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## G = (C2×C4).24D8order 128 = 27

### 17th non-split extension by C2×C4 of D8 acting via D8/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).24D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×D4⋊C4 — (C2×C4).24D8
 Lower central C1 — C2 — C22×C4 — (C2×C4).24D8
 Upper central C1 — C23 — C2×C42 — (C2×C4).24D8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4).24D8

Generators and relations for (C2×C4).24D8
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=ac-1 >

Subgroups: 344 in 137 conjugacy classes, 50 normal (28 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×D4, C22.7C42, C22.4Q16 [×2], C429C4, C24.3C22, C2×D4⋊C4 [×2], (C2×C4).24D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], SD16 [×2], C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.11D4, C4⋊D8, D4.D4, C22.D8, C23.46D4, C4.4D8, C42.28C22, (C2×C4).24D8

Smallest permutation representation of (C2×C4).24D8
On 64 points
Generators in S64
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(1 22 51 30)(2 9 52 59)(3 24 53 32)(4 11 54 61)(5 18 55 26)(6 13 56 63)(7 20 49 28)(8 15 50 57)(10 41 60 33)(12 43 62 35)(14 45 64 37)(16 47 58 39)(17 42 25 34)(19 44 27 36)(21 46 29 38)(23 48 31 40)
(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)
(2 46)(3 7)(4 44)(6 42)(8 48)(9 15)(10 28)(11 13)(12 26)(14 32)(16 30)(17 19)(18 62)(20 60)(21 23)(22 58)(24 64)(25 27)(29 31)(33 37)(34 56)(36 54)(38 52)(40 50)(41 45)(49 53)(57 59)(61 63)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D8 SD16 C4○D4 C8⋊C22 C8.C22 kernel (C2×C4).24D8 C22.7C42 C22.4Q16 C42⋊9C4 C24.3C22 C2×D4⋊C4 C22×C4 C2×D4 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 1 1 2 2 2 4 4 10 1 1

Matrix representation of (C2×C4).24D8 in GL6(𝔽17)

 16 0 0 0 0 0 0 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
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 8 0 0 0 0 13 13
,
 12 5 0 0 0 0 12 12 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 0 0 0 0 1 2 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,1,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

(C2×C4).24D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{24}D_8
% in TeX

G:=Group("(C2xC4).24D8");
// GroupNames label

G:=SmallGroup(128,803);
// by ID

G=gap.SmallGroup(128,803);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,2028,1027,124]);
// Polycyclic

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

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