D8.5D4 - GroupNames

G = D₈.₅D₄ order 128 = 2⁷

2^nd non-split extension by D₈ of D₄ acting via D₄/C₄=C₂

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D₈.₅D₄, C₄².₁₄₀D₄, C₄⋊C₁₆⋊₇C₂, (C₄×D₈)⋊₁₉C₂, (C₂×C₄).₃₄D₈, C₈.₇₄(C₂×D₄), (C₂×D₁₆).₂C₂, (C₂×C₈).₁₇₄D₄, C₂.D₁₆⋊₁₂C₂, C₂.Q₃₂⋊₆C₂, (C₂×SD₃₂)⋊₁₄C₂, C₈.₉₂(C₄○D₄), C₂.₈(C₄○D₁₆), C₈.₁₂D₄⋊₁C₂, C₄.₅₃(C₄⋊D₄), C₂.₂₂(C₄⋊D₈), C₄.₁₈(C₈⋊C₂²), (C₂×C₁₆).₄₁C₂², (C₂×C₈).₅₁₈C₂³, (C₄×C₈).₁₀₃C₂², C₂².₁₀₄(C₂×D₈), (C₂×Q₁₆).₄C₂², C₂.₁₁(C₁₆⋊C₂²), (C₂×D₈).₁₁₁C₂², C₂.D₈.₁₆₀C₂², (C₂×C₄).₇₈₆(C₂×D₄), SmallGroup(128,942)

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

Derived series

C₁ — C₂×C₈ — D₈.₅D₄

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×D₈ — C₄×D₈ — D₈.₅D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — D₈.₅D₄

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — D₈.₅D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — D₈.₅D₄

Generators and relations for D₈.₅D₄
G = < a,b,c,d | a⁸=b²=c⁴=1, d²=a⁴, bab=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a⁵b, dcd^-1=a⁴c^-1 >

Subgroups: 252 in 85 conjugacy classes, 32 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄×C₈, D₄⋊C₄, C₂.D₈, C₂×C₁₆, D₁₆, SD₃₂, C₄×D₄, C₄.₄D₄, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₂.D₁₆, C₂.Q₃₂, C₄⋊C₁₆, C₄×D₈, C₈.₁₂D₄, C₂×D₁₆, C₂×SD₃₂, D₈.₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₈⋊C₂², C₄⋊D₈, C₄○D₁₆, C₁₆⋊C₂², D₈.₅D₄

Character table of D₈.₅D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	8	8	16	2	2	2	2	4	8	8	16	2	2	2	2	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	-1	1	1	1	1	1	1	1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	-1	1	-1	-1	1	-1	1	1	1	1	1	1	1	-1	-1	1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₄	1	1	1	1	-1	-1	1	1	-1	-1	1	-1	1	1	-1	1	1	1	1	-1	-1	-1	1	1	-1	1	-1	-1	1	linear of order 2
ρ₅	1	1	1	1	1	1	-1	1	-1	-1	1	-1	-1	-1	1	1	1	1	1	-1	-1	-1	1	1	-1	1	-1	-1	1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₇	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₈	1	1	1	1	-1	-1	-1	1	1	1	1	1	-1	-1	-1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₉	2	-2	-2	2	2	-2	0	2	0	0	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	2	-2	-2	2	-2	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	0	2	2	2	2	2	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	2	-2	2	0	2	0	0	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	0	0	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	-√2	√2	√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	0	0	0	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	√2	√2	√2	-√2	-√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	2	2	2	0	0	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	√2	√2	-√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₆	2	2	2	2	0	0	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	-√2	-√2	√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₇	2	-2	-2	2	0	0	0	2	0	0	-2	0	2i	-2i	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	0	0	0	2	0	0	-2	0	-2i	2i	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	-√2	√2	-√2	√2	-√-2	√-2	ζ₁₆⁷-ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₀	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	-√2	√2	-√2	√2	√-2	-√-2	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from C₄○D₁₆
ρ₂₁	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	√2	-√2	√2	-√2	-√-2	√-2	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷-ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₂	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	√2	-√2	√2	-√2	√-2	-√-2	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₃	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	√2	-√2	√2	-√2	√-2	-√-2	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from C₄○D₁₆
ρ₂₄	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	√2	-√2	√2	-√2	-√-2	√-2	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from C₄○D₁₆
ρ₂₅	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	-√2	√2	-√2	√2	√-2	-√-2	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₆	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	-√2	√2	-√2	√2	-√-2	√-2	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from C₄○D₁₆
ρ₂₇	4	-4	-4	4	0	0	0	-4	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₈	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	-2√2	-2√2	2√2	2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²
ρ₂₉	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	2√2	2√2	-2√2	-2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²

Smallest permutation representation of D₈.₅D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

Matrix representation of D₈.₅D₄ ►in GL₄(𝔽₁₇) generated by

3	14	0	0
3	3	0	0
0	0	1	0
0	0	0	1

13	11	0	0
11	4	0	0
0	0	1	0
0	0	0	1

4	0	0	0
0	4	0	0
0	0	1	15
0	0	1	16

4	0	0	0
0	13	0	0
0	0	1	15
0	0	0	16

G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,1,0,0,0,0,1],[13,11,0,0,11,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;

D₈.₅D₄ in GAP, Magma, Sage, TeX

D_8._5D_4

% in TeX

G:=Group("D8.5D4");

// GroupNames label

G:=SmallGroup(128,942);

// by ID

G=gap.SmallGroup(128,942);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,1684,438,242,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*c^-1>;

// generators/relations

Export

Character table of D₈.₅D₄ in TeX

G = D8.5D4 order 128 = 27

2nd non-split extension by D8 of D4 acting via D4/C4=C2

G = D₈.₅D₄ order 128 = 2⁷

2^nd non-split extension by D₈ of D₄ acting via D₄/C₄=C₂