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G = C2×C4⋊SD16  order 128 = 27

Direct product of C2 and C4⋊SD16

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

Aliases: C2×C4⋊SD16, C42.208D4, C42.317C23, Q8⋊1(C2×D4), (C2×Q8)⋊27D4, C4⋊4(C2×SD16), C4⋊C8⋊73C22, (C2×C4)⋊16SD16, (C4×Q8)⋊77C22, C4.63(C22×D4), C4.66(C4⋊D4), C4⋊C4.373C23, (C2×C8).305C23, (C2×C4).236C24, (C2×D4).46C23, C23.855(C2×D4), (C22×C4).791D4, D4⋊C4⋊74C22, C2.8(C22×SD16), (C22×SD16)⋊22C2, (C2×SD16)⋊73C22, (C2×Q8).357C23, C22.83(C2×SD16), C4⋊1D4.135C22, (C22×C8).338C22, (C2×C42).805C22, C22.496(C22×D4), C22.168(C4⋊D4), C22.114(C8⋊C22), (C22×C4).1526C23, (C22×D4).336C22, (C22×Q8).469C22, (C2×C4⋊C8)⋊36C2, (C2×C4×Q8)⋊34C2, C4.146(C2×C4○D4), C2.54(C2×C4⋊D4), C2.14(C2×C8⋊C22), (C2×D4⋊C4)⋊38C2, (C2×C4).1416(C2×D4), (C2×C4⋊1D4).21C2, (C2×C4).903(C4○D4), (C2×C4⋊C4).917C22, SmallGroup(128,1764)

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

Derived series
C1 — C2×C4 — C2×C4⋊SD16
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — C2×C4⋊SD16
Lower central
C1 — C2 — C2×C4 — C2×C4⋊SD16
Upper central
C1 — C23 — C2×C42 — C2×C4⋊SD16
Jennings
C1 — C2 — C2 — C2×C4 — C2×C4⋊SD16

Subgroups: 652 in 292 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×12], D4 [×28], Q8 [×4], Q8 [×6], C23, C23 [×16], C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×6], C2×Q8 [×3], C24 [×2], D4⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C4⋊1D4 [×4], C4⋊1D4 [×2], C22×C8 [×2], C2×SD16 [×8], C2×SD16 [×8], C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×D4⋊C4 [×2], C2×C4⋊C8, C4⋊SD16 [×8], C2×C4×Q8, C2×C4⋊1D4, C22×SD16 [×2], C2×C4⋊SD16

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], SD16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×SD16 [×6], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C4⋊SD16 [×4], C2×C4⋊D4, C22×SD16, C2×C8⋊C22, C2×C4⋊SD16

Generators and relations
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c3 >

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽17)

160000
01000
00100
000160
000016
,
10000
01200
0161600
000160
000016
,
10000
016000
01100
000107
00050
,
160000
01000
0161600
00010
000116

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,16,0,0,0,2,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,0,10,5,0,0,0,7,0],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,1,0,0,0,0,16] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4R8A···8H
order12···222224···44···48···8
size11···188882···24···44···4

38 irreducible representations

dim1111111222224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4SD16C4○D4C8⋊C22
kernelC2×C4⋊SD16C2×D4⋊C4C2×C4⋊C8C4⋊SD16C2×C4×Q8C2×C4⋊1D4C22×SD16C42C22×C4C2×Q8C2×C4C2×C4C22
# reps1218112224842

In GAP, Magma, Sage, TeX 

C_2\times C_4\rtimes SD_{16}
 
% in TeX
 
G:=Group("C2xC4:SD16");
 
// GroupNames label
 
G:=SmallGroup(128,1764);
 
// by ID
 
G=gap.SmallGroup(128,1764);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,4037,1027,124]);
 
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
 
// generators/relations
 

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