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G = C8.30D8  order 128 = 27

7th non-split extension by C8 of D8 acting via D8/D4=C2

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

Aliases: C8.30D8, C8.35SD16, C42.35D4, C4⋊C16⋊3C2, C8⋊2C8⋊1C2, (C2×D8).2C4, (C2×C4).104D8, (C2×C8).359D4, (C2×Q16).2C4, (C4×C8).99C22, (C2×C4).15SD16, C4.4(D4⋊C4), C2.6(D8⋊2C4), C8.12D4.1C2, C2.6(C4.D8), C4.1(C4.D4), C2.6(D8.C4), C22.59(D4⋊C4), (C2×C8).21(C2×C4), (C2×C4).221(C22⋊C4), SmallGroup(128,92)

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

Derived series
C1 — C2×C8 — C8.30D8
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C8.12D4 — C8.30D8
Lower central
C1 — C2 — C2×C4 — C2×C8 — C8.30D8
Upper central
C1 — C22 — C42 — C4×C8 — C8.30D8
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.30D8

Generators and relations for C8.30D8
 G = < a,b,c | a8=b8=1, c2=a, bab-1=a3, ac=ca, cbc-1=ab-1 >

C1
C2
C2
C2
16C2
C22
C4
C4
2C4
2C4
8C22
8C4
8C22
8C22
C2×C4
C2×C4
C8
C8
C2×C4
2C8
4D4
4D4
4Q8
4C2×C4
4C23
4Q8
8C8
C2×C8
C2×C8
C42
2C2×Q8
2C2×D4
4Q16
4C16
4D8
4C22⋊C4
4C22⋊C4
4SD16
4SD16
4C2×C8
C2×Q16
C4×C8
C2×D8
2C2×C16
2C4⋊C8
2C2×SD16
2C4.4D4
C8⋊2C8
C4⋊C16
C8.12D4
C8.30D8

Character table of C8.30D8

 class 12A2B2C2D4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1111162222416222244888844444444
ρ111111111111111111111111111111    trivial
ρ21111-111111-1111111-1-1-1-111111111    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-111111-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-111-1-11-1-1-1-111-i-iii-ii-i-iiii-i    linear of order 4
ρ61111-1-111-1-11-1-1-1-111ii-i-ii-iii-i-i-ii    linear of order 4
ρ711111-111-1-1-1-1-1-1-111-i-iiii-iii-i-i-ii    linear of order 4
ρ811111-111-1-1-1-1-1-1-111ii-i-i-ii-i-iiii-i    linear of order 4
ρ922220-222-2-202222-2-2000000000000    orthogonal lifted from D4
ρ1022220222220-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ11222202-2-22-200000000000√2√2√2-√2-√2√2-√2-√2    orthogonal lifted from D8
ρ122-22-2002-20002-2-2200-√2√2√2-√200000000    orthogonal lifted from D8
ρ13222202-2-22-200000000000-√2-√2-√2√2√2-√2√2√2    orthogonal lifted from D8
ρ142-22-2002-20002-2-2200√2-√2-√2√200000000    orthogonal lifted from D8
ρ152-22-2002-2000-222-200√-2-√-2√-2-√-200000000    complex lifted from SD16
ρ1622220-2-2-2-2200000000000√-2-√-2√-2-√-2√-2-√-2√-2-√-2    complex lifted from SD16
ρ1722220-2-2-2-2200000000000-√-2√-2-√-2√-2-√-2√-2-√-2√-2    complex lifted from SD16
ρ182-22-2002-2000-222-200-√-2√-2-√-2√-200000000    complex lifted from SD16
ρ192-2-2202i00-2i00√-2-√-2√-2-√-2-√2√20000ζ1615+ζ1613ζ163+ζ16ζ167+ζ165ζ1611+ζ16ζ1615+ζ165ζ1611+ζ169ζ1613+ζ167ζ169+ζ163    complex lifted from D8.C4
ρ202-2-220-2i002i00-√-2√-2-√-2√-2-√2√20000ζ163+ζ16ζ1615+ζ1613ζ1611+ζ169ζ1615+ζ165ζ1611+ζ16ζ167+ζ165ζ169+ζ163ζ1613+ζ167    complex lifted from D8.C4
ρ212-2-220-2i002i00√-2-√-2√-2-√-2√2-√20000ζ1613+ζ167ζ169+ζ163ζ1615+ζ165ζ163+ζ16ζ1615+ζ1613ζ1611+ζ16ζ167+ζ165ζ1611+ζ169    complex lifted from D8.C4
ρ222-2-220-2i002i00√-2-√-2√-2-√-2√2-√20000ζ1615+ζ165ζ1611+ζ16ζ1613+ζ167ζ1611+ζ169ζ167+ζ165ζ169+ζ163ζ1615+ζ1613ζ163+ζ16    complex lifted from D8.C4
ρ232-2-220-2i002i00-√-2√-2-√-2√-2-√2√20000ζ1611+ζ169ζ167+ζ165ζ163+ζ16ζ1613+ζ167ζ169+ζ163ζ1615+ζ1613ζ1611+ζ16ζ1615+ζ165    complex lifted from D8.C4
ρ242-2-2202i00-2i00-√-2√-2-√-2√-2√2-√20000ζ169+ζ163ζ1613+ζ167ζ1611+ζ16ζ1615+ζ1613ζ163+ζ16ζ1615+ζ165ζ1611+ζ169ζ167+ζ165    complex lifted from D8.C4
ρ252-2-2202i00-2i00-√-2√-2-√-2√-2√2-√20000ζ1611+ζ16ζ1615+ζ165ζ169+ζ163ζ167+ζ165ζ1611+ζ169ζ1613+ζ167ζ163+ζ16ζ1615+ζ1613    complex lifted from D8.C4
ρ262-2-2202i00-2i00√-2-√-2√-2-√-2-√2√20000ζ167+ζ165ζ1611+ζ169ζ1615+ζ1613ζ169+ζ163ζ1613+ζ167ζ163+ζ16ζ1615+ζ165ζ1611+ζ16    complex lifted from D8.C4
ρ274-44-400-44000000000000000000000    orthogonal lifted from C4.D4
ρ2844-4-400000002√-22√-2-2√-2-2√-200000000000000    complex lifted from D8⋊2C4
ρ2944-4-40000000-2√-2-2√-22√-22√-200000000000000    complex lifted from D8⋊2C4

Smallest permutation representation of C8.30D8
►On 64 points
Generators in S64
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)

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

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

Matrix representation of C8.30D8 ►in GL4(𝔽17) generated by

121200
51200
0010
0001
,
151400
14200
0033
00143
,
151400
31500
0033
00314
G:=sub<GL(4,GF(17))| [12,5,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[15,14,0,0,14,2,0,0,0,0,3,14,0,0,3,3],[15,3,0,0,14,15,0,0,0,0,3,3,0,0,3,14] >;
 

C8.30D8 in GAP, Magma, Sage, TeX 

C_8._{30}D_8
 
% in TeX
 
G:=Group("C8.30D8");
 
// GroupNames label
 
G:=SmallGroup(128,92);
 
// by ID
 
G=gap.SmallGroup(128,92);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,192,2804,1411,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^8=1,c^2=a,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C8.30D8 in TeX
Character table of C8.30D8 in TeX

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