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G = C22⋊C44C8order 128 = 27

3rd semidirect product of C22⋊C4 and C8 acting via C8/C4=C2

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

Aliases: C22⋊C44C8, C2.13(C8×D4), (C2×C8).327D4, C23.7(C2×C8), C2.3(C86D4), C2.7(C89D4), C24.60(C2×C4), C22.101(C4×D4), (C2×C4).37M4(2), C4.191(C4⋊D4), C4.84(C4.4D4), C22.32(C8○D4), C22.42(C22×C8), (C23×C4).22C22, (C22×C8).46C22, C4.48(C422C2), C2.C42.11C4, (C2×C42).998C22, C23.271(C22×C4), C22.7C428C2, C22.53(C2×M4(2)), (C22×C4).1633C23, C2.10(C42.12C4), C22.59(C42⋊C2), C4.140(C22.D4), C2.5(C42.7C22), C2.4(C24.C22), (C2×C4×C8)⋊9C2, (C2×C4⋊C8)⋊14C2, (C2×C4).21(C2×C8), (C2×C4).1535(C2×D4), (C4×C22⋊C4).15C2, (C2×C22⋊C4).30C4, (C2×C22⋊C8).24C2, (C2×C4).939(C4○D4), (C22×C4).123(C2×C4), SmallGroup(128,655)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22⋊C44C8
C1C2C4C2×C4C22×C4C2×C42C2×C4×C8 — C22⋊C44C8
C1C22 — C22⋊C44C8
C1C22×C4 — C22⋊C44C8
C1C2C2C22×C4 — C22⋊C44C8

Generators and relations for C22⋊C44C8
 G = < a,b,c,d | a2=b2=c4=d8=1, cac-1=ab=ba, dad-1=ac2, bc=cb, bd=db, cd=dc >

Subgroups: 252 in 146 conjugacy classes, 68 normal (52 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×8], C22 [×7], C22 [×10], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×16], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×2], C2×C8 [×4], C2×C8 [×10], C22×C4 [×6], C22×C4 [×6], C24, C2.C42 [×2], C4×C8 [×2], C22⋊C8 [×4], C4⋊C8 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×4], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C4×C8, C2×C22⋊C8 [×2], C2×C4⋊C8, C22⋊C44C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C22×C8, C2×M4(2), C8○D4 [×2], C24.C22, C42.12C4, C42.7C22, C8×D4 [×2], C89D4, C86D4, C22⋊C44C8

Smallest permutation representation of C22⋊C44C8
On 64 points
Generators in S64
(1 52)(2 6)(3 54)(4 8)(5 56)(7 50)(9 28)(10 19)(11 30)(12 21)(13 32)(14 23)(15 26)(16 17)(18 42)(20 44)(22 46)(24 48)(25 47)(27 41)(29 43)(31 45)(33 37)(34 61)(35 39)(36 63)(38 57)(40 59)(49 53)(51 55)(58 62)(60 64)
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(1 29 56 19)(2 30 49 20)(3 31 50 21)(4 32 51 22)(5 25 52 23)(6 26 53 24)(7 27 54 17)(8 28 55 18)(9 58 42 35)(10 59 43 36)(11 60 44 37)(12 61 45 38)(13 62 46 39)(14 63 47 40)(15 64 48 33)(16 57 41 34)
(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)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q···4V8A···8P8Q···8X
order12···2224···44···44···48···88···8
size11···1441···12···24···42···24···4

56 irreducible representations

dim1111111112222
type+++++++
imageC1C2C2C2C2C2C4C4C8D4M4(2)C4○D4C8○D4
kernelC22⋊C44C8C22.7C42C4×C22⋊C4C2×C4×C8C2×C22⋊C8C2×C4⋊C8C2.C42C2×C22⋊C4C22⋊C4C2×C8C2×C4C2×C4C22
# reps12112144164488

Matrix representation of C22⋊C44C8 in GL5(𝔽17)

10000
016000
00100
00010
000016
,
10000
016000
001600
000160
000016
,
160000
001600
016000
000016
00010
,
20000
02000
00200
00008
00090

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,16,0],[2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,9,0,0,0,8,0] >;

C22⋊C44C8 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\rtimes_4C_8
% in TeX

G:=Group("C2^2:C4:4C8");
// GroupNames label

G:=SmallGroup(128,655);
// by ID

G=gap.SmallGroup(128,655);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,124]);
// Polycyclic

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

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