroupNames

G = (C₂×D₄)⋊Q₈ order 128 = 2⁷

1^st semidirect product of C₂×D₄ and Q₈ acting via Q₈/C₂=C₂²

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

Aliases: (C₂×D₄)⋊₁Q₈, C₄⋊C₄.₈₂D₄, (C₂×C₄).₂₀D₈, (C₂×C₄).₃₂SD₁₆, C₂².₈₂(C₂×D₈), C₂.₆(D₄⋊Q₈), C₂.₆(D₄⋊₂Q₈), C₂.₅(C₄.₄D₈), C₂³.₉₀₅(C₂×D₄), (C₂²×C₄).₃₁₁D₄, C₄.₂₉(C₂²⋊Q₈), C₂.₂₀(C₂²⋊D₈), C₂².₄Q₁₆⋊₁₉C₂, C₂.₂₀(Q₈⋊D₄), C₄.₃₀(C₄.₄D₄), (C₂²×C₈).₆₅C₂², C₂.₄(C₂³⋊Q₈), C₂².₉₃(C₂×SD₁₆), C₂².₂₀₇C₂²≀C₂, C₂².₇C₄²⋊₉C₂, (C₂×C₄²).₃₅₃C₂², (C₂²×D₄).₇₂C₂², C₂².₁₃₂(C₈⋊C₂²), (C₂²×C₄).₁₄₃₉C₂³, C₂².₈₆(C₄.₄D₄), C₂².₁₀₂(C₂²⋊Q₈), C₂².₁₂₁(C₈.C₂²), C₂⁴.₃C₂².₁₃C₂, C₂.₆(C₄².₂₈C₂²), (C₂×C₄⋊Q₈)⋊₃C₂, (C₂×C₄).₂₇₉(C₂×Q₈), (C₂×C₄).₁₀₂₉(C₂×D₄), (C₂×D₄⋊C₄).₁₁C₂, (C₂×C₄).₆₁₆(C₄○D₄), (C₂×C₄⋊C₄).₁₁₂C₂², SmallGroup(128,755)

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

Derived series

C₁ — C₂²×C₄ — (C₂×D₄)⋊Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂⁴.₃C₂² — (C₂×D₄)⋊Q₈

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂×D₄)⋊Q₈

Upper central

C₁ — C₂³ — C₂×C₄² — (C₂×D₄)⋊Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (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=ebe^-1=b^-1, bd=db, dcd^-1=ab²c, ece^-1=abc, ede^-1=d^-1 >

Subgroups: 400 in 163 conjugacy classes, 54 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, D₄⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊Q₈, C₂²×C₈, C₂²×D₄, C₂²×Q₈, C₂².₇C₄², C₂².₄Q₁₆, C₂⁴.₃C₂², C₂×D₄⋊C₄, C₂×C₄⋊Q₈, (C₂×D₄)⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, SD₁₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₈.C₂², C₂³⋊Q₈, C₂²⋊D₈, Q₈⋊D₄, D₄⋊Q₈, D₄⋊₂Q₈, C₄.₄D₈, C₄².₂₈C₂², (C₂×D₄)⋊Q₈

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

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	-	+			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	Q₈	D₈	SD₁₆	C₄○D₄	C₈⋊C₂²	C₈.C₂²
kernel	(C₂×D₄)⋊Q₈	C₂².₇C₄²	C₂².₄Q₁₆	C₂⁴.₃C₂²	C₂×D₄⋊C₄	C₂×C₄⋊Q₈	C₄⋊C₄	C₂²×C₄	C₂×D₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	2	1	2	1	4	2	2	4	4	6	1	1

Matrix representation of (C₂×D₄)⋊Q₈ ►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	16	0
0	0	0	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	2	0	0
0	0	16	1	0	0
0	0	0	0	16	0
0	0	0	0	0	16

16	0	0	0	0	0
1	1	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

1	2	0	0	0	0
16	16	0	0	0	0
0	0	16	2	0	0
0	0	16	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

13	0	0	0	0	0
4	4	0	0	0	0
0	0	0	7	0	0
0	0	12	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

(C₂×D₄)⋊Q₈ in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes Q_8

% in TeX

G:=Group("(C2xD4):Q8");

// GroupNames label

G:=SmallGroup(128,755);

// by ID

G=gap.SmallGroup(128,755);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,2804,718,172]);

// 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=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^2*c,e*c*e^-1=a*b*c,e*d*e^-1=d^-1>;

// generators/relations

G = (C2×D4)⋊Q8 order 128 = 27

1st semidirect product of C2×D4 and Q8 acting via Q8/C2=C22

G = (C₂×D₄)⋊Q₈ order 128 = 2⁷

1^st semidirect product of C₂×D₄ and Q₈ acting via Q₈/C₂=C₂²