C8.28D8 - GroupNames

G = C₈.₂₈D₈ order 128 = 2⁷

5^th non-split extension by C₈ of D₈ acting via D₈/D₄=C₂

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

Aliases: D₄⋊₁Q₁₆, C₈.₂₈D₈, C₄².₂₂₀C₂³, C₈⋊₁C₈⋊₄C₂, (C₈×D₄).₄C₂, C₄.₆₀(C₂×D₈), C₄⋊C₄.₁₉₅D₄, C₄⋊Q₁₆⋊₃C₂, (C₂×C₈).₃₄₁D₄, C₄.₂₃(C₂×Q₁₆), (C₂×D₄).₁₈₈D₄, C₄.₄₀(C₄○D₈), C₄.₁₀D₈⋊₅C₂, C₂.₈(C₄⋊D₈), C₄⋊C₈.₂₀C₂², (C₄×C₈).₅₁C₂², D₄⋊Q₈.₄C₂, C₄⋊Q₈.₄₃C₂², C₄.₁₁₈(C₈⋊C₂²), C₂.₆(D₄.₅D₄), C₂.₆(C₈.₁₈D₄), (C₄×D₄).₂₇₆C₂², C₂².₁₈₁(C₄⋊D₄), (C₂×C₄).₅(C₄○D₄), (C₂×C₄).₁₂₅₅(C₂×D₄), SmallGroup(128,401)

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

Derived series

C₁ — C₄² — C₈.₂₈D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈×D₄ — C₈.₂₈D₈

Lower central

C₁ — C₂² — C₄² — C₈.₂₈D₈

Upper central

C₁ — C₂² — C₄² — C₈.₂₈D₈

Jennings

C₁ — C₂² — C₂² — C₄² — C₈.₂₈D₈

Generators and relations for C₈.₂₈D₈
G = < a,b,c | a⁸=b⁸=1, c²=a⁴, bab^-1=cac^-1=a^-1, cbc^-1=a⁴b^-1 >

Subgroups: 192 in 86 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, C₄⋊C₈, C₄⋊C₈, C₂.D₈, C₄×D₄, C₄⋊Q₈, C₂²×C₈, C₂×Q₁₆, C₄.₁₀D₈, C₈⋊₁C₈, C₈×D₄, D₄⋊Q₈, C₄⋊Q₁₆, C₈.₂₈D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₄⋊D₈, C₈.₁₈D₄, D₄.₅D₄, C₈.₂₈D₈

Character table of C₈.₂₈D₈

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	4	4	2	2	2	2	4	4	4	16	16	2	2	2	2	4	4	4	4	4	4	8	8	8	8
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	-2	2	-2	2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	-2	2	-2	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	2	-2	2	-2	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	2	-2	2	-2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₄	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₆	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	-2	2	0	2	0	-2	0	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	-√2	√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₈	2	2	-2	-2	2	-2	0	2	0	-2	0	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	√2	-√2	√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₉	2	2	-2	-2	-2	2	0	2	0	-2	0	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	√2	√2	-√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₂₀	2	2	-2	-2	2	-2	0	2	0	-2	0	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	-√2	√2	-√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₂₁	2	2	2	2	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	-2	-2	0	0	0	-2	0	2	-2i	0	2i	0	0	√2	-√2	-√2	√2	-√-2	√-2	-√-2	√2	-√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₃	2	2	2	2	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₄	2	2	-2	-2	0	0	0	-2	0	2	2i	0	-2i	0	0	-√2	√2	√2	-√2	-√-2	√-2	-√-2	-√2	√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₅	2	2	-2	-2	0	0	0	-2	0	2	2i	0	-2i	0	0	√2	-√2	-√2	√2	√-2	-√-2	√-2	√2	-√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₆	2	2	-2	-2	0	0	0	-2	0	2	-2i	0	2i	0	0	-√2	√2	√2	-√2	√-2	-√-2	√-2	-√2	√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₇	4	-4	4	-4	0	0	4	0	-4	0	0	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	symplectic lifted from D₄.₅D₄, Schur index 2
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	symplectic lifted from D₄.₅D₄, Schur index 2

Smallest permutation representation of C₈.₂₈D₈
►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 26 51 20 47 14 39 62)(2 25 52 19 48 13 40 61)(3 32 53 18 41 12 33 60)(4 31 54 17 42 11 34 59)(5 30 55 24 43 10 35 58)(6 29 56 23 44 9 36 57)(7 28 49 22 45 16 37 64)(8 27 50 21 46 15 38 63)

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

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

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

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

Matrix representation of C₈.₂₈D₈ ►in GL₄(𝔽₁₇) generated by

8	0	0	0
0	15	0	0
0	0	16	0
0	0	0	16

0	1	0	0
1	0	0	0
0	0	11	11
0	0	3	0

0	1	0	0
16	0	0	0
0	0	11	11
0	0	3	6

G:=sub<GL(4,GF(17))| [8,0,0,0,0,15,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,11,3,0,0,11,0],[0,16,0,0,1,0,0,0,0,0,11,3,0,0,11,6] >;

C₈.₂₈D₈ in GAP, Magma, Sage, TeX

C_8._{28}D_8

% in TeX

G:=Group("C8.28D8");

// GroupNames label

G:=SmallGroup(128,401);

// by ID

G=gap.SmallGroup(128,401);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈.₂₈D₈ in TeX

G = C8.28D8 order 128 = 27

5th non-split extension by C8 of D8 acting via D8/D4=C2

G = C₈.₂₈D₈ order 128 = 2⁷

5^th non-split extension by C₈ of D₈ acting via D₈/D₄=C₂