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G = C2×C22.D8  order 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.D8⋊51C22, C22⋊C8⋊52C22, (C2×C4).283C24, (C2×C8).140C23, (C2×D4).75C23, C23.661(C2×D4), (C22×C4).434D4, D4⋊C4⋊65C22, 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), (C2×C22⋊C8)⋊21C2, (C22×C4⋊C4)⋊33C2, (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

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

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

Generators and relations
 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 >

Smallest permutation representation
►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 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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