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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C22.SD32
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8⋊7D4 — C22.SD32
 Lower central C1 — C2 — C2×C4 — C2×C8 — C22.SD32
 Upper central C1 — C22 — C22×C4 — C22×C8 — C22.SD32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, 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, C4, C22, C2×C4, D4, 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 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 2 2 16 2 2 4 8 8 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 -i i -i i -1 -1 -1 -1 -1 1 1 -i i -i -i i i i -i linear of order 4 ρ6 1 1 1 1 -1 -1 -1 1 1 -1 -i i -i i 1 -1 -1 -1 -1 1 1 i -i i i -i -i -i i linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 1 -1 i -i i -i 1 -1 -1 -1 -1 1 1 -i i -i -i i i i -i linear of order 4 ρ8 1 1 1 1 -1 -1 1 1 1 -1 i -i i -i -1 -1 -1 -1 -1 1 1 i -i i i -i -i -i i linear of order 4 ρ9 2 2 2 2 -2 -2 0 2 2 -2 0 0 0 0 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 0 2 2 2 0 0 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 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 -ζ167+ζ16 ζ167-ζ16 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 orthogonal lifted from D16 ρ12 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -ζ165+ζ163 ζ165-ζ163 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 orthogonal lifted from D16 ρ13 2 2 2 2 2 2 0 -2 -2 -2 0 0 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 2 2 0 -2 -2 -2 0 0 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 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 ζ165-ζ163 -ζ165+ζ163 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 orthogonal lifted from D16 ρ16 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 ζ167-ζ16 -ζ167+ζ16 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 orthogonal lifted from D16 ρ17 2 -2 2 -2 0 0 0 2 -2 0 -1+i 1+i 1-i -1-i 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ18 2 -2 2 -2 0 0 0 2 -2 0 -1-i 1-i 1+i -1+i 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ19 2 2 2 2 -2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ20 2 2 2 2 -2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ21 2 -2 2 -2 0 0 0 2 -2 0 1+i -1+i -1-i 1-i 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ22 2 -2 2 -2 0 0 0 2 -2 0 1-i -1-i -1+i 1+i 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4≀C2 ρ23 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 ζ165+ζ163 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ1615+ζ169 complex lifted from SD32 ρ24 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 ζ1615+ζ169 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ1613+ζ1611 complex lifted from SD32 ρ25 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 ζ1613+ζ1611 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ167+ζ16 complex lifted from SD32 ρ26 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 ζ167+ζ16 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ165+ζ163 complex lifted from SD32 ρ27 4 -4 4 -4 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 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 complex lifted from D8⋊2C4 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 complex lifted from D8⋊2C4

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

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

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

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

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

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 2 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 11 0 0 6 13 0 0 0 0 8 9 0 0 0 9
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 5 13
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

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