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G = C22.SD32order 128 = 27

1st non-split extension by C22 of SD32 acting via SD32/Q16=C2

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

Aliases: C22.2D16, C23.31D8, C22.7SD32, (C2×D8)⋊1C4, C2.D81C4, C4.18C4≀C2, C22⋊C163C2, C87D4.1C2, (C2×C8).300D4, C4.5(C23⋊C4), (C2×C4).13SD16, C2.3(C2.D16), C2.4(D82C4), (C22×C4).185D4, C22.4Q1632C2, (C22×C8).97C22, C22.55(D4⋊C4), C2.12(C22.SD16), (C2×C8).17(C2×C4), (C2×C4).217(C22⋊C4), SmallGroup(128,79)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C22.SD32
C1C2C4C2×C4C22×C4C22×C8C87D4 — C22.SD32
C1C2C2×C4C2×C8 — C22.SD32
C1C22C22×C4C22×C8 — C22.SD32
C1C2C2C2C2C4C2×C4C22×C8 — C22.SD32

Generators and relations for C22.SD32
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=acd7 >

Subgroups: 204 in 63 conjugacy classes, 22 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22 [×3], C22 [×5], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C16, C22⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8, D8, C22×C4, C22×C4, C2×D4 [×2], D4⋊C4, C2.D8, C2×C16, C2×C4⋊C4, C4⋊D4, C22×C8, C2×D8, C22.4Q16, C22⋊C16, C87D4, C22.SD32
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, C23⋊C4, D4⋊C4, C4≀C2, D16, SD32, C22.SD16, C2.D16, D82C4, C22.SD32

Character table of C22.SD32

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111221622488881622224444444444
ρ111111111111111111111111111111    trivial
ρ2111111-11111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1111-1-1-1-1-111111111111111    linear of order 2
ρ51111-1-1111-1-ii-ii-1-1-1-1-111-ii-i-iiii-i    linear of order 4
ρ61111-1-1-111-1-ii-ii1-1-1-1-111i-iii-i-i-ii    linear of order 4
ρ71111-1-1-111-1i-ii-i1-1-1-1-111-ii-i-iiii-i    linear of order 4
ρ81111-1-1111-1i-ii-i-1-1-1-1-111i-iii-i-i-ii    linear of order 4
ρ92222-2-2022-2000002222-2-200000000    orthogonal lifted from D4
ρ10222222022200000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112-2-222-20000000002-22-22-216716ζ16716ζ16716165163ζ16516316716165163ζ165163    orthogonal lifted from D16
ρ122-2-222-2000000000-22-22-22165163ζ165163ζ165163ζ1671616716165163ζ1671616716    orthogonal lifted from D16
ρ132222220-2-2-200000000000-2-2-222-222    orthogonal lifted from D8
ρ142222220-2-2-200000000000222-2-22-2-2    orthogonal lifted from D8
ρ152-2-222-2000000000-22-22-22ζ16516316516316516316716ζ16716ζ16516316716ζ16716    orthogonal lifted from D16
ρ162-2-222-20000000002-22-22-2ζ167161671616716ζ165163165163ζ16716ζ165163165163    orthogonal lifted from D16
ρ172-22-20002-20-1+i1+i1-i-1-i02i-2i-2i2i0000000000    complex lifted from C4≀C2
ρ182-22-20002-20-1-i1-i1+i-1+i0-2i2i2i-2i0000000000    complex lifted from C4≀C2
ρ192222-2-20-2-2200000000000--2-2--2-2--2-2--2-2    complex lifted from SD16
ρ202222-2-20-2-2200000000000-2--2-2--2-2--2-2--2    complex lifted from SD16
ρ212-22-20002-201+i-1+i-1-i1-i0-2i2i2i-2i0000000000    complex lifted from C4≀C2
ρ222-22-20002-201-i-1-i-1+i1+i02i-2i-2i2i0000000000    complex lifted from C4≀C2
ρ232-2-22-22000000000-22-222-2ζ165163ζ165163ζ16131611ζ16716ζ16716ζ16131611ζ1615169ζ1615169    complex lifted from SD32
ρ242-2-22-220000000002-22-2-22ζ1615169ζ1615169ζ16716ζ165163ζ165163ζ16716ζ16131611ζ16131611    complex lifted from SD32
ρ252-2-22-22000000000-22-222-2ζ16131611ζ16131611ζ165163ζ1615169ζ1615169ζ165163ζ16716ζ16716    complex lifted from SD32
ρ262-2-22-220000000002-22-2-22ζ16716ζ16716ζ1615169ζ16131611ζ16131611ζ1615169ζ165163ζ165163    complex lifted from SD32
ρ274-44-4000-4400000000000000000000    orthogonal lifted from C23⋊C4
ρ2844-4-400000000000-2-2-2-22-22-20000000000    complex lifted from D82C4
ρ2944-4-4000000000002-22-2-2-2-2-20000000000    complex lifted from D82C4

Smallest permutation representation of C22.SD32
On 32 points
Generators in S32
(2 27)(4 29)(6 31)(8 17)(10 19)(12 21)(14 23)(16 25)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 17)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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)
(2 25 27 16)(3 24)(4 14 29 23)(5 13)(6 21 31 12)(7 20)(8 10 17 19)(11 32)(15 28)(22 30)

G:=sub<Sym(32)| (2,27)(4,29)(6,31)(8,17)(10,19)(12,21)(14,23)(16,25), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,17), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (2,25,27,16)(3,24)(4,14,29,23)(5,13)(6,21,31,12)(7,20)(8,10,17,19)(11,32)(15,28)(22,30)>;

G:=Group( (2,27)(4,29)(6,31)(8,17)(10,19)(12,21)(14,23)(16,25), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,17), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (2,25,27,16)(3,24)(4,14,29,23)(5,13)(6,21,31,12)(7,20)(8,10,17,19)(11,32)(15,28)(22,30) );

G=PermutationGroup([(2,27),(4,29),(6,31),(8,17),(10,19),(12,21),(14,23),(16,25)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,17)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(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)], [(2,25,27,16),(3,24),(4,14,29,23),(5,13),(6,21,31,12),(7,20),(8,10,17,19),(11,32),(15,28),(22,30)])

Matrix representation of C22.SD32 in GL4(𝔽17) generated by

1000
0100
0010
00216
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
131100
61300
0089
0009
,
1000
01600
0010
00513
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,2,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[13,6,0,0,11,13,0,0,0,0,8,0,0,0,9,9],[1,0,0,0,0,16,0,0,0,0,1,5,0,0,0,13] >;

C22.SD32 in GAP, Magma, Sage, TeX

C_2^2.{\rm SD}_{32}
% in TeX

G:=Group("C2^2.SD32");
// GroupNames label

G:=SmallGroup(128,79);
// by ID

G=gap.SmallGroup(128,79);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,2804,1411,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^7>;
// generators/relations

Export

Character table of C22.SD32 in TeX

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