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G = C8×C22⋊C4order 128 = 27

Direct product of C8 and C22⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C8×C22⋊C4, C23.35C42, C2.1(C8×D4), C222(C4×C8), C84(C22⋊C8), C22⋊C814C4, (C22×C8)⋊13C4, C4.158(C4×D4), (C2×C8).399D4, (C23×C8).6C2, C24.93(C2×C4), C23.18(C2×C8), (C2×C4).45C42, C22.83(C4×D4), C82(C2.C42), C22.22(C8○D4), C22.20(C22×C8), C22.28(C2×C42), C4.70(C42⋊C2), C2.4(C82M4(2)), C2.C42.27C4, C8(C22.7C42), C23.253(C22×C4), (C2×C42).990C22, (C22×C8).589C22, (C23×C4).630C22, C22.7C4243C2, (C22×C4).1606C23, (C2×C4×C8)⋊7C2, C2.7(C2×C4×C8), (C2×C4)⋊7(C2×C8), C8(C2×C22⋊C8), C2.3(C4×C22⋊C4), C22⋊C8(C22×C8), (C2×C8)2(C22⋊C8), (C2×C8).203(C2×C4), (C2×C4).1496(C2×D4), (C2×C22⋊C8).49C2, (C4×C22⋊C4).76C2, (C2×C22⋊C4).53C4, C4.106(C2×C22⋊C4), (C2×C4).916(C4○D4), (C2×C8)(C2.C42), (C22×C4).376(C2×C4), (C2×C4).596(C22×C4), C2.C42(C22×C8), (C2×C8)(C22.7C42), (C22×C8)(C22.7C42), (C2×C8)(C2×C22⋊C4), (C2×C8)(C4×C22⋊C4), (C2×C8)(C2×C22⋊C8), (C22×C8)(C4×C22⋊C4), SmallGroup(128,483)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8×C22⋊C4
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C8×C22⋊C4
C1C2 — C8×C22⋊C4
C1C22×C8 — C8×C22⋊C4
C1C2C2C22×C4 — C8×C22⋊C4

Generators and relations for C8×C22⋊C4
 G = < a,b,c,d | a8=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 276 in 192 conjugacy classes, 108 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×14], C2×C4 [×18], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C2×C8 [×12], C2×C8 [×14], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C4×C8 [×4], C22⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×2], C22×C8 [×6], C22×C8 [×4], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C4×C8 [×2], C2×C22⋊C8, C23×C8, C8×C22⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C2×C8 [×12], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4×C8 [×4], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C22×C8 [×2], C8○D4 [×2], C4×C22⋊C4, C2×C4×C8, C82M4(2), C8×D4 [×4], C8×C22⋊C4

Smallest permutation representation of C8×C22⋊C4
On 64 points
Generators in S64
(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)
(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 61 27 41)(2 62 28 42)(3 63 29 43)(4 64 30 44)(5 57 31 45)(6 58 32 46)(7 59 25 47)(8 60 26 48)(9 17 33 52)(10 18 34 53)(11 19 35 54)(12 20 36 55)(13 21 37 56)(14 22 38 49)(15 23 39 50)(16 24 40 51)

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

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

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

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB8A···8P8Q···8AN
order12···222224···44···48···88···8
size11···122221···12···21···12···2

80 irreducible representations

dim11111111111222
type+++++++
imageC1C2C2C2C2C2C4C4C4C4C8D4C4○D4C8○D4
kernelC8×C22⋊C4C22.7C42C4×C22⋊C4C2×C4×C8C2×C22⋊C8C23×C8C2.C42C22⋊C8C2×C22⋊C4C22×C8C22⋊C4C2×C8C2×C4C22
# reps121211484832448

Matrix representation of C8×C22⋊C4 in GL4(𝔽17) generated by

2000
01300
0020
0002
,
16000
01600
00116
00016
,
1000
0100
00160
00016
,
1000
01300
00161
00151
G:=sub<GL(4,GF(17))| [2,0,0,0,0,13,0,0,0,0,2,0,0,0,0,2],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,16,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,13,0,0,0,0,16,15,0,0,1,1] >;

C8×C22⋊C4 in GAP, Magma, Sage, TeX

C_8\times C_2^2\rtimes C_4
% in TeX

G:=Group("C8xC2^2:C4");
// GroupNames label

G:=SmallGroup(128,483);
// by ID

G=gap.SmallGroup(128,483);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,172]);
// Polycyclic

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

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