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G = C2×C22.D8order 128 = 27

Direct product of C2 and C22.D8

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

Aliases: C2×C22.D8, C23.52D8, C24.179D4, C2.8(C22×D8), C4⋊C4.48C23, C22.21(C2×D8), C2.D851C22, C22⋊C852C22, (C2×C8).140C23, (C2×C4).283C24, (C2×D4).75C23, (C22×C4).434D4, C23.661(C2×D4), D4⋊C465C22, C4⋊D4.151C22, (C22×C8).145C22, (C23×C4).553C22, C22.543(C22×D4), (C22×C4).1002C23, C4.55(C22.D4), (C22×D4).356C22, C22.109(C8.C22), C22.106(C22.D4), (C2×C2.D8)⋊23C2, C4.93(C2×C4○D4), (C22×C4⋊C4)⋊33C2, (C2×C22⋊C8)⋊21C2, (C2×C4).845(C2×D4), (C2×D4⋊C4)⋊23C2, (C2×C4⋊C4)⋊116C22, (C2×C4⋊D4).55C2, C2.25(C2×C8.C22), (C2×C4).841(C4○D4), C2.48(C2×C22.D4), SmallGroup(128,1817)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C22.D8
C1C2C4C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — C2×C22.D8
C1C2C2×C4 — C2×C22.D8
C1C23C23×C4 — C2×C22.D8
C1C2C2C2×C4 — C2×C22.D8

Generators and relations for C2×C22.D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 540 in 256 conjugacy classes, 108 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×30], D4 [×14], C23, C23 [×6], C23 [×12], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×15], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×4], D4⋊C4 [×8], C2.D8 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×3], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×D4, C22×D4, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C2.D8 [×2], C22.D8 [×8], C22×C4⋊C4, C2×C4⋊D4, C2×C22.D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C2×D8 [×6], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.D8 [×4], C2×C22.D4, C22×D8, C2×C8.C22, C2×C22.D8

Smallest permutation representation of C2×C22.D8
On 64 points
Generators in S64
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 22)(2 51)(3 24)(4 53)(5 18)(6 55)(7 20)(8 49)(9 48)(10 59)(11 42)(12 61)(13 44)(14 63)(15 46)(16 57)(17 32)(19 26)(21 28)(23 30)(25 54)(27 56)(29 50)(31 52)(33 41)(34 60)(35 43)(36 62)(37 45)(38 64)(39 47)(40 58)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)
(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 16)(2 38)(3 14)(4 36)(5 12)(6 34)(7 10)(8 40)(9 28)(11 26)(13 32)(15 30)(17 62)(18 43)(19 60)(20 41)(21 58)(22 47)(23 64)(24 45)(25 35)(27 33)(29 39)(31 37)(42 55)(44 53)(46 51)(48 49)(50 57)(52 63)(54 61)(56 59)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4N4O4P8A···8H
order12···222222244444···4448···8
size11···122228822224···4884···4

38 irreducible representations

dim111111122224
type++++++++++-
imageC1C2C2C2C2C2C2D4D4C4○D4D8C8.C22
kernelC2×C22.D8C2×C22⋊C8C2×D4⋊C4C2×C2.D8C22.D8C22×C4⋊C4C2×C4⋊D4C22×C4C24C2×C4C23C22
# reps112281131882

Matrix representation of C2×C22.D8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
000014
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1430000
14140000
0014300
00141400
000040
00001513
,
1600000
010000
001000
0001600
0000160
000091

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,4,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,4,15,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,9,0,0,0,0,0,1] >;

C2×C22.D8 in GAP, Magma, Sage, TeX

C_2\times C_2^2.D_8
% in TeX

G:=Group("C2xC2^2.D8");
// GroupNames label

G:=SmallGroup(128,1817);
// by ID

G=gap.SmallGroup(128,1817);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,4037,1027,124]);
// Polycyclic

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

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