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G = C23.22D8order 128 = 27

1st non-split extension by C23 of D8 acting via D8/C8=C2

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

Aliases: C23.22D8, C24.134D4, C23.12Q16, C87(C22⋊C4), (C22×C8)⋊14C4, (C2×C8).347D4, C2.1(C87D4), (C23×C8).13C2, C22.32(C2×D8), C222(C2.D8), (C22×C4).90Q8, C23.68(C4⋊C4), C22.4Q164C2, (C22×C4).545D4, C23.745(C2×D4), C4.70(C22⋊Q8), C22.25(C2×Q16), C2.1(C8.18D4), C22.44(C4○D8), C4.53(C42⋊C2), C23.7Q8.7C2, (C23×C4).671C22, (C22×C8).523C22, C22.110(C4⋊D4), (C22×C4).1330C23, C2.8(C23.25D4), C2.17(C23.7Q8), (C2×C2.D8)⋊1C2, C2.6(C2×C2.D8), (C2×C4).84(C4⋊C4), (C2×C8).209(C2×C4), C4.87(C2×C22⋊C4), C22.91(C2×C4⋊C4), (C2×C4).188(C2×Q8), (C2×C4).1319(C2×D4), (C2×C4⋊C4).37C22, (C2×C4).552(C4○D4), (C22×C4).481(C2×C4), (C2×C4).528(C22×C4), SmallGroup(128,540)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.22D8
C1C2C22C23C22×C4C23×C4C23×C8 — C23.22D8
C1C2C2×C4 — C23.22D8
C1C23C23×C4 — C23.22D8
C1C2C2C22×C4 — C23.22D8

Generators and relations for C23.22D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 316 in 168 conjugacy classes, 76 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×10], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, C2.C42 [×2], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C23×C4, C22.4Q16 [×2], C23.7Q8 [×2], C2×C2.D8 [×2], C23×C8, C23.22D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2.D8 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×D8, C2×Q16, C4○D8 [×2], C23.7Q8, C2×C2.D8, C23.25D4, C87D4 [×2], C8.18D4 [×2], C23.22D8

Smallest permutation representation of C23.22D8
On 64 points
Generators in S64
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(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 58 53 20)(2 57 54 19)(3 64 55 18)(4 63 56 17)(5 62 49 24)(6 61 50 23)(7 60 51 22)(8 59 52 21)(9 46 31 33)(10 45 32 40)(11 44 25 39)(12 43 26 38)(13 42 27 37)(14 41 28 36)(15 48 29 35)(16 47 30 34)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···28···82···2

44 irreducible representations

dim11111122222222
type+++++++-++-
imageC1C2C2C2C2C4D4D4Q8D4C4○D4D8Q16C4○D8
kernelC23.22D8C22.4Q16C23.7Q8C2×C2.D8C23×C8C22×C8C2×C8C22×C4C22×C4C24C2×C4C23C23C22
# reps12221841214448

Matrix representation of C23.22D8 in GL5(𝔽17)

10000
01000
00100
00010
000016
,
10000
01000
00100
000160
000016
,
160000
01000
00100
00010
00001
,
10000
031400
03300
00020
00009
,
40000
061300
0131100
00009
00020

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,2,0,0,0,0,0,9],[4,0,0,0,0,0,6,13,0,0,0,13,11,0,0,0,0,0,0,2,0,0,0,9,0] >;

C23.22D8 in GAP, Magma, Sage, TeX

C_2^3._{22}D_8
% in TeX

G:=Group("C2^3.22D8");
// GroupNames label

G:=SmallGroup(128,540);
// by ID

G=gap.SmallGroup(128,540);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2019,248]);
// Polycyclic

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

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