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G = C23.24D8order 128 = 27

3rd non-split extension by C23 of D8 acting via D8/C8=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.24D8, C4○D81C4, D8.7(C2×C4), C8.92(C2×D4), C4(C2.D16), (C22×C16)⋊5C2, (C2×C8).277D4, (C2×C4).168D8, Q16.7(C2×C4), C2.D1620C2, C4(C2.Q32), C2.1(C4○D16), C8.29(C22×C4), (C2×C4).77SD16, C4.10(C2×SD16), C22.53(C2×D8), C2.Q3220C2, C8.28(C22⋊C4), (C2×C16).63C22, (C2×C8).494C23, (C22×C4).584D4, C4.56(D4⋊C4), (C2×D8).100C22, C23.25D41C2, (C2×Q16).99C22, C2.D8.143C22, C22.7(D4⋊C4), (C22×C8).551C22, (C2×C4○D8).3C2, (C2×C4)(C2.D16), (C2×C8).179(C2×C4), (C2×C4).756(C2×D4), C4.50(C2×C22⋊C4), (C2×C4)(C2.Q32), C2.28(C2×D4⋊C4), (C2×C4).271(C22⋊C4), SmallGroup(128,870)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.24D8
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — C23.24D8
C1C2C4C8 — C23.24D8
C1C2×C4C22×C4C22×C8 — C23.24D8
C1C2C2C2C2C4C4C2×C8 — C23.24D8

Generators and relations for C23.24D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd7 >

Subgroups: 260 in 114 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C4.Q8, C2.D8 [×2], C2×C16 [×2], C2×C16 [×2], C42⋊C2, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, C2.D16 [×2], C2.Q32 [×2], C23.25D4, C22×C16, C2×C4○D8, C23.24D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C4○D16 [×2], C23.24D8

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L8A···8H16A···16P
order122222224444444···48···816···16
size111122881111228···82···22···2

44 irreducible representations

dim1111111222222
type++++++++++
imageC1C2C2C2C2C2C4D4D4D8SD16D8C4○D16
kernelC23.24D8C2.D16C2.Q32C23.25D4C22×C16C2×C4○D8C4○D8C2×C8C22×C4C2×C4C2×C4C23C2
# reps12211183124216

Matrix representation of C23.24D8 in GL3(𝔽17) generated by

100
044
0913
,
1600
010
001
,
100
0160
0016
,
400
0106
0515
,
1300
0104
057
G:=sub<GL(3,GF(17))| [1,0,0,0,4,9,0,4,13],[16,0,0,0,1,0,0,0,1],[1,0,0,0,16,0,0,0,16],[4,0,0,0,10,5,0,6,15],[13,0,0,0,10,5,0,4,7] >;

C23.24D8 in GAP, Magma, Sage, TeX

C_2^3._{24}D_8
% in TeX

G:=Group("C2^3.24D8");
// GroupNames label

G:=SmallGroup(128,870);
// by ID

G=gap.SmallGroup(128,870);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,e*a*e^-1=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=b*c*d^7>;
// generators/relations

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