C2xD4:Q8 - GroupNames

G = C₂×D₄⋊Q₈ order 128 = 2⁷

Direct product of C₂ and D₄⋊Q₈

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

Aliases: C₂×D₄⋊Q₈, C₄².₂₁₃D₄, C₄².₃₂₆C₂³, D₄⋊₁(C₂×Q₈), (C₂×D₄)⋊₂₀Q₈, C₄.₇₀(C₂×D₈), C₄⋊C₈⋊₅₈C₂², (C₂×C₄).₁₅₅D₈, C₄⋊Q₈⋊₅₄C₂², C₂.₇(C₂²×D₈), C₄⋊C₄.₃₃C₂³, C₂².₇₁(C₂×D₈), C₂.D₈⋊₄₈C₂², C₄.₂₁(C₂²×Q₈), (C₂×C₄).₂₆₈C₂⁴, (C₂×C₈).₁₃₆C₂³, C₂³.₈₆₄(C₂×D₄), (C₂²×C₄).₇₉₅D₄, C₄.₆₁(C₂²⋊Q₈), (C₂×D₄).₃₉₀C₂³, (C₄×D₄).₃₁₂C₂², (C₂×C₄²).₈₁₄C₂², (C₂²×C₈).₁₄₁C₂², C₂².₅₂₈(C₂²×D₄), C₂².₉₆(C₂²⋊Q₈), D₄⋊C₄.₁₅₇C₂², (C₂²×C₄).₁₅₃₈C₂³, (C₂²×D₄).₅₆₈C₂², C₂².₁₀₆(C₈.C₂²), (C₂×C₄⋊C₈)⋊₂₃C₂, (C₂×C₄⋊Q₈)⋊₃₁C₂, (C₂×C₄×D₄).₈₀C₂, (C₂×C₂.D₈)⋊₂₀C₂, C₄.₇₈(C₂×C₄○D₄), (C₂×C₄).₃₁₆(C₂×Q₈), C₂.₄₉(C₂×C₂²⋊Q₈), (C₂×C₄).₁₄₃₁(C₂×D₄), C₂.₂₀(C₂×C₈.C₂²), (C₂×D₄⋊C₄).₂₅C₂, (C₂×C₄).₈₃₄(C₄○D₄), (C₂×C₄⋊C₄).₅₉₇C₂², SmallGroup(128,1802)

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

Derived series

C₁ — C₂×C₄ — C₂×D₄⋊Q₈

Chief series

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

Lower central

C₁ — C₂ — C₂×C₄ — C₂×D₄⋊Q₈

Upper central

C₁ — C₂³ — C₂×C₄² — C₂×D₄⋊Q₈

Jennings

C₁ — C₂ — C₂ — C₂×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, cd=dc, ece^-1=bc, ede^-1=d^-1 >

Subgroups: 476 in 240 conjugacy classes, 116 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, D₄⋊C₄, C₄⋊C₈, C₂.D₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄×D₄, C₄⋊Q₈, C₄⋊Q₈, C₂²×C₈, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×D₄⋊C₄, C₂×C₄⋊C₈, C₂×C₂.D₈, D₄⋊Q₈, C₂×C₄×D₄, C₂×C₄⋊Q₈, C₂×D₄⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₂×D₈, C₈.C₂², C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, D₄⋊Q₈, C₂×C₂²⋊Q₈, C₂²×D₈, C₂×C₈.C₂², C₂×D₄⋊Q₈

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

Generators in S₆₄

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4N	4O	4P	4Q	4R	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	4	4	2	···	2	4	···	4	8	8	8	8	4	···	4

38 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+	-	+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	Q₈	D₈	C₄○D₄	C₈.C₂²
kernel	C₂×D₄⋊Q₈	C₂×D₄⋊C₄	C₂×C₄⋊C₈	C₂×C₂.D₈	D₄⋊Q₈	C₂×C₄×D₄	C₂×C₄⋊Q₈	C₄²	C₂²×C₄	C₂×D₄	C₂×C₄	C₂×C₄	C₂²
# reps	1	2	1	2	8	1	1	2	2	4	8	4	2

Matrix representation of C₂×D₄⋊Q₈ ►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	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

16	0	0	0	0	0
6	1	0	0	0	0
0	0	16	0	0	0
0	0	13	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	13	0	0	0
0	0	1	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

6	2	0	0	0	0
8	11	0	0	0	0
0	0	4	15	0	0
0	0	0	13	0	0
0	0	0	0	14	3
0	0	0	0	3	3

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,6,0,0,0,0,0,1,0,0,0,0,0,0,16,13,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,8,0,0,0,0,2,11,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;

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(128,1802);

// by ID

G=gap.SmallGroup(128,1802);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,4037,1027,124]);

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

// generators/relations

G = C2×D4⋊Q8 order 128 = 27

Direct product of C2 and D4⋊Q8

G = C₂×D₄⋊Q₈ order 128 = 2⁷

Direct product of C₂ and D₄⋊Q₈