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G = C2×D8⋊C4  order 128 = 27

Direct product of C2 and D8⋊C4

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

Aliases: C2×D8⋊C4, C42.204D4, C42.271C23, (C2×D8)⋊17C4, D8⋊10(C2×C4), C8⋊4(C22×C4), C4.47(C4×D4), D4⋊2(C22×C4), (C4×D4)⋊79C22, C4.19(C23×C4), C8⋊C4⋊35C22, C4.Q8⋊44C22, C4⋊C4.359C23, (C2×C4).199C24, (C2×C8).410C23, (C22×D8).15C2, (C22×C4).709D4, C23.845(C2×D4), C22.120(C4×D4), D4⋊C4⋊87C22, (C2×D4).368C23, (C2×D8).155C22, (C22×C8).438C22, (C2×C42).764C22, C22.143(C22×D4), C22.108(C8⋊C22), (C22×C4).1515C23, (C22×D4).558C22, (C2×C4×D4)⋊57C2, C2.59(C2×C4×D4), (C2×C8)⋊12(C2×C4), (C2×C4.Q8)⋊7C2, (C2×C8⋊C4)⋊5C2, C4.7(C2×C4○D4), (C2×D4)⋊33(C2×C4), C2.5(C2×C8⋊C22), (C2×D4⋊C4)⋊51C2, (C2×C4).1211(C2×D4), (C2×C4).691(C4○D4), (C2×C4⋊C4).911C22, (C2×C4).470(C22×C4), SmallGroup(128,1674)

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

Derived series
C1 — C4 — C2×D8⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C2×D8⋊C4
Lower central
C1 — C2 — C4 — C2×D8⋊C4
Upper central
C1 — C23 — C2×C42 — C2×D8⋊C4
Jennings
C1 — C2 — C2 — C2×C4 — C2×D8⋊C4

Subgroups: 604 in 304 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×28], D4 [×8], D4 [×12], C23, C23 [×20], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×2], D8 [×16], C22×C4, C22×C4 [×2], C22×C4 [×18], C2×D4 [×12], C2×D4 [×6], C24 [×2], C8⋊C4 [×4], D4⋊C4 [×8], C4.Q8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×8], C4×D4 [×4], C22×C8 [×2], C2×D8 [×12], C23×C4 [×2], C22×D4 [×2], C2×C8⋊C4, C2×D4⋊C4 [×2], C2×C4.Q8, D8⋊C4 [×8], C2×C4×D4 [×2], C22×D8, C2×D8⋊C4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8⋊C22 [×4], C23×C4, C22×D4, C2×C4○D4, D8⋊C4 [×4], C2×C4×D4, C2×C8⋊C22 [×2], C2×D8⋊C4

Generators and relations
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
016000000
10000000
0016150000
00110000
0000015215
000011510
000011602
000090162
,
160000000
01000000
001600000
00110000
00001000
000011600
000000160
000000161
,
160000000
016000000
001300000
000130000
000011520
000011602
000010162
000001161

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,1,1,9,0,0,0,0,15,15,16,0,0,0,0,0,2,1,0,16,0,0,0,0,15,0,2,2],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,15,16,0,1,0,0,0,0,2,0,16,16,0,0,0,0,0,2,2,1] >;
 

44 conjugacy classes

class 1 2A···2G2H···2O4A···4L4M···4T8A···8H
order12···22···24···44···48···8
size11···14···42···24···44···4

44 irreducible representations

dim111111112224
type++++++++++
imageC1C2C2C2C2C2C2C4D4D4C4○D4C8⋊C22
kernelC2×D8⋊C4C2×C8⋊C4C2×D4⋊C4C2×C4.Q8D8⋊C4C2×C4×D4C22×D8C2×D8C42C22×C4C2×C4C22
# reps1121821162244

In GAP, Magma, Sage, TeX 

C_2\times D_8\rtimes C_4
 
% in TeX
 
G:=Group("C2xD8:C4");
 
// GroupNames label
 
G:=SmallGroup(128,1674);
 
// by ID
 
G=gap.SmallGroup(128,1674);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,520,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
 
// generators/relations
 

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