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## G = C4×SD32order 128 = 27

### Direct product of C4 and SD32

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C4×SD32
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×Q16 — C4×SD32
 Lower central C1 — C2 — C4 — C8 — C4×SD32
 Upper central C1 — C2×C4 — C42 — C4×C8 — C4×SD32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C4×SD32

Generators and relations for C4×SD32
G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Subgroups: 196 in 81 conjugacy classes, 42 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×3], C23, C16 [×2], C16, C42, C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], D8 [×2], D8, Q16 [×2], Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C2.D8 [×2], C2×C16 [×2], SD32 [×4], C4×D4, C4×Q8, C2×D8, C2×Q16, C4×C16, C2.D16, C2.Q32, C164C4, C4×D8, C4×Q16, C2×SD32, C4×SD32
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, SD32 [×2], C4×D4, C2×D8, C4○D8, C4×D8, C2×SD32, C4○D16, C4×SD32

Smallest permutation representation of C4×SD32
On 64 points
Generators in S64
(1 38 27 62)(2 39 28 63)(3 40 29 64)(4 41 30 49)(5 42 31 50)(6 43 32 51)(7 44 17 52)(8 45 18 53)(9 46 19 54)(10 47 20 55)(11 48 21 56)(12 33 22 57)(13 34 23 58)(14 35 24 59)(15 36 25 60)(16 37 26 61)
(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 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)(33 59)(34 50)(35 57)(36 64)(37 55)(38 62)(39 53)(40 60)(41 51)(42 58)(43 49)(44 56)(45 63)(46 54)(47 61)(48 52)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 16 ··· 16 size 1 1 1 1 8 8 1 1 1 1 2 2 2 2 8 ··· 8 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 D8 SD32 C4○D8 C4○D16 kernel C4×SD32 C4×C16 C2.D16 C2.Q32 C16⋊4C4 C4×D8 C4×Q16 C2×SD32 SD32 C42 C2×C8 C8 C2×C4 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 4 8 4 8

Matrix representation of C4×SD32 in GL4(𝔽17) generated by

 13 0 0 0 0 13 0 0 0 0 4 0 0 0 0 4
,
 14 14 0 0 3 14 0 0 0 0 1 7 0 0 10 1
,
 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[14,3,0,0,14,14,0,0,0,0,1,10,0,0,7,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;

C4×SD32 in GAP, Magma, Sage, TeX

C_4\times {\rm SD}_{32}
% in TeX

G:=Group("C4xSD32");
// GroupNames label

G:=SmallGroup(128,905);
// by ID

G=gap.SmallGroup(128,905);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,100,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

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