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## G = (C2×C4)⋊9SD16order 128 = 27

### 1st semidirect product of C2×C4 and SD16 acting via SD16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C4)⋊9SD16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — (C2×C4)⋊9SD16
 Lower central C1 — C2 — C2×C4 — (C2×C4)⋊9SD16
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊9SD16
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊9SD16

Generators and relations for (C2×C4)⋊9SD16
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=ab-1, dcd=c3 >

Subgroups: 388 in 174 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, C24.3C22, C23.67C23, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C22×SD16, (C2×C4)⋊9SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C41D4, C2×SD16, C4○D8, C24.3C22, C4×SD16, C88D4, C85D4, C8.12D4, (C2×C4)⋊9SD16

Smallest permutation representation of (C2×C4)⋊9SD16
On 64 points
Generators in S64
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 24 33 55)(2 17 34 56)(3 18 35 49)(4 19 36 50)(5 20 37 51)(6 21 38 52)(7 22 39 53)(8 23 40 54)(9 57 32 42)(10 58 25 43)(11 59 26 44)(12 60 27 45)(13 61 28 46)(14 62 29 47)(15 63 30 48)(16 64 31 41)
(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 7)(3 5)(4 8)(10 12)(11 15)(14 16)(17 57)(18 60)(19 63)(20 58)(21 61)(22 64)(23 59)(24 62)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 53)(42 56)(43 51)(44 54)(45 49)(46 52)(47 55)(48 50)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 4M ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 SD16 C4○D4 C4○D8 kernel (C2×C4)⋊9SD16 C24.3C22 C23.67C23 C2×C4×C8 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.Q8 C22×SD16 C2×SD16 C2×C8 C22×C4 C2×C4 C2×C4 C22 # reps 1 1 1 1 1 1 1 1 8 6 2 8 4 8

Matrix representation of (C2×C4)⋊9SD16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 15 0 0 0 0 16 13
,
 0 10 0 0 0 0 12 10 0 0 0 0 0 0 10 10 0 0 0 0 12 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 4 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,16,0,0,0,0,15,13],[0,12,0,0,0,0,10,10,0,0,0,0,0,0,10,12,0,0,0,0,10,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,16] >;

(C2×C4)⋊9SD16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("(C2xC4):9SD16");
// GroupNames label

G:=SmallGroup(128,700);
// by ID

G=gap.SmallGroup(128,700);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a*b^-1,d*c*d=c^3>;
// generators/relations

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