C4oD4.5Q8 - GroupNames

G = C₄○D₄.₅Q₈ order 128 = 2⁷

3^rd non-split extension by C₄○D₄ of Q₈ acting via Q₈/C₄=C₂

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

Aliases: C₈○D₄⋊₃C₄, C₄○D₄.₅Q₈, D₄.₄(C₄⋊C₄), (C₂×C₈).₁₀₆D₄, Q₈.₄(C₄⋊C₄), (C₂×D₄).₂₀₃D₄, (C₂×Q₈).₁₆₁D₄, C₈.₃₉(C₂²⋊C₄), C₄.₇₃(C₂²⋊Q₈), C₄.₁₂₄(C₄⋊D₄), C₂.₁(D₄.₄D₄), C₂.₁(D₄.₅D₄), M₄(2).₃₂(C₂×C₄), C₂².C₄²⋊₁₄C₂, C₂³.₂₆₂(C₄○D₄), (C₂²×C₄).₆₆₆C₂³, (C₂²×C₈).₂₁₈C₂², C₂³.₃₆D₄.₆C₂, C₂².₁₄(C₂²⋊Q₈), C₂².₁₁₃(C₄⋊D₄), C₂.₂₅(C₂³.₇Q₈), C₂².₁₆(C₄²⋊C₂), (C₂×M₄(2)).₁₅₉C₂², C₄.₈(C₂×C₄⋊C₄), (C₂×C₄).₆(C₂×Q₈), (C₂×C₈).₆₂(C₂×C₄), (C₂×C₈○D₄).₅C₂, (C₂×C₂.D₈)⋊₃₃C₂, (C₂×C₈.C₄)⋊₆C₂, C₄○D₄.₂₉(C₂×C₄), (C₂×C₄).₂₃₃(C₂×D₄), C₄.₉₅(C₂×C₂²⋊C₄), (C₂×C₄).₄₈(C₄○D₄), (C₂×C₄⋊C₄).₄₀C₂², (C₂×C₄).₁₈₀(C₂²×C₄), (C₂×C₄○D₄).₂₆₀C₂², SmallGroup(128,548)

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

Derived series

C₁ — C₂×C₄ — C₄○D₄.₅Q₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×C₈○D₄ — C₄○D₄.₅Q₈

Lower central

C₁ — C₂ — C₂×C₄ — C₄○D₄.₅Q₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₄○D₄.₅Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₄○D₄.₅Q₈

Generators and relations for C₄○D₄.₅Q₈
G = < a,b,c,d,e | a⁴=c²=1, b²=d⁴=a², e²=abd², ab=ba, ac=ca, ad=da, eae^-1=a^-1, cbc=a²b, bd=db, be=eb, cd=dc, ece^-1=bc, ede^-1=a²d³ >

Subgroups: 244 in 132 conjugacy classes, 60 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, D₄⋊C₄, Q₈⋊C₄, C₂.D₈, C₈.C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×M₄(2), C₈○D₄, C₈○D₄, C₂×C₄○D₄, C₂².C₄², C₂³.₃₆D₄, C₂×C₂.D₈, C₂×C₈.C₄, C₂×C₈○D₄, C₄○D₄.₅Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄⋊D₄, C₂²⋊Q₈, C₂³.₇Q₈, D₄.₄D₄, D₄.₅D₄, C₄○D₄.₅Q₈

Smallest permutation representation of C₄○D₄.₅Q₈
►On 64 points

Generators in S₆₄

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

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

(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 13)(10 14)(11 15)(12 16)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(57 61)(58 62)(59 63)(60 64)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+		+	+	+	-			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	D₄	D₄	Q₈	C₄○D₄	C₄○D₄	D₄.₄D₄	D₄.₅D₄
kernel	C₄○D₄.₅Q₈	C₂².C₄²	C₂³.₃₆D₄	C₂×C₂.D₈	C₂×C₈.C₄	C₂×C₈○D₄	C₈○D₄	C₂×C₈	C₂×D₄	C₂×Q₈	C₄○D₄	C₂×C₄	C₂³	C₂	C₂
# reps	1	2	2	1	1	1	8	4	1	1	2	2	2	2	2

Matrix representation of C₄○D₄.₅Q₈ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	8	16	13	0
0	0	8	0	0	13

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	15	0	0
0	0	1	1	0	0
0	0	0	0	0	1
0	0	0	0	16	0

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

1	15	0	0	0	0
1	16	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	10	3	15	0
0	0	10	0	0	15

9	2	0	0	0	0
10	8	0	0	0	0
0	0	0	13	2	15
0	0	2	4	15	0
0	0	12	12	15	15
0	0	0	5	2	15

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,8,8,0,0,0,4,16,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,8,0,10,10,0,0,0,8,3,0,0,0,0,0,15,0,0,0,0,0,0,15],[9,10,0,0,0,0,2,8,0,0,0,0,0,0,0,2,12,0,0,0,13,4,12,5,0,0,2,15,15,2,0,0,15,0,15,15] >;

C₄○D₄.₅Q₈ in GAP, Magma, Sage, TeX

C_4\circ D_4._5Q_8

% in TeX

G:=Group("C4oD4.5Q8");

// GroupNames label

G:=SmallGroup(128,548);

// by ID

G=gap.SmallGroup(128,548);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2019,248,718,172,1027,124]);

// Polycyclic

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

// generators/relations

G = C4○D4.5Q8 order 128 = 27

3rd non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

G = C₄○D₄.₅Q₈ order 128 = 2⁷

3^rd non-split extension by C₄○D₄ of Q₈ acting via Q₈/C₄=C₂